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+{"text":"\\section{Introduction} \\label{sec:intro}\n\n\n\n\n\\par\n\nThe W Ursa Majoris (W~UMa)-type binary, also known as an EW-type binary, is a contact system where the two components fill their Roche lobes and share a common envelope. \nThe spectral types of EW binaries usually range from F to K \\citep{2017RAA....17...87Q}.\nTheir variability amplitudes are generally less than 1 magnitude, while the typical orbital period ranges from 0.2 to 1.0 day, but strongly\npeaked between 0.2-0.5 days \\citep[see][Figure~1]{2017RAA....17...87Q}.\nThe iconic feature of their optical light curves is the continuous variation of luminosity with nearly equal depths of the\nprimary and secondary minima, which indicates that the two components are in thermal contact, characterized by nearly identical temperatures. Furthermore, EWs can be subsequently divided into two subtypes based on mass and temperature, namely: the W subtype and A subtype. The primary star of the former is a more-massive and hotter component, while that of the latter is a more-massive and cooler one.\nA fraction of W~UMa systems are X-ray sources, which we refer to as EW-type binaries with X-ray emission (hereafter EWXs).\nAlthough the X-ray emission mechanism remains puzzling, the stellar dynamo magnetic activity generated by rapid rotation and envelope convection is usually in consideration \\citep{2004A&A...415.1113G}.\nThe studies of BH~Cas \\citep{2019PASP..131h4202L} and 2MASS~J11201034$-$2201340 \\citep{2016AJ....151..170H} indicate that the X-ray light curves do not show any obvious occultation or modulation as optical light curves would. The X-ray spectra of the former can be described by thermal models, while those of the latter can be both fitted by a thermal or a power-law model. The X-ray grating spectra of VW~Cep \\citep{2006ApJ...650.1119H} reveal that the compact corona is mainly located in the polar region of the primary star.\n\n\\par\n\n\\citet{2001A&A...370..157S} examined a sample of 102 W~UMa systems and found 57 of them were X-ray sources detected by the ROSAT All Sky Survey (RASS), which indicates a high fraction of W~UMa binaries having X-ray emission. \n\\citet{2006AJ....131..990C} also obtained X-ray fluxes for 34 W~UMa  systems from the RASS database. In these studies, they calculated the hardness ratio between the X-ray counts in the hard (H; 0.5-2.0 keV) and soft (S; 0.1-0.4 keV) bands, and then the X-ray flux and X-ray luminosity based on the energy conversion factor derived from hardness ratio \\citep{1996A&A...310..801H}. The X-ray luminosities of W~UMa type binaries within 400 pc range from $4.4 \\times 10^{29}$ to $2.3 \\times 10^{31}$ erg s$^{-1}$ \\citep{2006AJ....131..990C}. Combining the samples of contact binaries observed by ROTSE-1 and the RASS catalog, \\citet{2006AJ....131..633G}  found that 140 contact binaries have X-ray emission with typical luminosities of $\\sim 1.0 \\times 10^{30}$erg s$^{-1}$. \\citet{2008AcA....58..405S} compiled a catalog containing 379 X-ray emitting contact eclipsing binaries for which they applied somewhat different selection criteria from the widely-used EW classification. They found evidence of an X-ray saturation effect, while their sample exhibits large scatter in the X-ray activity and period relation. \n\n\n\\par\n\nFor late-type main sequence stars (G- to F-types), their X-ray luminosities, $L_{\\textrm{X}}$, tend to reach a maximum value at 10$^{-3}$ of the star's bolometric luminosity $L_{\\textrm{bol}}$, namely, log($L_{\\textrm{X}}$\/$L_{\\textrm{bol}}$) $\\sim -3$, which is known as the \"saturation limit,\" while the $\\log(L_{\\textrm{X}}$\/$L_{\\textrm{bol}}$) can be used to describe the X-ray activity level \\citep{1984A&A...133..117V, 1987ApJ...321..958V, 1993ApJ...410..387F}. For single stars with $P<$~0.4~days, the phenomenon of log($L_{\\textrm{X}}$\/$L_{\\textrm{bol}}$) decreasing with the shortening period is referred to as \"supersaturation\"\\ \\citep{1984ApJ...277..263C, 2001A&A...370..157S}. The $L_{\\textrm{X}}$\/$L_{\\textrm{bol}}$ ratio also represents the X-ray activity level of binary systems; this value increases as the orbital period becomes shorter \\citep{1996AJ....112.1570P, 2001A&A...370..157S}. Previous studies of the relationship between period and X-ray emission of EWs often suffer from a limited sample size and large scattering of the data, making it difficult to describe them quantitatively, thus impeding further studies of the physical mechanism. Therefore, enlarging the sample size and improving the X-ray coverage and data quality are very important for elucidating the origins and properties of X-ray emission and further investigating the evolution of binary systems. \n\n\\par\n\nIn this paper, we identify a large number of EW-type binaries with X-ray emission by cross-matching the All-Sky Automated Survey for Supernovae Variable Stars Database (AVSD) with the 4XMM-Newton Data Release 9 (4XMM-DR9) and the RASS catalogs. Combining with the spectral parameters from the seventh data release of the Large Sky Area Multi object Fiber Spectroscopic Telescope (LAMOST DR7) and binary absolute parameters collected from lectures, we investigate the possible mechanisms of X-ray radiation. In Section~\\ref{sec:dataselection}, we describe the selection of our sample. In Section~\\ref{sec:data_analysis}, we mainly provide the correlation analyses for X-ray luminosity and activity level versus stellar rotation, along with the spectral and component parameters. We discuss the physical implications of our results in Section~\\ref{sec:Discu} and provide a summary of this work in Section~\\ref{sec:Summary}.\n\n\\par\n\n\\section{Sample selection} \\label{sec:dataselection}\n\n\\subsection{EWXs in 4XMM-DR9 }\\label{sec:Sample_selection_XMM}\n\n\\par\n\nThe All-Sky Automated Survey for Supernovae (ASAS-SN) is a ground-based optical survey regularly scanning the full visible sky with a cadence of between two and three days, with a sensitivity limit down to $V$ $\\lesssim$ 17 mag \\citep{2018MNRAS.477.3145J}. ASAS-SN discovered new EW binaries by using V-band light curves with a random forest classifier based on 16 Fourier features and 10 other features describing the statistical and mathematical characteristics that the EW binaries are expected to exhibit \\citep{2019MNRAS.486.1907J}. Through a cross-matching the variable stars with Gaia DR2 \\citep{2018yCat.1345....0G}, 2MASS \\citep{2006AJ....131.1163S}, and ALLWISE \\citep{2010AJ....140.1868W, 2014yCat.2328....0C}, the AVSD \\footnote{https:\/\/asas-sn.osu.edu\/variables} provides V-bands light curves, the parallaxes, proper motions, photometry, and color or reddening information for most variable sources. Up until September 2021, 76378 objects have been classified as EW-type binaries from the analysis of $\\sim$ 660000 variable stars listed in AVSD. We chose these EW-type binaries as our primary catalog (hereafter, ASAS-SN-EW). The 4XMM-DR9 released 550124 unique X-ray sources detected over the 11204 pointed XMM-Newton EPIC observations \\citep{2020A&A...641A.136W}. The long period of data accumulation, high sensitivity, and deep exposures make XMM-Newton very suitable for searches of the X-ray counterparts of EW-type binaries, which are often weak X-ray sources. Firstly, we cross-match the ASAS-SN-EW catalog with the 4XMM-DR9 full catalog with a matching radius of $6^{\\prime\\prime}$. This process leads to 723 unique X-ray sources (1205 observations in total) with XMM-Newton detections, defined as the Parent Group. Secondly, in order to further purify the Parent Group, we cross-matched it with the ATLAS \\citep{2018AJ....156..241H} and WISE \\citep{2018ApJS..237...28C} catalogs that provide classifications for binary systems. We use a matching radius of $3^{\\prime\\prime}$, and only the closest object from multiple matches is selected as the counterpart (less than 1.7\\%). There are 433 and 261 unique counterparts from these two catalogs, respectively. We further divided these counterparts into Group A and Group B. \nGroup A includes 407 objects labeled as close binaries in one or both of the two catalogs.\\footnote{CBF or CBH types in ATLAS, EW or EW\/EA types in WISE.} The 123 objects in Group B are those labeled as types other than close binaries in both catalogs. The remaining 193 objects from the Parent Group that have counterparts in neither catalogs are named as Group C.  \n\n\n\\par\n \nWe consider the Group A objects as having reliable classifications, since they are consistent in at least two catalogs (one is the ASAS-SN-EW, while the other is either ATLAS or WISE). The objects in Groups B and C were screened again via a visual inspection of their light curves, which is crucial for distinguishing genuine W~UMa binaries from other types of variables that may be misclassified by the mathematical screening criteria used in the production of different catalogs. We calculated the distance of each system in the three groups using Gaia DR2 parallax data and eliminated those with a period $>$ 1~day or distance $>$ 1~kpc (see below), or an uncertainty of distance $>$ 20\\% (parallax error\/parallax $>$ 20\\%). There are 255, 39, and 82 objects retained in Groups A, B, and C, respectively. All these objects combined constitute the main sample of this work (STW hereafter). The full process of this sample compilation is illustrated in the flowchart in Figure~\\ref{fig:Flowchart}. The STW contains 376 objects in total, listed in Table~\\ref{table:Properties_EWXs}. The columns are organized as ASAS-SN name, right ascension (R.A.; J2000), declination (DEC.; J2000), period, parallax, distance, 4XMM-DR9 name, log$L_{\\textrm{X}}$, ATLAS classification, and WISE classification. All the X-ray fluxes of our objects are taken from the 4XMM-DR9, which provides an average unabsorbed flux value in cases of multiple observations for a given source. The X-ray luminosity (log$L_{\\textrm{X}}$) is calculated from the full-band X-ray flux in 0.2-12~keV and the Gaia DR2 distance. Almost all of the sources in the STW have $L_{\\textrm{X}} > 10^{29}$~erg~s$^{-1}$. \n\n\\par\n\n\\begin{sidewaystable*}[!htpb]\n  \\centering\n  \\caption{Properties of EWXs in STW (Groups A, B, and C)}\n  \\label{table:Properties_EWXs}\n  \\small\n  \\begin{tabular}{rrrcccrccccrr}\n  \\hline\\noalign{\\smallskip}\n  ASAS-SN Name & R.A. (J2000) & DEC. (J2000) & Period& Parallax & Distance & 4XMM-DR9 Name & log$L_{\\textrm{X}}$ & ATLAS Type & WISE Type & $A_{\\rm G}$ & log$L_{\\textrm{bol}}$ & log($L_{\\textrm{X}}\/L_{\\textrm{bol}}$) \\\\\n(ASASSN-V) & ($^{\\circ}$) & ($^{\\circ}$) & (days) &  & (pc) & (4XMM) & (erg s$^{-1}$) &  & & mag & (erg s$^{-1}$) & \\\\\n  \\hline\\noalign{\\smallskip}\n                & & & & &  &  Group A & & & & & & \\\\\n        J001322.67+054009.6 & 3.34447 & 5.66933 & 0.28948 & 1.9764 & 498.85 & J001322.7+054008 & 29.738  & EW\/EA    & CBF & 0.525 & 33.538 & -3.800\\\\ \n                J001445.74-391435.4 & 3.69058 & -39.24317 & 0.36436 & 5.3361 & 186.40 & J001445.7-391435 & 30.255  & EW     &   & 0.997 & 33.812 & -3.557\\\\ \n                ... & ...&...&...&...&...&...&...&...&...&...&...&...\\\\\n                J184128.52+622409.4 & 280.36883  & 62.40270 & 0.28545 & 1.0850 & 898.01 &J184128.5+622408 & 30.222 & EW  &dubious  & 0.600* & 33.655 & -3.433\\\\\n                J195923.73+225703.6   & 299.84887 & 22.95099 & 0.38876 & 1.4855 & 660.76 & J195923.6+225702 & 29.800  & EW               & MSINE& 0.467  & 34.275 & -4.475 \\\\ \n                J232739.41-004346.6   & 351.91420 & -0.72962 & 0.41252 & 1.8069 & 544.93 & J232739.2-004345 & 30.204  & EW               & SINE  & 0.393 & 34.202 & -3.998\\\\ \n                 & & & & & & Group B & & & & &    &  \\\\\n                J000525.84-084035.2   & 1.35768 & -8.67644 & 0.26309 & 1.8841 & 523.14 & J000525.8-084035 & 29.879  &      & MPULSE & 0.566 & 33.856 & -3.977\\\\ \n                J013143.20+302327.9   & 22.93000 & 30.39107 & 0.26660 & 2.2050 & 447.81 & J013143.1+302329 & 29.773  &      & MSINE &  0.702  & 33.464 & -3.691\\\\ \n                ... & ...&...&...&...&...&...&...&...&...&...&...&... \\\\\n                J232008.39+241055.1   & 350.03494 & 24.18197 & 0.32222 & 1.7312 & 568.21 & J232008.3+241055 & 30.036  &      & MSINE  &0.637  & 33.856 & -3.820\\\\ \n                J232907.52+145731.2   & 352.28132 & 14.95866 & 0.26678 & 1.8248 & 547.99 & J232907.4+145731 & 29.714  &      & MSINE  &0.720  & 33.516 & -3.802\\\\\n                & & & & & & Group C & & & &  &    & \\\\\n                J002150.83-704642.5   & 5.46179 & -70.77846 & 0.27176 & 1.9298 & 510.58 & J002150.5-704640 & 29.963 &  &  &0.639  & 34.025 & -4.062\\\\ \n                J002234.47+614417.2   & 5.64364 & 61.73811 & 0.35770 & 1.3647 & 717.89 &J002234.3+614417 & 30.032 &  &  &1.208 &  34.107 & -4.075\\\\ \n                ... & ...&...&...&...&...&...&...&...&...&...&...&...\\\\\n                J234658.78-545310.2   & 356.74490 & -54.88617 & 0.28150 & 2.6239 & 376.96 & J234658.8-545308 & 29.719 &  &   &0.531  & 33.475 & -3.756\\\\ \n                J235142.03-395949.8   & 357.92512 & -39.99716 & 0.28395 & 1.2965 & 754.65 & J235142.1-395946 & 29.892 &  &   &0.475  & 33.777 & -3.885\\\\ \n\n  \\hline\\noalign{\\smallskip}\n  \\end{tabular}  \n  \\flushleft\n      \\begin{tablenotes}\n     \\footnotesize\n    \\item[1] (This table is available in its entirety in the online machine-readable form.)\n    \\end{tablenotes}\n \\end{sidewaystable*}\n\n\n\\par\n\n\\begin{figure}\n\\centering\n\\includegraphics[width = 7cm]{Fig1.png}\n\\caption{Flowchart of compiling STW sample.}\n\\label{fig:Flowchart}\n\\end{figure}\n\n\\par\n\nOne factor which could potentially affect the accuracy of X-ray luminosity is the presence of X-ray flares. Since there has been sparse mention in the literature on this aspect of EWXs, we refer to the occurrence frequency of X-ray flares for main sequence stars. \\citet{2019A&A...628A..41P} selected 102 X-ray emitting main sequence stars with spectral types ranging from A to M to investigate their magnetic activities and X-ray flares. They detected six flares on five stars, indicating a low X-ray flare rate. Therefore, we did not consider the effects of X-ray flares in our analysis.\n\n\\par\n\n\\subsection{EWXs in RASS} \\label{sec:sec:Sample_selection_RASS}\n\n\n\\par\n\nBy reprocessing the RASS data, \\citet{2016A&A...588A.103B} released an updated catalog with 135118 X-ray sources (the 2RXS catalog). We compiled a sample of 190 EWXs detected by RASS following the same procedure as that in Section~\\ref{sec:Sample_selection_XMM}, but with the X-ray-to-optical matching radius set to $20^{\\prime\\prime}$. \nWe follow \\citet{2001A&A...370..157S} and \\citet{2006AJ....131..990C} to convert the count rate to X-ray flux using the energy conversion factor.  The X-ray luminosity is calculated with the Gaia DR2 distance.\n\n\\par\n\n\nIn addition, we collected the EWXs identified with RASS from the literature and updated their distances and X-ray luminosities using the Gaia DR2 distances. \\citet{2001A&A...370..157S} found that log($L_{\\textrm{X}}$\/$L_{\\textrm{bol}}$) declines with the decreasing color index $B-V < 0.6$. \\citet{2006AJ....131..990C} investigated the relationship between the X-ray luminosity and the stellar parameters, such as the rotation period and color index. In combining their sample with those of \\citet{2001A&A...370..157S} and \\citet{1996MNRAS.280..627M}, these authors reinforced a relation in which the shorter the rotation period, the weaker the X-ray emission when the rotational period is shorter than $\\sim0.5$ days. \\citet{2006AJ....131..633G} also used the RASS database to estimate the X-ray emissivity of contact binaries in our Galaxy. These previous studies utilized a variety of approaches to calculate the distance of each binary. \\citet{2006AJ....131..633G} used a period-color-luminosity relation with $J-H$ colors. The distances of the \\citet{2006AJ....131..990C} sample were obtained via the absolute magnitude calibrated with period and $B-V$ colors, while those of \\citet{2001A&A...370..157S} were derived from the parallaxes released by the Hipparcos satellite. In total, they identified 231 EWXs (including duplicates; for more details,  see below). \nWe cross-matched these 231 objects with Gaia DR2 using the $3^{\\prime\\prime}$ radius to obtain the updated parallaxes. For cases with multiple matches (about 18 \\%), the closest object is selected as the counterpart. In order to ensure the consistency across different samples, we only kept objects with parallax values in Gaia DR2. In total, there are 54\/57 objects from \\citet{2001A&A...370..157S}, 117\/140 from \\citet{2006AJ....131..633G}, and 9\/34 from \\citet{2006AJ....131..990C} remaineing. The X-ray luminosity of each object in these three groups was updated with the new distance from Gaia parallax. Duplicated objects among these three samples were removed. We finally obtained 165 EWXs from the literature (54 from \\citealt{2001A&A...370..157S}, 106 from \\citealt{2006AJ....131..633G} and 8 from \\citealt{2006AJ....131..990C}).\n\n\\par\n\n\nFinally, after combining the EWXs we obtained from 2RXS catalog and those from the literature and then removing 36 duplicated objects, we obtained a sample of 319 objects consistently selected from RASS, named as the SRASS sample (listed in Table~\\ref{table:SRASS_EWXs}). All the EWXs in SRASS are within 1~kpc from us. The STW objects are also limited within 1~kpc for consistency. Because of the substantially different X-ray detection sensitivity between 4XMM and RASS, we could not merge these two samples. Instead, we performed the same analyses for the two samples separately and compared their results, as we present in the following sections. Eight sources are duplicated in STW and SRASS, and each of them has similar X-ray luminosities in both catalogs. In order to maintain the completeness of the samples, we kept them in their respective datasets in the analysis. \n\n\n\\par\n \\begin{table*}[!htpb]\n   \\caption{Properties of objects in SRASS}\n     \\label{table:SRASS_EWXs}\n   \\begin{center}\n   \\begin{tabular}{rrccclccc}\\hline \\hline\n\nR.A. (J2000)  &DEC. (J2000) & Period & Distance & log$L_{\\textrm{X}}$ & $A_{\\rm G}$ & log$L_{\\textrm{bol}}$  & log($L_{\\textrm{X}}$\/$L_{\\textrm{bol}}$) & Reference   \\\\\n($^{\\circ}$) & ($^{\\circ}$)       & (days) & (pc)         & (erg s$^{-1}$)            & mag          & (erg s$^{-1}$)       & &    \\\\ \\hline\n324.70584 &  26.69278 & 0.28040 & 152.54 & 29.880 & 0.612 & 33.689 & -3.809 & (1)   \\\\\n150.41876 &  17.40888 & 0.28410  & 62.87 & 30.188  & 1.211  & 33.883 & -3.695 & (1)   \\\\\n...&...&...&...&...&...&...&...&... \\\\\n330.69898 & -12.31137 & 0.30678 & 182.27 & 30.261 & 0.454 & 33.732 & -3.471 & (5)    \\\\\n347.74778 & 21.71198 & 0.25817 & 105.88 & 30.168 & 1.058 & 33.505 & -3.337 & (5)  \\\\\n\\hline\\noalign{\\smallskip}\n  \\end{tabular}\n  \\end{center}\n  \\begin{tablenotes}\n     \\footnotesize\n    \\item[1] Ref.(1): \\citet{2001A&A...370..157S}, (2):\\citet{2006AJ....131..633G}, (3):\\citet{2006AJ....131..990C}, (4): Duplications in STW, (5): This Work.\n    \\item[2] (This table is available in its entirety in the online machine-readable form.)\n    \\end{tablenotes}\n\\end{table*}\n\n\\par\n\n\\subsection{Spectra from LAMOST}\\label{sec:Spectra_of_LAMOST}\n\n\\par\n\nWe utilized the LAMOST stellar spectral database to investigate the relationships between the optical spectral parameters and X-ray properties of EW-type binaries. \nThe LAMOST is a spectroscopic survey telescope located at the Xinglong station, National Astronomical Observation of China, with a $\\sim4$-meter effective aperture and a field of view of $5^\\circ$ \\citep{2012RAA....12.1197C}. LAMOST can obtain 4,000 spectra covering wavelength from 3700 to 9000 {\\AA} with a resolving power of 1800 in a single exposure. We input the coordinates of STW and SRASS samples into the LAMOST DR7 database\\footnote{http:\/\/dr7.lamost.org} with a matching radius of $3^{\\prime\\prime}$ to obtain stellar spectral parameters, including the effect temperature, $T_{\\rm eff}$, surface gravity, $\\log g$, metallicity [Fe\/H], and radial velocity, $V_{\\rm r}$; these values were automatically derived by the LAMOST stellar parameters pipeline \\citep{2011RAA....11..924W, 2014IAUS..306..340W}.  A total of 139 unique sources are matched in LAMOST DR7 (hereafter, the Spec-EWX sample), which are listed in Table~\\ref{table:Spectra_EWXs}. The designation of each column is defined as follows: right ascension (J2000), declination (J2000), log$L_{\\textrm{X}}$, period, effective temperature ($T_{\\rm eff}$), error of $T_{\\rm eff}$, metallicity ([Fe\/H]), error of [Fe\/H], surface gravity ($\\log g$), and its error.\n\n\\par\n\\begin{table*}[!htpb]\n  \\centering\n  \\caption{Stellar spectral parameters of EWXs in the Spec-EWX sample}\n  \\label{table:Spectra_EWXs}\n  \\small\n  \\begin{tabular}{rrcccccccc}\n  \\hline\\noalign{\\smallskip}\nR.A. (J2000) & DEC. (J2000) & log$L_{\\textrm{X}}$ & Period  &$T_{\\rm eff}$ & $T_{\\rm eff}$ error & [Fe\/H] &  [Fe\/H] error & $\\log g$  &$\\log g$ error \\\\\n($^{\\circ}$) & ($^{\\circ}$) & (erg s$^{-1}$) & (days) & (K) & (K) & (dex) &  (dex) & (dex) & (dex) \\\\\n  \\hline\\noalign{\\smallskip}\n283.39054 &47.43649 &30.213 &0.31503 &5589.50&21.82&-0.10&0.02&4.29& 0.04 \\\\ \n8.52526 &39.69754 &29.078 &0.28669 &5220.21&63.34&0.25&0.06&4.43& 0.10 \\\\ \n...&...&...&...&...&...&...\\\\\n349.62904 &42.92814 &30.072 &0.34180 &5818.81&25.68&0.09&0.02&4.20& 0.04 \\\\ \n350.03494 &24.18197 &30.036 &0.32222 &5416.84&32.06&0.19&0.03&4.291& 0.05 \\\\\n  \\hline\\noalign{\\smallskip}\n  \\end{tabular}  \n  \\flushleft\n    \\begin{tablenotes}\n     \\footnotesize\n    \\item[1] (This table is available in its entirety in the online machine-readable form.)\n    \\end{tablenotes}\n\\end{table*}\n\n\n\n\\par\n\n\n\n\\subsection{Upper limits for X-ray nondetections} \\label{sec:sec:upper limits}\n\nFor the completeness of the sample selection, we determined that EW binaries with X-ray nondetections should also be considered. We screened the ASAS-SN-EW samples following the same procedures as in Section~\\ref{sec:Sample_selection_XMM} (corresponding to the ``Group A\" objects after applying the cuts on period, distance, and parallax precision) without matching X-ray source catalogs. A total of 14096 reliably-classified EW binaries are obtained. After removing the X-ray detected objects we already obtained in Section~\\ref{sec:Sample_selection_XMM}, we calculated the X-ray flux upper limits using the web client {\\it HIgh-energy LIght curve GeneraTor} (\\citealt{2022A&C....3800531S})\\footnote{http:\/\/xmmuls.esac.esa.int\/hiligt\/scripts\/\nhiligt.py} which can poll individual servers for the chosen X-ray missions, and return the X-ray flux and\/or upper limits for given targets or coordinates.\nThe default parameter settings were adopted, namely: a 2$\\sigma$ upper limit significance, an absorbed power law spectral model for flux conversion with a photon index  of $\\Gamma=2,$ and the hydrogen column density of $N_{\\rm H} =3 \\times 10^{20}\\ \\rm cm^{-2}$. The XMM-Newton observations in both slew mode \\citep{2012A&A...548A..99W} and pointed mode were searched, while the entire RASS was utilized. When multiple upper limits were returned for the same coordinate, we chose the lower value (to achieve a tighter constraint). In particular, we used the upper limits from pointed observations for XMM-Newton rather than those from the slew mode when both exist. Finally, we get 12279 (39 from pointed mode and 12240 from slew mode) and 13769 X-ray flux upper limits from the XMM-Newton and ROSAT servers, respectively. Combining the distance data of Gaia DR2, we further calculated the X-ray luminosity upper limits, listed in Tables~\\ref{table:XMMnewtonupperlimits} and \\ref{table:ROSATupperlimits} for XMM-Newton and ROSAT, respectively. These X-ray upper limits are also considered in the subsequent analyses.\n\n\\par\n\n \\begin{table*}[!htpb]\n   \\caption{Properties of EW-type binaries with X-ray luminosity upper limits from XMM-Newton pointed and slew surveys}\n   \\begin{center}\n   \\begin{tabular}{rrcccc}\\hline \\hline\n\nR.A. (J2000) & DEC. (J2000) & Modes & Period & Distance & log$L_{\\textrm{X}}$\\\\\n($^{\\circ}$)   & ($^{\\circ}$)     &             & (days)    & (pc)       & (erg s$^{-1}$)  \\\\ \\hline\n  0.01355 & 69.37062 & slew & 0.37280 & 902.25 & $<$32.072 \\\\\n  0.14513 & 53.81984 & slew  & 0.23166  & 583.38 & $<$31.536 \\\\\n... & ... & ... &... &... & ...  \\\\\n359.97490 & 66.15284 & slew  & 0.32750 & 835.48 & $<$31.985 \\\\\n359.98450 & 51.50363  & slew & 0.29762 & 579.29 & $<$31.561 \\\\\n\n\\hline\\noalign{\\smallskip}\n  \\end{tabular}\n  \\end{center}\n  \\label{table:XMMnewtonupperlimits}\n    \\begin{tablenotes}\n     \\footnotesize\n    \\item[1] (This table is available in its entirety in the online machine-readable form.)\n    \\end{tablenotes}\n\\end{table*}\n\n\\par\n \\begin{table*}[!htpb]\n   \\caption{Properties of EW-type binaries with X-ray luminosity upper limits from the ROSAT survey}\n   \\begin{center}\n   \\begin{tabular}{rrcccc}\\hline \\hline\n\nR.A. (J2000) & DEC. (J2000) & Modes & Period & Distance & log$L_{\\textrm{X}}$\\\\\n($^{\\circ}$)   & ($^{\\circ}$)     &             & (days)    & (pc)       & (erg s$^{-1}$)  \\\\ \\hline\n   0.01355 & 69.37062& RosatSurvey& 0.37280 &902.25 & $<$31.385 \\\\\n    0.14513  &53.81984 &RosatSurvey &0.23166 &583.38 & $<$30.956 \\\\\n... & ... & ... &... &... & ...  \\\\\n359.97490 & 66.15284& RosatSurvey &0.32750 &835.48 & $<$31.492 \\\\\n359.98450  &51.50363 &RosatSurvey &0.29762 &579.29 & $<$31.047 \\\\\n\n\\hline\\noalign{\\smallskip}\n  \\end{tabular}\n  \\end{center}\n  \\label{table:ROSATupperlimits}\n    \\begin{tablenotes}\n     \\footnotesize\n    \\item[1] (This table is available in its entirety in the online machine-readable form.)\n    \\end{tablenotes}\n\\end{table*}\n\n\\par\n\n\n\\section{Data analysis} \\label{sec:data_analysis}\n\n\\subsection{X-ray emission versus the period} \\label{sec:X-ray_P}\n\n\\par\n\nIn this section, we investigate the relationship between the X-ray emission and rotation for EW-type binaries. Contact binaries have circular orbits and synchronous co-rotation, which mean that the orbital period of the binary equals to the spin period of each component. In Figure~\\ref{fig:Lx_P} (1) and (2), we plot the period against log$L_{\\textrm{X}}$ of the EWXs in STW and SRASS, respectively. Objects with X-ray detections and upper limits are both plotted. Most of the upper limits do not provide physically meaningful constraints because of the low sensitivity. We discuss those X-ray nondetections in Section~\\ref{sec:completeness}. For the X-ray detected EW binaries, the X-ray luminosity has a significant positive correlation with the orbital period of the binary stars between $\\sim$0.2 and $\\sim$0.45 days. Meanwhile, in the period interval [0.4,0.5], there appears to be a break point (designated ``P1\") after which the correlations between the period and log$L_{\\textrm{X}}$ no longer hold, and the data points become more scarce and scattered. \n\n\\par\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width = 15cm]{Fig2.png}\n\\caption{Orbital period vs. log$L_{\\textrm{X}}$: for STW (1) shown as black points. The magenta and gray `$tri\\_down$' symbols represent the X-ray upper limits for some EW-type binaries from XMM-Newton pointed and slew survey, respectively;  for SRASS (2)  shown in black points. The gray `$tri\\_right$' symbols represent the X-ray upper limits for some EW-type binaries from ROSAT survey. The period vs. slopes with their corresponding error bars of STW (black) and SRASS (red) with different limiting period ranges (3). The black dashed lines are at $P=0.44$ days. The solid blue lines stand for best-fit linear correlations, and the light blue dashed lines represent 95\\% uncertainty ranges.}\n\\label{fig:Lx_P}\n\\end{figure*}\n\n\\par\n\nAs shown in panel (3) of Figure~\\ref{fig:Lx_P}, the value of each point on the black and red lines represents the slope $a$ of the least-squares quadratic fitting (in the form of $\\log L_{\\rm X} = a \\times P + b $) to all objects with a period less than its abscissa value  \\textit{P}. The black broken line with error bars is for STW, while the red one line is for SRASS. The fitting range is from 0.3 to 0.8 days with the step of 0.01 days. We performed the Kendall$^{\\prime}$s $\\tau$ test \\citep{1990BJpc....25..86} between \\textit{P} and log$L_{\\textrm{X}}$ for objects in these two samples. The null-hypothesis (i.e., no correlation exists) probability is always less than 10$^{-5}$ for the period range of [0.36, 0.80], which suggests that the periods of EWXs show a strong linear correlation ($>5\\sigma$) with the log$L_{\\textrm{X}}$ for both samples.\nWith the increasing period (and thus the number of objects), the slopes of the two samples gradually converge. The fitting slopes of STW and SRASS are both within the 1$\\sigma$ uncertainty range of each other, showing that these two samples are highly consistent with each other in the relationship between \\textit{P} and $\\log L_{\\rm X}$. Especially in the period range of [0.38, 0.44], the difference between the slopes is less than 1$\\sigma$. Beyond $P=0.44$ days, the slope difference starts to increase as the period gets longer, which means that P1 is likely to be at 0.44 days. Meanwhile, only a few data points have a period greater than 0.44 days. Therefore, we took 0.44~days as the upper bound for the period of STW ($\\sim$96\\% of its objects) and SRASS ($\\sim$90\\%) samples for the correlation analysis; that is, data points with period values less than 0.44 days were selected for the least-squares fitting. The blue lines in panels (1) and (2) of Figure~\\ref{fig:Lx_P} represent the best-fit relation and the light blue dashed lines represent the 95\\% uncertainty range. We formulate, for the first time, a clear linear correlation between the period $P$ (log$P$) and X-ray luminosity log$L_{\\textrm{X}}$ for EWXs, which is described by the following Equations~(\\ref{equ:STW_LX_P})--(\\ref{equ:SRASS_LX_logP}) for the STW and SRASS: \n\n\\par\n\n\\begin{equation}\\label{equ:STW_LX_P}\n    \\log L_{X_{STW}} = 2.81(27) \\times P + 29.10(9),\n\\end{equation}\n\n\\begin{equation}\\label{equ:SRASS_LX_P}\n    \\log L_{X_{SRASS}} = 3.17(35) \\times P + 29.17(12),\n\\end{equation}\n\n\\begin{equation}\\label{equ:STW_LX_logP}\n    \\log L_{X_{STW}} = 2.14(24) \\times \\log P + 31.08(10),\n\\end{equation}\n\n\\begin{equation}\\label{equ:SRASS_LX_logP}\n   \\log L_{X_{SRASS}} = 2.45(27) \\times \\log P + 31.41(13).\n\\end{equation}\n\n\\par \n\nWhile the dependency of X-ray luminosity upon period has been pointed out by previous studies (e.g., \\citealt{2001A&A...370..157S}, \\citealt{2006AJ....131..990C}, and \\citealt{2006AJ....131..633G}), quantitative descriptions had not been provided. The linear correlations between orbital period and X-ray luminosity for the STW and SRASS samples we present here are consistent with each other ($\\simlt1\\sigma$). The use of Gaia's more accurate parallax information, which results in more accurate log$L_{\\textrm{X}}$ values in our work help reduce the scatter of the data, making the correlations more clear and better defined. The linear correlations shown in both the STW and SRASS have high statistical significance: the probabilities of null-hypothesis (i.e., no correlation exists) in both cases are $<10^{-5}$, corresponding to a $>5\\sigma$ confidence level. For EWs (regardless of their X-ray emission), \\citet{2018ApJ...859..140C} has established the positive correlation between period and luminosity in the optical and mid-infrared bands. The period of their sample ranges from 0.25 to 0.56 days. As a comparison, the positive correlation between period and X-ray luminosity is well maintained at $P=0.2$ to 0.44 days. \n\n\\par\n\nBased on the method provided by \\citet{2018A&A...616A...8A}, we can calculate the bolometric luminosity $L_{\\textrm{bol}}$ of the EWXs with the following equations:\n\\begin{equation}\\label{equ:bolometricL}\n    -2.5~\\textrm{log}_{10}~(L_{\\textrm{bol}}\/L_{\\odot}) = M_{\\textrm{G}} + BC - M_{\\rm sun},\n\\end{equation}\n \\begin{equation}\\label{equ:Mg}\n    M_{\\textrm{G}} = G + 5 - 5\\log_{10} D - A_{\\rm G},\n\\end{equation}\nwhere $M_{\\textrm{G}}$ is the absolute $G$-band magnitude, $BC$ is the temperature-dependent bolometric correction (based on the effective temperatures in Gaia DR2 estimated from the $BP$-band, $RP$-band, and $G$-band magnitudes; see \\citealt{2018A&A...616A...8A} for details), $M_{\\rm sun}$ is the solar bolometric magnitude 4.74~mag, $G$ is the apparent magnitude in $G$-band, $D$ is the distance, and $A_{\\rm G}$ is the $StarHorse$ extinction \\citep{2019A&A...628A..94A} in the $G$-band.\\footnote{For sources without this value, we set it to 0.600 mag according to the statistical distributions of extinction and distance of our samples.} The values of $A_{\\rm G}$, log$L_{\\rm bol}$, and log$(L_{\\rm X}\/L_{\\rm bol})$ of 370 EWXs from STW are listed in the last three columns of Table~\\ref{table:Properties_EWXs}, while those of 318 EWXs from SRASS are listed in the Columns (6)--(8) of Table~\\ref{table:SRASS_EWXs}; for the remaining 6 sources in STW and 1 source in SRASS, these three parameters are not calculated because of the lack of $BP$\/$RP$-band coverage. In Figure~\\ref{fig:Lx_Lbol_STW_sub_SRASS} (1) and (2), we plot the period against log$L_{\\textrm{bol}}$ and log$(L_{\\rm X}\/L_{\\rm bol})$ of the EWXs in STW and SRASS with blue and red symbols, respectively. The blue and red solid lines represent the best-fit linear relations ($P<0.44$~days), while the light blue and light red dashed lines represent the 95\\% uncertainty ranges for these two samples, respectively.  The Kendall$^{\\prime}$s $\\tau$ tests suggest that the confidence levels of $P$ correlated with both the log$L_{\\textrm{X}}$ and log$(L_{\\rm X}\/L_{\\rm bol})$ are all greater than 5~$\\sigma$. The fitting parameters are listed in Table~\\ref{table:ste_sub_srassLX\/Lbol}.\n\n\\par\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width = 15cm]{Fig3.png}\n\\caption{Orbital period vs. log$L_{\\textrm{bol}}$: for STW in blue points and SRASS in red points (1).  Orbital period vs. log$(L_{\\rm X}\/L_{\\rm bol})$ for STW in blue points and SRASS in red points (2). Black dashed lines are at $P=0.44$ days. Solid blue and red lines stand for best-fit linear correlations ($P<0.44$~days) for STW and SRASS respectively, while the light blue and red dashed lines are their corresponding 95\\% uncertainty ranges.}\n\\label{fig:Lx_Lbol_STW_sub_SRASS}\n\\end{figure*}\n\n\\par\n\n\\begin{table}[!htpb]\n   \\caption{Best-fit parameters for $P$-log$L_{\\textrm{bol}}$ and $P$-\\-log($L_{\\textrm{X}}$\/$L_{\\textrm{bol}}$) of STW and SRASS}\n   \\begin{center}\n   \\begin{tabular}{llrrrr}\\hline \\hline\nRelations &  Samples & Slope & Intercept  \\\\\n\\hline \\\\\n  $P$-log$L_{\\textrm{bol}}$               & STW              &    4.19 $\\pm$ 0.21 &  32.47 $\\pm$ 0.07  \\\\\n                          & SRASS &  4.57 $\\pm$ 0.23 &  32.34 $\\pm$ 0.08  \\\\ \n$P$-log($L_{\\textrm{X}}$\/$L_{\\textrm{bol}}$)               &   STW     &  -1.36 $\\pm$ 0.29 &  -3.37 $\\pm$ 0.10  \\\\\n                          & SRASS&   -1.39 $\\pm$ 0.37 &  -3.17 $\\pm$ 0.13  \\\\      \n\\hline\\noalign{\\smallskip}\n  \\end{tabular}\n  \\end{center}\n  \\label{table:ste_sub_srassLX\/Lbol}\n\\end{table}\n\n\n\n\n\\par\n\n\n\\subsection{Spectral parameters analysis}\\label{sec:Spectral_parameters}\n\n\\par\n\nThe Spec-EWX sample has stellar spectral parameters measured from LAMOST spectra. First, we sought to address whether they indeed constitute a representative sample of the STW and SRASS.  \nWe performed K-S tests on the distributions of period between Spec-EWX and these two samples; the probability $P_{\\rm K-S}$ are 93.2\\% and 22.1\\%, respectively, which suggest they follow the same distributions. As shown in Figure \\ref{fig:LAMOST_P_Lx}, we also provide the linear fitting for the period versus log$L_{\\textrm{X}}$ relation (black points) for Spec-EWX sample with $P<0.44$~days. The best-fit slope and intercept values are 2.53(46) and 29.31(16), respectively, which is consistent within $\\sim 1.5\\sigma$ error of those values in Equations (\\ref{equ:STW_LX_P}) and (\\ref{equ:SRASS_LX_P}).\n\n\\par\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width = 15cm]{Fig4.png}\n\\caption{Orbital period vs. log$L_{\\textrm{X}}$ of Spec-EWX sample (black points) with LAMOST stellar parameters. Solid black line shows the least-squares linear fitting for the data with a period less than 0.44 days. Light black dashed lines represents 95\\% uncertainty range. The size of the light blue circles represents the binary temperature ($\\sim4500$--6600~K). }\n\\label{fig:LAMOST_P_Lx}\n\\end{figure*}\n\n\\par\n\nThe statistical distributions of binary temperature $\\log T_{\\rm eff}$, metallicity  ([Fe\/H]), and the surface gravity $\\log g$ are shown in the upper panels of Figure~\\ref{fig:LAMOST_logT_logg_FeH_Lx_N} (1-1, 2-1, 3-1). For all of the Spec-EWX sample, the effective temperature, $T_{\\rm eff}$, ranges from $\\sim4450$~K (K5) to $\\sim6600$~K (F2), while the majority are distributed between 5150~K and 6000~K (corresponding to $\\sim$ K0V to G0V spectral type), indicating that they are solar-like main sequence stars. The metallicity [Fe\/H] ranges from $\\sim-1.00$ to $\\sim0.6$ dex, with a mean value of -0.05 dex. The distribution peaks at $-0.05$ dex which is slightly lower than that of Sun, that is, [Fe\/H] = 0. The surface gravity $\\log g$ ranges from $\\sim3.4$ to $\\sim4.7$, with the mean value of 4.22 and the distribution peak at $4.15$.  We also plot the normalized distributions of $\\log T_{\\rm eff}$, [Fe\/H], and $\\log g$ of EW-type binaries collected by \\citet{2017RAA....17...87Q} in panel (1-1), (2-1) and (3-1) of Figure~\\ref{fig:LAMOST_logT_logg_FeH_Lx_N}, respectively. K-S tests between our Spec-EWX  and their sample were performed for these three parameters. The values of $P_{\\textrm{K-S}}$ are 2.72 $\\times 10^{-4}$, 7.25$\\times 10^{-10}$, and 4.63$\\times 10^{-5}$, respectively, suggesting that for each of the three parameters, the statistical distributions of these two samples differ at the $> 3\\sigma$ level. \n\n\\par\n\n\\begin{table*}[!htpb]\n   \\caption{Best-fit parameters for log$T$, [Fe\/H] and $\\log g$ with log$L_{\\textrm{X}}$ and log($L_{\\textrm{X}}$\/$L_{\\textrm{bol}}$) of Figure \\ref{fig:LAMOST_logT_logg_FeH_Lx_N}, and corresponding correlation probabilities from the Kendall$^{\\prime}$s $\\tau$ tests}\n   \\begin{center}\n   \\begin{tabular}{llrrrrr}\\hline \\hline\nRelations &Parameters&  Slope & Intercept &  $\\tau$ & $1-P_{\\tau} $ \\\\\n\\hline \\\\\nlog$L_{\\textrm{X}}$& log$T$ &     3.78 $\\pm$ 0.65 &  16.01 $\\pm$ 2.44  &   0.320  &   $> 99.99\\%$  \\\\\n                            & $[\\textrm{Fe}\/\\textrm{H}]$&    0.26 $\\pm$ 0.09 &  30.19 $\\pm$ 0.02  &      0.188   & 99.89\\%    \\\\ \n                            & $\\log g$&    $-$0.49 $\\pm$ 0.17 &  32.26 $\\pm$ 0.72     &    $-$0.198 & 99.94\\%      \\\\\n                            \nlog~($L_{\\textrm{X}}$\/$L_{\\textrm{bol}}$)& log$T$ &     $-$2.70 $\\pm$ 0.67 &  6.41 $\\pm$ 2.50  &  $-$0.232  &   $> 99.99\\%$  \\\\\n                            & $[\\textrm{Fe}\/\\textrm{H}]$&    0.06 $\\pm$ 0.09 &  $-$3.73 $\\pm$ 0.02  &      0.047  & 59.09\\%    \\\\ \n                            & $\\log g$&    0.23 $\\pm$ 0.17 &  $-$4.69 $\\pm$ 0.72  &     0.083 & 84.79\\%      \\\\                           \n\\hline\\noalign{\\smallskip}\n  \\end{tabular}\n  \\end{center}\n  \\label{table:leastsquare_parameters}\n\\end{table*}\n\n\n\n\\par\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width = 18cm]{Fig5.png}\n\\caption{Normalized distributions of three spectral parameters ($\\log T_{\\rm eff}$, [Fe\/H] , and $\\log g$) of EWXs, and their relationships with log$L_{\\textrm{X}}$ and log($L_{\\textrm{X}}$\/$L_{\\textrm{bol}}$). Upper panels: Normalized distributions of the binary temperature (1-1), metallicity [Fe\/H] (2-1), and surface gravity $\\log g$ (3-1). The red and black broken lines represent the EW-type binaries collected by \\citet{2017RAA....17...87Q} and our Spec-EWX sample, respectively. \nMiddle panels: Relationships between  log$L_{\\textrm{X}}$ and the binary temperature (1-2), metallicity [Fe\/H] (2-2), and surface gravity $\\log g$ (3-2). \nLower panels: Relationships between log($L_{\\textrm{X}}$\/$L_{\\textrm{bol}}$) and the binary temperature (1-3), metallicity [Fe\/H] (2-3), and surface gravity $\\log g$ (3-3).\nSolid black lines are the least-squares linear fits and the light black dashed lines represent the 95\\% uncertainty ranges.}\n\\label{fig:LAMOST_logT_logg_FeH_Lx_N}\n\\end{figure*}\n\n\\par\n\nThe component stars of an EW binary, especially the more massive primary, still maintain the mass-temperature relation of single main sequence stars \\citep{2005ApJ...629.1055Y, 2018MNRAS.473.5043E}. Therefore, it is reasonable to use spectral temperature to infer the mass of the primary star of an EW. In terms of temperature, as shown in Figure \\ref{fig:LAMOST_logT_logg_FeH_Lx_N} (1-1), both the EWXs and EWs are mainly distributed ($\\sim$67\\% vs. 48\\%) between $\\sim5150$~K (corresponding to $\\sim$0.79 M$_{\\odot}$ for a main sequence star) and $\\sim6000$~K ($\\sim$1.1 M$_{\\odot}$). In particular, the ratio of EWXs with spectral temperatures below 6000~K is 82.7\\%, which may indicate that a large fraction of EWXs have primary stars with masses lower than 1.1~M$_{\\odot}$. However, EWXs have a higher percentage of objects in this range ($T<6000$~K) than the full EW-binary population has, while the opposite is true for the case of $T>6000$~K. Meanwhile, only a few EWXs have temperatures higher than 6500~K, which indicates that few EWXs have primary stars heavier than $\\sim1.4$ M$_{\\odot}$.\n\n\\par\n\nIn Figure~\\ref{fig:LAMOST_P_Lx}, the size of the light blue circles around each data point represents the effective temperature of Spec-EWX sample from LAMOST spectroscopy. This plot suggests the positive correlations between X-ray luminosity and both period and effective temperature. However, some sources with a period from $\\sim0.40$ to 0.44 days have high temperature but low X-ray luminosity. This can be explained by the general trend of X-ray luminosity exhibited by single main sequence stars. With the ascending temperatures from K to G type stars, the typical X-ray luminosity increases gradually from $\\sim$10$^{29}$ erg s$^{-1}$ to $\\sim$10$^{30}$ erg s$^{-1}$. However, this trend does not continue to F-type stars as their typical X-ray luminosity ($\\sim$10$^{30}$ erg s$^{-1}$) is comparable to that of G-type stars \\citep[see][the left panel of Figure 6]{2020ApJ...902..114W}.\n\n\\par\n\nAs shown in Figure \\ref{fig:LAMOST_logT_logg_FeH_Lx_N} (2-1), the peak value of Spec-EWX metallicity distribution is at $\\sim-0.05$. About 77\\% of the Spec-EWX objects have [Fe\/H] $> -0.25$. In contrast, for general EWs (represented by the red line), the distribution peak is at  $\\sim-0.25$; the population is distributed almost evenly around this peak value (53\\% vs. 47\\%). This indicates that EWs with higher metallicity are more likely to produce X-ray emission. \n\n\\par\n\nIn Figure~\\ref{fig:LAMOST_logT_logg_FeH_Lx_N} (3-1), we can see that the surface gravity of EWXs and of EWs shows nearly the same distribution peaks ($\\sim$4.15). At the interval of $4.2\\leq \\log g \\leq 4.5$, the percentage of EWXs is slightly higher than that of EWs. \n\n\\par\n\n\nThe middle and bottom rows of Figure~\\ref{fig:LAMOST_logT_logg_FeH_Lx_N} demonstrates the correlations between the above three stellar parameters and X-ray luminosity log$L_{\\textrm{X}}$, and X-ray activity level log($L_{\\textrm{X}}$\/$L_{\\textrm{bol}}$) of EWXs, respectively. Linear regressions and Kendall$^{\\prime}$s $\\tau$ tests were performed and the results are listed in Table~\\ref{table:leastsquare_parameters}. The binary temperature $\\log T_{\\rm eff}$ has a strong positive correlation with log$L_{\\textrm{X}}$ at $>5\\sigma$. The metallicity is also positively correlated with log$L_{\\textrm{X}}$, while the surface gravity $\\log g$ is negatively correlated with log$L_{\\textrm{X}}$. The latter two correlations are somewhat weaker, but still at the $ >3\\sigma$ level. Regarding the X-ray activity level log($L_{\\textrm{X}}$\/$L_{\\textrm{bol}}$), it has a negative correlation with the temperature $\\log T_{\\rm eff}$ ($>5\\sigma$), while being almost independent of [Fe\/H] and $\\log g$. \n\n\\par\n\n\n\\subsection{Binary components parameters analysis}\\label{sec:binaryparameters}\n\n\\par\n\nFor the SRASS sample, we collected 23 sources (hereafter, sub-SRASS) that feature measurements of their absolute stellar parameters for each of the two individual components, that is: mass ($M_{1}$ and $M_{2}$), radius ($R_{1}$ and $R_{2}$), and temperature ($T_{1}$ and $T_{2}$), as well as the orbital inclination ($i$) from the literature. Here, the more massive component of the binary is defined as Star$_{1}$, and its physical parameters have the subscript of 1, while the parameters with the subscript of 2 represent the less massive component Star$_{2}$. These parameters are listed in Table~\\ref{table:SRASSsamples}. The K-S tests between the sub-SRASS and the whole SRASS samples are performed upon the orbital period and log$L_{\\textrm{X}}$. The null-hypothesis probabilities, $P_{\\rm K-S}$,  are 79.5\\% and 37.9\\%, respectively, suggesting that they follow the same distributions. For the STW sample, there are no such measurements for the absolute stellar parameters for each of the binary components. \n\n\\par\n\n \\begin{table*}[!htpb]\n   \\caption{Absolute stellar parameters for each component of the EWXs in sub-SRASS}\n   \\begin{center}\n   \\begin{tabular}{lcccccccccccc}\\hline \\hline\n\nName &  log$L_{\\textrm{X}}$ & Period & $M_{1}$ & $M_{2}$  &  $R_{1}$ & $R_{2}$  & $T_{1}$  & $T_{2}$ & $i$ &  Subtype &  Reference \\\\\n & (erg s$^{-1}$) & (days) & (M$_{\\odot}$) & (M$_{\\odot}$) &  (R$_{\\odot}$) & (R$_{\\odot}$) & (K) & (K) & ($^{\\circ}$) & &  \\\\ \\hline\nV523 Cas & 29.707 & 0.2337 & 0.740 & 0.380 & 0.770 & 0.590 & 5104 & 5076 & 84.36  &  A & (1) \\\\\nBX Peg   & 29.880 & 0.2804 & 1.020 & 0.380 & 0.966 & 0.623 & 5872 & 5300 & 87.00  &  A & (2) \\\\\nSX Crv   & 30.376 & 0.3166 & 1.246 & 0.098 & 1.347 & 0.409 & 6340 & 6160 & 61.21  &  A & (3) \\\\\nV508 Oph & 29.983 & 0.3448 & 1.010 & 0.520 & 1.060 & 0.800 & 5980 & 5893 & 83.78  &  A & (4) \\\\\nGR Vir   & 30.081 & 0.3470 & 1.370 & 0.170 & 1.420 & 0.610 & 6300 & 6163 & 83.36  &  A & (5) \\\\\nAH Cnc   & 30.488 & 0.3604 & 1.188 & 0.185 & 1.332 & 0.592 & 6300 & 6151 & 83.11  &  A & (6) \\\\\nU Peg    & 30.161 & 0.3748 & 1.149 & 0.379 & 1.224 & 0.744 & 5860 & 5785 & 77.51  &  A & (7) \\\\\nV566 Oph & 29.859 & 0.4096 & 1.500 & 0.380 & 1.490 & 0.810 & 6456 & 6247 & 80.40  &  A & (8) \\\\\nAQ Psc   & 30.470 & 0.4756 & 1.260 & 0.280 & 1.220 & 1.180 & 6445 & 5946 & 68.90  &  A & (9) \\\\\nSW Lac   & 30.453 & 0.3207 & 1.207 & 0.991 & 1.090 & 1.000 & 5371 & 5529 & 80.95  &  W & (10)\\\\\nRW Com   & 29.638 & 0.2373 & 0.800 & 0.380 & 0.770 & 0.540 & 4720 & 4900 & 74.90  &  W & (11)\\\\\nRW Dor   & 29.919 & 0.2854 & 0.820 & 0.520 & 0.881 & 0.703 & 5238 & 5560 & 76.90  &  W & (12)\\\\\nBW Dra   & 30.198 & 0.2923 & 0.920 & 0.260 & 0.980 & 0.550 & 5980 & 6164 & 74.42  &  W & (13)\\\\\nTW Cet   & 30.241 & 0.3117 & 1.060 & 0.610 & 0.990 & 0.760 & 5450 & 5600 & 83.70  &  W & (14)\\\\\nFG Hya   & 30.139 & 0.3278 & 1.444 & 0.161 & 1.405 & 0.591 & 5900 & 6012 & 82.25  &  W & (15)\\\\\nV781 Tau & 29.993 & 0.3449 & 1.060 & 0.430 & 1.130 & 0.760 & 5536 & 6000 & 65.89  &  W & (16)\\\\\nAC Boo   & 30.060 & 0.3524 & 1.200 & 0.360 & 1.190 & 0.690 & 6241 & 6250 & 86.03  &  W & (17)\\\\\nV752 Cen & 30.285 & 0.3702 & 1.310 & 0.390 & 1.300 & 0.770 & 6014 & 6138 & 82.07  &  W & (18)\\\\\nRT LMi   & 29.662 & 0.3749 & 1.290 & 0.490 & 1.280 & 0.840 & 6400 & 6513 & 84.10  &  W & (19)\\\\\nTX Cnc   & 30.182 & 0.3829 & 1.350 & 0.610 & 1.270 & 0.890 & 6250 & 6537 & 62.10  &  W & (20)\\\\\nUV Lyn   & 30.083 & 0.4150 & 1.430 & 0.550 & 1.400 & 0.920 & 5736 & 5960 & 66.13  &  W & (16)\\\\\nUX Eri   & 30.993 & 0.4453 & 1.450 & 0.540 & 1.450 & 0.910 & 6046 & 6100 & 76.89  &  W & (21)\\\\\nV502 Oph & 30.263 & 0.4534 & 1.370 & 0.460 & 1.510 & 0.940 & 5900 & 6140 & 76.40  &  W & (22)\\\\\n\n\\hline\\noalign{\\smallskip}\n  \\end{tabular}\n  \\end{center}\n  \\label{table:SRASSsamples}\n    \\begin{tablenotes}\n     \\footnotesize\n    \\item[1] Ref.(1):\\citet{2016NewA...44...78M}, (2):\\citet {2015NewA...41...17L}, (3): \\citet {2004AcA....54..299Z}, (4): \\citet{2015AJ....149...62X} , (5): \\citet{2004AJ....128.2430Q} , (6): \\citet{2016RAA....16..157P} , (7): \\citet{2002CoSka..32...79P} , (8): \\citet{2018Ap&SS.363...34S} , (9): \\citet{2020MNRAS.491.6065Z} ,  (10): \\citet{2014arXiv1402.2929E} , (11): \\citet{2011A&A...525A..66D} , (12): \\citet{2019PASJ...71...34S} , (13): \\citet{1986AJ.....92..666K} , (14): \\citet{1982A&AS...47..211R} , (15): \\citet{2005MNRAS.356..765Q} , (16): \\citet{2020ApJ...901..169L} , (17): \\citet{2010IBVS.5951....1N}, (18): \\citet{2019MNRAS.489.4760Z} , (19): \\citet{2008PASJ...60...77Q} , (20): \\citet{1987PNAS...84..610C},  (21): \\citet{2007AJ....134.1769Q} , (22): \\citet{2016PASJ...68..102X}\n    \\end{tablenotes}\n\\end{table*}\n\n\\par\n\nWe investigate the possible relationships between log$L_{\\textrm{X}}$ and the temperature and mass of the different components in Figures~\\ref{fig:DIS_T_Lx} and \\ref{fig:DIS_M_Lx}, respectively. The temperatures of the primary (more massive), secondary, and the binary (average value of the two components) are plotted in Figure~\\ref{fig:DIS_T_Lx} (1), (2) and (3), respectively. The primary and secondary components, as well as the W\/A-subtypes, are indicated in the subscripts of the labels in each panel. The black dashed lines are located at $P=0.44$ days, the break point determined in Section~\\ref{sec:X-ray_P}. For all three panels, the temperature increases along with the period when $P < 0.44$ days. The similarity in this tendency is expected since the temperature difference between the two components of an EW binary is small, about several hundred kelvins. The relation between the average temperature of two components and X-ray luminosity is also plotted in panel (4). We also plot Spec-EWX sample and its best-fit line (see Figure~\\ref{fig:LAMOST_logT_logg_FeH_Lx_N}, panel 1-2) in light blue points and gray line in the background, respectively. In this panel, the sub-SRASS objects are distributed in the region generally similar to where the Spec-EWX sample occupies. \n\n\\par\n\nSimilarly, we plot the masses of the primary and secondary from sub-SRASS in Figure~\\ref{fig:DIS_M_Lx} (1) and (2), respectively. The relation between log$L_{\\textrm{X}}$ and the masses of primary and secondary are plotted in panels (3) and (4). In the last two panels, the black dots with error bars represent the typical saturated X-ray luminosity of the main sequence stars within the mass range (0.22 to 1.29 M$_{\\odot}$). Each horizontal error bar indicates that stars with that mass range reach saturation when $P < 1$ day at the corresponding X-ray luminosity  \\citep[see][Table~3]{2003A&A...397..147P}. \nIn addition, for those EWXs with the mass of the primary star between $\\sim$1.29 and $\\sim$ 1.7 M$_{\\odot}$ (corresponding to F-type), we adopted a saturation X-ray luminosity value of $\\sim$3.2 $\\times$ 10$^{29}$ to $\\sim$3.2 $\\times$ 10$^{30}$ erg s$^{-1}$ \\citep{2019A&A...628A..41P, 2020ApJ...902..114W}. \n\n\n\\par\n\nThe mass ranges of the primary and secondary stars are  $\\sim$0.7 to $\\sim$1.5 M$_{\\odot}$ and $\\sim$ 0.1 to $\\sim$1 M$_{\\odot}$, respectively. It is clearly shown in panels (1) that the mass of the primary star is positively correlated with the period below $P=0.44$ days, while the mass of the secondary shows a much weaker correlation (panel 2). For all the relationships discussed in this subsection, it is clear that there is no significant difference between the behaviors of W-subtype and A-subtype systems.\n\n\\par\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width = 14cm]{Fig6.png}\n\\caption{Plots of the period vs. log$T$ for primary stars (1), secondary stars (2), and the binary systems (3) of the W-subtype (`+' symbols) and A-subtype (`x' symbols) from sub-SRASS. \n (4): Plot of log T vs. log$L_{\\textrm{X}}$. The dashed lines are at $P=0.44$ days. The light blue points and gray line in the background of panel (4) are the Spec-EWX sample points and its fitting line from Figure~\\ref{fig:LAMOST_logT_logg_FeH_Lx_N} (1-2).}\n\\label{fig:DIS_T_Lx}\n\\end{figure*}\n\n\\par\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width = 14cm]{Fig7.png}\n\\caption{Plots of the period vs. stellar mass for primary (1) and secondary (2) from sub-SRASS, and the mass of primary (3) and secondary (4) vs. log$L_{\\textrm{X}}$. The black dashed line in panels (1) \\& (2) is at 0.44 days. The small black points with dashed line in panels (3) \\& (4) indicate the X-ray luminosity of the main sequence star at saturation in the mass range (0.22 to 1.7 M$_{\\odot}$). In panel (1), the blue contours contain the data points from the 10,000 simulated samples (see text) with masses corresponding to the spectral temperatures from Spec-EWX (extra adding $0$--200~K random errors of uniform distribution for each object) against the period, while the gray line and  light blue dashed lines are the best-fit relation of these points with periods of $\\leq$~0.44~days and its 95\\% uncertainty range. The blue and red histograms (with 0.02-day steps) with axis on the right represent the period distributions of the primary stars with masses less than 1.1~M$_{\\odot}$ and greater than 1.1~M$_{\\odot}$, respectively (the average value of 10,000 simulations). }\n\\label{fig:DIS_M_Lx}\n\\end{figure*}\n\n\\par\n\n\\section{Discussion}\\label{sec:Discu}\n\n\\subsection{Sample completeness}\\label{sec:completeness}\n\nIn this section, we discuss the completeness of our EWX sample, especially with regard to how the X-ray nondetections would affect our results. For our reliably classified EW binaries without X-ray detections, most of the X-ray luminosity upper limits have been obtained from the XMM Slew Survey or the RASS (see the gray symbols in the upper two panels of Figure~\\ref{fig:Lx_P}). These upper limits are generally substantially higher than the X-ray luminosity range of detected EWXs because of the low sensitivity of the two surveys. For the correlation analyses we  carried out (presented in Section~\\ref{sec:data_analysis}), adding these upper limits will not provide further constraints.\n\n\n\nFrom the pointed XMM-Newton observations that were utilized in the 4XMM-DR9 catalog, we obtained X-ray luminosity upper limits for 39 EWXs. These objects and the 255 EWXs detected by XMM-Newton in Group A are all from the 14096 EW binaries that we selected using the same criteria as those of the Group A (after applying the cuts on period, distance, and parallax precision; see Section~\\ref{sec:sec:upper limits}). Therefore, the covering fraction (the number of EWXs over the number of EWs) of pointed XMM-Newton observations for EWXs in the whole sky region is estimated to be $2.09\\%$, which is similar to the $2.85\\%$ sky coverage of XMM-Newton pointed observations (1152 deg$^{2}$).\n\n\nWe repeated the correlation analysis between the period and X-ray luminosity by taking those 39 upper limits into account. We employed the software package ASURV Rev 1.2 \\citep{1990BAAS...22..917I, 1992ASPC...25..245L}, which implements the methodology proposed by \\citet{1986ApJ...306..490I} and contains the Expectation-Maximization regression algorithm for astronomical data with detection limits, that is, censored data. The best-fit correlations of $P$-$\\rm log L_{\\rm X}$, $P$-$\\rm log L_{\\rm bol}$, and $P$-$L_{\\textrm{X}}$\/$L_{\\textrm{bol}}$ for the objects with periods of less than 0.44~days are $\\log L_{\\rm X} = 2.84(27) \\times P + 29.07(9)$, $\\log L_{\\rm bol} = 4.34(31) \\times P + 32.38(10),$ and $\\log L_{\\textrm{X}}\/L_{\\textrm{bol}}= -1.42(29) \\times P -3.37(10)$, which are values that are highly consistent with previous results on X-ray detections only shown in Equation~(\\ref{equ:STW_LX_P}) and Table~\\ref{table:ste_sub_srassLX\/Lbol}. This indicates that whether or not including the upper limits from pointed XMM-Newton observations would not affect the correlation studies of EWXs. We do not include these upper limits in the discussions in Sections~\\ref{sec:magnetic}--\\ref{sec:primary_Xemission}.\n\nThere are four objects having tight X-ray upper limits from the pointed XMM-Newton observations that are well below the best-fit period-luminosity relation of EWXs. Labeled as bigger magenta `$tri\\_down$' symbols in Figure~\\ref{fig:Lx_P}~(1) from left to right, these objects are ASASSN-V J152949.56-445113.1, J023156.36+605519.0,  J074108.82+251600.7, and J071826.31-243845.6. \nThe exposure times of their corresponding observations are 14ks, 7ks, 9ks, and 8ks, respectively, suggesting that they do not suffer from insufficient exposure depth. They do indeed have lower X-ray luminosity than typical EWXs. The estimated temperatures from Gaia DR2 for these four objects (4198K, 4864K, 6655K, and 6948K) may provide clues to the reason. Based on the temperature and period information from Spec-EWX sample in Table~\\ref{table:Spectra_EWXs}, we calculate the average temperature of the six objects with closest periods to each of the four sources. The mean temperature values are 5500 $\\pm$ 77K, 5831 $\\pm$ 84K, 6113 $\\pm$ 102K, and 6098 $\\pm$ 61K, respectively. The temperatures of the four sources are $> 3\\sigma$ below or above their respective mean temperature. This may indicate that the temperature deviation of EW-type binaries may affect their X-ray emission. As discussed in Section~\\ref{subsubsec:Lx_Lbol_P} (below), EW-type binaries with lower temperature may generate less overall radiation across the full spectrum, while stars hotter than typical would also produce weaker X-ray radiation due to their thinner convection zone. Deeper X-ray observations on these sources are needed to replace the current upper limits with definite X-ray luminosity level and, thus, to facilitate further investigations on their lower X-ray emission level compared to typical EW binaries.\n\n\n\n\\subsection{$L_{\\rm X}$ and $L_{\\textrm{X}}$\/$L_{\\textrm{bol}}$ of whole EWX system}\\label{sec:magnetic}\n\n\\subsubsection{Linear relationships with $P$}\\label{subsubsec:Lx_Lbol_P}\n\n\\par\n\nAs shown in Figures~\\ref{fig:Lx_P} and \\ref{fig:Lx_Lbol_STW_sub_SRASS}, we obtained clear correlations ($>5\\sigma$) of $P$-log$L_{\\textrm{X}}$ and $P$-log($L_{\\textrm{X}}$\/$L_{\\textrm{bol}}$) for EWXs with $P<0.44$~days using the high-quality X-ray and optical data of 4XMM-DR9 and Gaia DR2, and we provide the first linear parametrization of these relationships in this work. The same correlations for the SRASS sample are generally consistent ($\\lower 2pt \\hbox{$\\, \\buildrel {\\scriptstyle <}\\over {\\scriptstyle \\sim}\\,$} 1\\sigma$) with those of the STW sample (see Equations~\\ref{equ:STW_LX_P} \\& \\ref{equ:SRASS_LX_P} and Table~\\ref{table:ste_sub_srassLX\/Lbol}). Since the STW has a larger sample size and higher-quality X-ray data from XMM-Newton, the  analyses presented in the following are mainly based on the correlation analysis results of the STW objects with periods of less than 0.44~days. As the number of EWXs with $P >0.44$~days is very small, only a qualitative description can be given, which is that the above relationships may appear to remain flat or weakly negative. \n\n\\par\n\nBased on the best-fit linear relationships we have obtained, the period can be treated as a good predictor of the X-ray luminosity (Section \\ref{sec:X-ray_P}) and activity level for EWXs. The slope of the $P$-$\\log(L_{\\textrm{X}}$\/$L_{\\textrm{bol}}$) relation is $-1.36\\pm0.29$, which is fully consistent with the difference between the slopes of the $P$-$\\log L_{\\textrm{X}}$ and $P$-$\\log L_{\\textrm{bol}}$ relationships ($2.81\\pm0.27$ and $4.19\\pm0.21,$ respectively). When the period of an EWX is shorter than 0.44~days, both its X-ray luminosity and bolometric luminosity rise with the increasing period, but the growth rate of former is slower than the latter, making the $P$-log($L_{\\textrm{X}}$\/$L_{\\textrm{bol}}$) relation display a downward trend. The linear relationships of $P$-log$L_{\\textrm{X}}$, $P$-log$L_{\\textrm{bol}}$, and $P$-log($L_{\\textrm{X}}$\/$L_{\\textrm{bol}}$), as listed in Equations~(\\ref{equ:STW_LX_P})--(\\ref{equ:SRASS_LX_logP}), and Table~\\ref{table:ste_sub_srassLX\/Lbol}, provide a convenient check of whether a given EWX has X-ray luminosity, bolometric luminosity, and X-ray activity level that are consistent with the typical population.\n\n\n\\par\n\nWith the period increasing from 0.2 to 0.44 days, the average X-ray luminosity, $L_{\\rm X}$, of EWXs increases from 4.60$\\times 10^{29}$ erg s$^{-1}$ to 2.17$\\times 10^{30}$ erg s$^{-1}$, while the average bolometric luminosity, $L_{\\rm bol}$, rises from 2.03$\\times 10^{33}$ erg s$^{-1}$ to 2.04$\\times 10^{34}$ erg s$^{-1}$. This makes that the average ratio of the X-ray radiation flux to the total radiation flux $L_{\\rm X}\/L_{\\rm bol}$ decreases from $2.2 \\times 10^{-4}$ to $1.0 \\times 10^{-4}$. If we assume that the X-ray emission region is proportional to the total surface region of the EWXs, one reason for this phenomenon may be that the EWXs with short periods have higher X-ray activity level due to their thicker convective zone on its surface. Given the parallel correlation of increasing effective temperature with increasing period (i.e., $P$-$\\log L_{\\textrm{bol}}$), as the temperature rises, the convection zone on the surface will be thinner to generate less X-ray per unit area, while convection and rotation are the essential ingredients to power the magnetic dynamos \\citep{1966ApJ...144..695W, 1967ApJ...150..551K, 2003A&A...397..147P}. Therefore, the X-ray emitting area of EWXs with shorter periods is smaller than that of EWXs with longer period, which makes the total X-ray luminosity increase along with the period. However, the convection zone of EWXs with short periods is thicker, providing higher X-ray activity level than that of longer-period EWXs. \n\n\n\n\\par\n\n\n\\subsubsection{Plateau of saturation}\\label{subsubsec:Lx_Lbol_platreau}\n\n\\par\n\nAs shown in Figure~\\ref{fig:LAMOST_logT_logg_FeH_Lx_N}~(1-3), for our Spec-EWX sample, the log($L_{\\textrm{X}}$\/$L_{\\textrm{bol}}$) shows a monotonically decreasing trend with the increasing temperature, that is, the higher the temperature, the lower the X-ray activity level. This finding supports the magnetic dynamo theory we discuss in Section~\\ref{subsubsec:Lx_Lbol_P}. However, based on the $B-V$ color index, \\citet{2001A&A...370..157S} divided EWXs into cool ($>$ 0.6) and hotter ($<$ 0.6) stars, and concluded that the log($L_{\\textrm{X}}$\/$L_{\\textrm{bol}}$) (also called normalized X-ray flux) of cool variables reaches a plateau while that of hotter objects decreases with the decreasing color index.\nThis means that at $B-V=0.6$ ($T_{\\textrm{eff}}~\\approx~5880$~K and log$T_{\\textrm{eff}}\\approx~3.77$), there is a \"break\" in the log$T_{\\textrm{eff}}$-log($L_{\\textrm{X}}$\/$L_{\\textrm{bol}}$) relation.\\footnote{One should note that the $B-V$ color index corresponds almost linearly with temperatures between 0.30 and 1.15 ($T=$ 7300~K and 4410~K, respectively; \\citealt{2000asqu.book.....C}).} We suggest that the difference between our results and that of \\citet{2001A&A...370..157S} is due to the data completeness on the cool end. \n\\citet{2001A&A...370..157S} stressed that it was implausible to determine the positive correlation between the X-ray activity level and color index with their available data  \\citep[see][Section 3.4]{2001A&A...370..157S}. \n\n\n\\par\n\nSimilarly, the STW objects also show a negative $P$-log($L_{\\textrm{X}}$\/$L_{\\textrm{bol}}$) correlation, namely,  objects with shorter periods generally have higher X-ray activity level, as shown in Figure~\\ref{fig:Lx_Lbol_STW_sub_SRASS} (2). The nature of this similarity is that EWXs has a strong period-temperature (or bolometric luminosity) correlation. \\citet{2001A&A...370..157S} invoked a surface horizontal flow to explain the X-ray emission level of EWXs.\nAs the period gradually reaches $\\sim$0.2~days, the activity level of EWXs gradually approaches the state of saturation limit (log($L_{\\textrm{X}}$\/$L_{\\textrm{bol}})=-3$), but there is no saturation plateau similar to that of ultra fast rotating stars (UFRs). We further discuss the interpretation of this relation in the last paragraph of Section~\\ref{sec:primary_Xemission}.\n\n\\par\n\nAt $P<$0.44~days, the monotonously decreasing trend of the X-ray activity level (i.e., no plateau) may indicate that the EWXs are all in the same state (i.e., regarding whether they are in X-ray saturation or supersaturation). In the case of a mixture of multiple states, it is likely that the correlation will exhibit a changing slope or no correlation exists at all. For late-type main-sequence stars, the X-ray radiation reaches saturation when the rotation period is less than one day \\citep{1984A&A...133..117V, 1987ApJ...321..958V, 1993ApJ...410..387F, 2003A&A...397..147P}. For EWs, the orbital period is synchronized with the rotational period of their individual components. These EWX objects have a period far less than one day, which suggests that their X-ray emission is likely to reach saturation as well. \n\n\\par\n\n\n\\subsection{Binary components versus X-ray emission}\\label{sec:primary-dominating}\n\n\\par\n\nIn this section, we compare the X-ray luminosity of the EWXs with that of the main sequence stars for given mass ranges when they reach saturation, as shown by the black points with error bars in Figure~\\ref{fig:DIS_M_Lx} (3) and (4). For the main sequence stars, the saturated X-ray luminosity gradually increases from $\\sim$ $1.2\\times10^{29}$ erg s$^{-1}$ to $\\sim$4.0 $\\times$ 10$^{30}$ erg s$^{-1}$ with increasing stellar mass from 0.6 to 1.1 M$_{\\odot}$.  However, as the mass of the stars continues to grow (in the range of 1.1 to 1.7 M$_{\\odot}$), the saturated X-ray luminosity remains at $\\sim10^{30}$ erg s$^{-1}$. The panel (3) demonstrates that in the range of $\\sim$0.6 to $\\sim$1.5 M$_{\\odot}$, the X-ray luminosity of an EWX, whether for A-subtype or W-subtype, is generally consistent with the saturated X-ray luminosity of a single main sequence star with the same mass value as that of the primary component of this EWX, but not the mass of the secondary (panel 4). More generally, we further infer that EWXs may inherit the changing trends of the X-ray luminosities of the single stars with different masses. These may be the results of a combination of the primary star dominating the X-ray radiation of the binary system \\citep{2004A&A...426.1035G, 2006ApJ...650.1119H} and the X-ray emission being saturated for the EWXs with $P<$0.44~days. \n\n\n\n\\par\n\nGiven that the X-ray emissions of single stars and EWXs are all determined by the coverage area of the X-ray emitting regions on their surface \\citep{2001A&A...370..157S}, the consistent X-ray luminosities of EWXs and single stars we mention above may imply that the coverage area of the EWXs is close to that of the single star with the mass of the primary component. In this scenario, it can be understood that the different log($L_{\\textrm{X}}$\/$L_{\\textrm{bol}}$) distributions exhibited by UFRs and EWXs with the same period \\citep{2001A&A...370..157S} is mainly related to the differences in the bolometric flux of UFRs and EWXs. The bolometric luminosity of an EWX binary is likely greater than that of an UFR (e.g., those from \\citealt{2003A&A...397..147P}) with similar mass to the primary star of that EWX, which will result in lower log($L_{\\textrm{X}}$\/$L_{\\textrm{bol}}$) of EWXs. Meanwhile, the difference between the bolometric luminosity of UFRs and EWXs also likely varies with changing periods (for $P<$~0.44 days). For EWXs, the $P$-$L_{\\textrm{bol}}$ relationship (Figure~\\ref{fig:Lx_Lbol_STW_sub_SRASS} panel 1) is essentially the same relationship between the period and mass (Section~\\ref{sec:primary_Xemission}). Based on this degeneracy, we can find that the $P$-log($L_{\\textrm{X}}$\/$L_{\\textrm{bol}}$) trend of our EWXs (all with primary mass greater than 0.5M$_\\odot$) is basically similar to the mass-log($L_{\\textrm{X}}$\/$L_{\\textrm{bol}}$) relation of single stars under saturation  in the middle panel of Figure 7 (we note the direction of the mass axis in that figure) in \\citet{2003A&A...397..147P}, which supports our above analyses.\n\n\\par\n\n\n\n\\subsection{Period and mass versus X-ray emission}\\label{sec:primary_Xemission}\n\n\\par\n\nSince the LAMOST DR7 survey used single-star models to fit unresolved binaries, \\citet{2018MNRAS.473.5043E} derived that the LAMOST temperatures derivation is systematically lower than the temperature from the spectrum of the primary star by  $\\lower 2pt \\hbox{$\\, \\buildrel {\\scriptstyle <}\\over {\\scriptstyle \\sim}\\,$} 200$~K. Therefore, we generated 10,000 simulated samples by adding random errors uniformly distributed in [0, 200]~K to the spectral temperature of each Spec-EWX object. Then we derived their corresponding masses \n(linear interpolated using Tables 15.7 and 15.8 of \\citealt{2000asqu.book.....C}) and performed linear fittings between the mass and period of the simulated samples with $P<$~0.44~days. The blue-colored contours in Figure \\ref{fig:DIS_M_Lx} (1) contain the 10,000 simulated samples, while the gray line and light blue dashed lines represent the best-fitting relations between period and mass, and its 95\\% uncertainty range, respectively.\\footnote{This blue bubble structure in the lower right corner is due to the simulation of one binary system ASASSN-V J141756.07+210554.7, which has a period of 0.61 days but a spectral temperature of only 5321K.}  As shown in this figure, the primary stars' masses of sub-SRASS are mostly distributed in the range of that of Spec-EWX. We find that $83.7\\pm$2.7~\\% (from statistical distribution and error of the 10,000 simulated samples) objects of EWXs have masses lower than 1.1~M$_{\\odot}$ in this sample. The best-fit gray line (slope is 1.82~$\\pm$~0.02, intercept is 0.35~$\\pm$~0.04) is intercepted by the black dashed line corresponding to $P=0.44$~days at mass value of $1.15\\pm$0.04~M$_{\\odot}$. Combined with the fact that the X-ray luminosity increases with mass until reaching $\\sim$~1.1~M$_{\\odot}$ (see Figure~\\ref{fig:DIS_M_Lx}, panel 3) and the X-ray emission of EWXs is dominated by the primary, this naturally explains the positive correlation between period and $\\log L_{\\rm X}$ until $P=0.44$~days (corresponding to $\\sim1.1$M$_{\\odot}$) that we found in Section~\\ref{sec:X-ray_P}. Considering each finer period interval (see the blue and red histograms with axis on the right in Figure~\\ref{fig:DIS_M_Lx} panel 1), the fraction of objects with a primary mass lower than 1.1~M$_{\\odot}$ generally decreases with increasing period. It is apparent that when the period approaching $0.44$~days, the objects with primary more massive than 1.1~M$_{\\odot}$ start to dominate. Meanwhile, the X-ray luminosity of the primary stars in this mass range ($>1.1$~M$_{\\odot}$) remains constant ($\\sim$ 10$^{30}$ erg s$^{-1}$). Therefore, the $P-\\log L_{\\rm X}$ correlation breaks at $P=0.44$~days.\n\n\\par\n\nCertainly, we cannot rule out the contribution of secondary stars to the binary X-ray luminosity. Long-period binaries also usually have larger secondary stars. This would reinforce the positive correlation between X-ray luminosity and period. On the other hand, this would not affect the correlation break. Based on the mass range of the secondary, their saturated X-ray luminosity is $\\geqslant 1.5$ dex lower than that of the primary. \nThe X-ray emission from the primary remains dominating. It is notable that the sub-SRASS sample has a higher fraction of objects with more massive primary ($>1.1$~M$_{\\odot}$) than that of the Spec-EWX sample, as shown in Figure~\\ref{fig:DIS_M_Lx} (1). This could be a selection effect since obtaining the spectral parameters for each of the binary component would require higher quality optical spectra from more luminous (thus more massive) stars.\n\n\\par\n\n\nAs shown in Figure~\\ref{fig:DIS_M_Lx} (1), the increase of the main mass is strongly correlated with the increase of the period of EWXs. Given the positive correlation between effective temperature and mass, at a certain mass, the temperature of primary star is so high that the convective zone on its surface would not have adequate material to support a normal magnetic dynamo, which is essentially driven by convection and rotation \\citep{1966ApJ...144..695W, 1967ApJ...150..551K, 2003A&A...397..147P}. This critical mass\/temperature relation occurs at the mass of a primary star is $\\geqslant 1.1$~M$_{\\odot}$ (P$\\sim$0.44 days), which may result in decreased X-ray luminosity for binaries with longer rotation periods. Most of EWXs in this work have short periods, which provide a unique sample to test the magnetic dynamo theories under the extreme physical conditions from fast-rotating stars.\n\n\\par\n\nIn combination with sections 4.1, 4.2, and 4.3, we can conclude that the physical nature of the $P$-log($L_{\\textrm{X}}$\/$L_{\\textrm{bol}}$) relation for single stars and for EWXs is quite different. For single stars, the log($L_{\\textrm{X}}$\/$L_{\\textrm{bol}}$) increases monotonically as the period decreases, meaning that they are not saturated with X-ray radiation. In a fixed mass range, their rotation periods are distributed over a wide range (e.g., 0.1 to 100 days). Only when the period is below a certain value will there be a plateau of X-ray luminosity and activity level \\citep{2003A&A...397..147P, 2019A&A...628A..41P}, and this plateau represents the saturated X-ray activity level. Single stars with different masses have different saturated $P$-log$L_{\\textrm{X}}$ and $P$-log($L_{\\textrm{X}}$\/$L_{\\textrm{bol}}$) plateaus. For EWXs, as we discuss in Section \\ref{subsubsec:Lx_Lbol_platreau}, although their $P$-log($L_{\\textrm{X}}$\/$L_{\\textrm{bol}}$) relationship has a slope, they are still likely to be all saturated with X-ray emission. Given the primary star dominating the X-ray radiation of an EWX, the X-ray saturation luminosities of the EWXs are likely to behave as the collection of the X-ray saturation luminosities from single stars with a range of masses and, thus, $Mass$ and log$L_{\\textrm{X}}$ are correlated with each other (Figure~\\ref{fig:DIS_M_Lx} panel 3). Since the period and mass of EWXs are highly degenerated, the $P$-log$L_{\\textrm{X}}$ relation in Figure~\\ref{fig:Lx_P} (1) is similar to the trend of $Mass$-log$L_{\\textrm{X}}$ in Figure~\\ref{fig:DIS_M_Lx} (3), which explains why EWXs have no X-ray luminosity plateau although they are all in saturation. At $P<$0.44~days, $P$-log$L_{\\textrm{X}}$ relation is the part that maintains a nearly monotonous increase. This, combined with the linear relationship of $P$-log$L_{\\textrm{bol}}$, naturally leads to our finding that the $P$-log($L_{\\textrm{X}}$\/$L_{\\textrm{bol}}$) relation maintains a monotony with no plateau. Essentially, for EWXs, the (primary) mass is probably the most fundamental physical parameter that determines the X-ray emission level, and the period and $L_{\\textrm{bol}}$ are the observed manifestations of the mass. More EWXs with measured stellar parameters (such as mass, temperature of each component) and more single star X-ray data with wider mass ranges will strongly facilitate the study of X-ray radiation mechanisms of binary and single stars.\n\n\\par\n\n\\section{Summary}\\label{sec:Summary}\n\n\\par\n\nIn this work, by cross-matching the AVSD with 4XMM-Newton DR9 and the 2RXS catalogs, we compile the largest sample to date of X-ray emitting EW-type binaries with periods of less than 1 day and distance less than 1~kpc. We also added in the RASS-selected EWXs from literature \\citep{2001A&A...370..157S,2006AJ....131..633G,2006AJ....131..990C} and updated their distance and X-ray luminosity with the Gaia DR2 parallax data. The full EWX sample in this work contains 376 objects detected by XMM-Newton and 319 objects detected by ROSAT. For EW binaries with X-ray coverage but without any detections, most of the upper limits are from the low-sensitivity RASS and XMM slew survey, which could not provide useful constraints on the X-ray emission. There are 39 objects having tight X-ray upper limits from pointed XMM-Newton observations. Adding these upper limits to the sample does not affect the results of the correlation analysis. Our sample of EWXs is sufficiently complete for studying the X-ray properties of EW binaries (see Section~\\ref{sec:completeness}).\n\nThe statistical distributions of parameters (period, temperature, mass, etc.) of EWXs and their relationships with the X-ray luminosity and X-ray activity level are investigated in Section~\\ref{sec:data_analysis}. We discuss the properties of $\\log L_{\\textrm{X}}$, $\\log L_{\\textrm{bol}}$, and log($L_{\\textrm{X}}$\/$L_{\\textrm{bol}}$) for the whole EWX systems in Section~\\ref{sec:magnetic}. Then, at the level of component stars, we further investigate the possible physical mechanisms of the observed characteristics of EWXs in Sections~\\ref{sec:primary-dominating} and \\ref{sec:primary_Xemission}. Our main results are detailed as follows.\n\n\\begin{enumerate}\n\n\n\\par\n\n\\item  We provide the quantitative formulation of the strong positive correlation ($>5\\sigma$) between the period, $P,$ and X-ray luminosity, log$L_{\\textrm{X}}$, for EWXs (Equations~\\ref{equ:STW_LX_P}--\\ref{equ:SRASS_LX_logP}) at $P<0.44$~days. We also present the best-fit $P$-log$L_{\\textrm{bol}}$ and $P$-log($L_{\\textrm{X}}$\/$L_{\\textrm{bol}}$) relations for EWXs. These linear relationships effectively constrain the model of X-ray emission mechanisms for contact binary stars. Based on the fitting results of $P$-$\\log L_{\\textrm{X}}$ and $P$-$\\log L_{\\textrm{bol}}$, we provide the quantitative description of EWX activity levels with different periods, which may relate to the thickness of the convection zone in the magnetic dynamo theories. An EWX with a short period has a smaller but thicker X-ray emitting region than a relative longer period EWX. The relationship between log$T_{\\rm eff}$-log($L_{\\textrm{X}}$\/$L_{\\textrm{bol}}$) shows the higher the temperature, the lower the X-ray activity level, which is the evidence to support magnetic dynamo theories adopted to interpret the $P$-log($L_{\\textrm{X}}$\/$L_{\\textrm{bol}}$) relation of EWXs.\n\n\\par\n\n\\item At $P<0.44$~days, neither the $P$-log($L_{\\textrm{X}}$\/$L_{\\textrm{bol}}$) nor the log$T_{\\rm eff}$-log($L_{\\textrm{X}}$\/$L_{\\textrm{bol}}$) relationship has a saturation plateau similar to that of single ultra fast rotating stars. The X-ray activity level decreases monotonically with the period, which, combined with the short periods of EWXs, may indicate that EWXs are all in X-ray emission saturated state. \n\n\\par\n\n\\item  We compiled the spectral parameters, including effective temperature, $T_{\\rm eff}$,  metallicity [Fe\/H], and surface gravity $\\log g$, for the Spec-EWX sample using LAMOST spectroscopy. We find that EWXs and the general EW population have different statistical distributions ($>3\\sigma$) of all the above three spectral parameters. In particular, the proportions of EWXs on the low mass (temperature) end is higher than that of EWs. The values of $T_{\\rm eff}$ and [Fe\/H] are positively correlated with the X-ray luminosity while $\\log g$ is anti-correlated with log$L_{\\textrm{X}}$. There is a high confidence ($>5\\sigma$) negative correlation between temperature and activity level log($L_{\\textrm{X}}$\/$L_{\\textrm{bol}}$).\n\n\\par\n\n\\item Based on the relation between saturated X-ray luminosity and the mass of main sequence stars, we find that the X-ray luminosity of an EWX is generally consistent with the saturated X-ray luminosity of a main sequence star with the same mass as the primary component of the EWX. The X-ray saturation luminosity values of the EWXs are similar to the collection of single stars with different masses when they all reach saturation. \n\n\n\\par\n\n\\item Most of our EWXs with $P < 0.44$~days have primary stars that are less massive than 1.1~M$_{\\odot}$. In this mass range, the X-ray luminosity increases with mass while it remains constant when the primary mass exceeds 1.1~M$_{\\odot}$. As the period increases, the temperature, primary mass and the X-ray luminosity of the binary systems change in concordance. The mass distribution of the primary stars may be the direct reason for the positive $P$-$\\log L_{\\textrm{X}}$ correlation and also contribute to break for this relation at $P\\sim$ 0.44~days. Because there is no plateau in $P$-$\\log L_{\\textrm{X}}$ and the $P$-$\\log L_{\\textrm{bol}}$ remains monotonically increasing for periods less than 0.44 days, we  strictly confirm that there is a decreasing tendency in $P$-log($L_{\\textrm{X}}$\/$L_{\\textrm{bol}}$) with no plateau. The degeneracy between the mass and the period of EWXs results in the monotonous relationships of $P$-log$L_{\\textrm{X}}$, $P$-log$L_{\\textrm{bol}}$ and $P$-log($L_{\\textrm{X}}$\/$L_{\\textrm{bol}}$). \n\n\\end{enumerate}\n\nIn conclusion, most, if not all, of the EWXs are in X-ray saturation state. The $P$-log($L_{\\textrm{X}}$\/$L_{\\textrm{bol}}$) relation for EWXs is the manifestation of the X-ray saturation and the degeneracy between mass, period, and temperature. The mass of the primary star is the most fundamental physical parameter determining the X-ray emission properties of EWXs.\n\n\n\n\\begin{acknowledgements}\n\nWe thank the anonymous referee very much for the constructive suggestions, which helped to improve the paper. This work is supported by the National Natural Science Foundation of China (NSFC) under the grant numbers U1938105 (J.L. and J.Wu), U2031209 (A.E.), 11925301 (W.-M.G), 11973002 (M.Y.S), and U1831205 (J.Wang). We acknowledge the data support from Guoshoujing Telescope. Guoshoujing Telescope (the Large Sky Area Multi-Object Fiber Spectroscopic Telescope LAMOST) is a National Major Scientific Project built by the Chinese Academy of Sciences. Funding for the project has been provided by the National Development and Reform Commission. LAMOST is operated and managed by the National Astronomical Observatories, Chinese Academy of Sciences.\nWe also acknowledge the support of X-ray data based on observations obtained with \\textit{XMM-Newton}, an ESA science mission with instruments and contributions directly funded by ESA Member States and NASA. This work has made use of data from the European Space Agency (ESA) mission \\textit{Gaia} (https:\/\/www.cosmos.esa.int\/gaia), processed by the \\textit{Gaia} Data Processing and Analysis Consortium. We thank the Las Cumbres Observatory and its staff for its continuing support of the ASAS-SN project. ASAS-SN is supported by the Gordon and Betty Moore Foundation through grant GBMF5490 to the Ohio State University, and NSF grants AST-1515927 and AST-1908570. Development of ASAS-SN has been supported by NSF grant AST-0908816, the Mt. Cuba Astronomical Foundation, the Center for Cosmology and AstroParticle Physics at the Ohio State University, the Chinese Academy of Sciences South America Center for Astronomy (CAS-SACA), the Villum Foundation, and George Skestos.\n\n\n\nSoftware: Matplotlib \\citep{2007CSE.....9...90H}, Numpy (https:\/\/numpy.org\/), Scipy (http:\/\/www.scipy.org), Seaborn \\citep{2021JOSS....6.3021W}.\n\n\\end{acknowledgements}\n\n\\par\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nReconstruction of non-linear dynamic processes based on sparse observations is an important and difficult problem. The problem traditionally requires knowledge of the governing equations or processes to be able to generalize from the the sparse observations to a wider area around, in-between and beyond the measurements. Alternatively it is possible to learn the underlying processes or equations based on data itself, so called data driven methods. In geophysics and environmental monitoring measurements is often only available at sparse locations. For instance, within the field of meteorology, atmospheric pressures, temperatures and wind are only measured at limited number of stations. To produce accurate and general weather predictions, requires methods that both forecast in the future, but also reconstruct where no data is available. Within oceanography one faces the same problem, that in-situ information about the ocean dynamics is only available at sparse locations such as buoys or sub-sea sensors. \\\\\n\nBoth the weather and ocean currents can be approximated with models that are governed by physical laws, e.g. the Navier-Stokes Equation. However, to get accurate reliable reconstructions and forecasts it is of crucial importance to incorporate observations. \\\\ \n\nReconstruction and inference based on sparse observations is important in many applications both in engineering and physical science \\cite{brunton2015closed, kong2018application, bolton2019applications, venturi2004gappy,callaham2018robust, manohar2018data}. Bolton et. al. \\cite{bolton2019applications}  used convolutional neural networks to hindcast ocean models, and in \\cite{yeo2019data} K. Yeo reconstructs time series of nonlinear dynamics from sparse observation.  Oikonomo et. al. \\cite{oikonomou2018novel} proposed a method for filling data gaps in groundwater level observations and Kong. et. al \\cite{kong2018application} used reconstruction techniques to modeling the characteristics of cartridge valves.  \\\\\n\nThe above mentioned applications are just some of the many examples of reconstruction of a dynamic process based on limited information. Here we focus on reconstruction of flow. This problem can be formulated as follows. Let $\\bm w \\in {\\mathbb R}^d,$ $d \\in {\\mathbb N},$ represent a state of the flow, for example velocity, pressure, temperature, etc. Here, we will focus on incompressible unsteady flows and ${\\bm w}=(u,v)\\in {\\mathbb R}^2$ where $u$ and $v$ are the horizontal and vertical velocities, respectively. The velocities ${\\bm w}$ are typically obtained from computational fluid dynamic simulations on a meshed spatial domain $\\mathcal{P}$ at discrete times $\\mathcal{T} = \\{t_1,...,t_K\\} \\subset {\\mathbb R}$. \\\\\n\nLet $ \\mathcal{P}=\\{p_1,...,p_N\\}$ consist of $N$ grid points $p_n,$ $n=1,...,N.$  \nThen the state of the flow $\\bm w$ evaluated on ${\\mathcal P}$ at a time $t_i \\in {\\mathcal T}$ can be represented as a vector \n$\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)} \\in {\\mathbb R}^{2N},$ \n\\begin{equation}\n\\label{eq:x_i}\n\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}=(u(p_1,t_i),...,u(p_N,t_i), v(p_1,t_i),...,v(p_N,t_i))^T.\n\\end{equation}\nThe collection of $\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)},$ $i=1,\\dots,K,$ constitutes the data set $\\ifmmode\\bm{X}\\else\\textbf{\\textit{X}}\\fi.$ In order to account for incompressibility, we introduce a discrete divergence operator $L_{div}$, which is given by a $N \\times 2N$ matrix associated with a finite difference scheme, and\n\\begin{equation}\n\\label{eq:L_div}\n(L_{div} \\, \\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi)_{k} \\approx (\\nabla \\cdot w)(p_k)=0.\n\\end{equation}\n\nFurther, we assume that the state can be measured only at specific points in ${\\mathcal P},$ that is, at ${\\mathcal Q}=\\{q_1,...,q_M\\} \\subset {\\mathcal P}$ where $M$ is typically much less than $N.$  Hence, there is $\\ifmmode\\bm{M}\\else\\textbf{\\textit{M}}\\fi=\\{\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)} \\in {\\mathbb R}^{2M}: \\, \\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)}=C\\,\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}, \\, \\forall \\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)} \\in \\ifmmode\\bm{X}\\else\\textbf{\\textit{X}}\\fi \\},$ where ${\\bm C} \\in {\\mathbb R}^{2M \\times 2N}$ is a sampling matrix. More specifically, $\\bm C$ is a two block matrix \n\n$$\n{\\bm C}=\\begin{pmatrix}\n{\\bm C}_{1\/2}& O\\\\\nO&{\\bm C}_{1\/2}\\\\\n\\end{pmatrix}, \\quad \n({\\bm C}_{1\/2})_{ij}= \\left\\{\n\\begin{array}{ll}\n1,& \\mbox{if } \\, q_i =p_j\\\\\n0,& \\mbox{otherwise}\n\\end{array} \\right. , \\quad i=1,...,N \\quad j=1,...,M,\n$$\nand ${\\bm O}\\in {\\mathbb R}^{M\\times N}$ is a zero matrix. The problem of reconstructing fluid flow $\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)} \\in \\ifmmode\\bm{X}\\else\\textbf{\\textit{X}}\\fi$ from $\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)}\\in \\ifmmode\\bm{M}\\else\\textbf{\\textit{M}}\\fi$ is presented as a schematic plot in \\cref{Sketch_map}.\n\\begin{figure}[h]\n\t\\centering\n\n\t\t\\includegraphics[width=0.70\\linewidth]{Subset_M_of_D_v2_new_notation_i.png}\n\t\t\\captionof{figure}{Sketch of reconstruction of $\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}$ from $\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)}$. The dots on the right side represent the grid ${\\mathcal P}$, and those on  the left side represent the measurement locations ${\\mathcal Q}.$}\\label{Sketch_map}\n\n\\end{figure}\nThere have been a wide range of methods for solving the problem, e.g. \\cite{sirovich1987turbulence, everson1995karhunen, donoho2006compressed, schmid2010dynamic, ELMS2018, raissi2019physics}. In particular, use of proper orthogonal decomposition (POD) \\cite{sirovich1987turbulence} techniques has been popular. \\\\ \n\nPOD \\cite{sirovich1987turbulence} is a traditional dimensional reduction technique where based on a data set, a number of basis functions are constructed. The key idea is that a linear combination of the basis functions can reconstruct the original data within some error margin, efficiently reducing the dimension of the problem. In a modified version of the POD, the Gappy POD (GPOD) \\citep{everson1995karhunen}, the aim is to fill the gap in-between sparse measurements. Given a POD basis one can minimize the $L_2$-error of the measurements and find a linear combination of the POD-basis that complements between the measurements. If the basis is not know, a iterative scheme can be formulated to optimize the basis based on the measurements. The original application of GPOD \\citep{everson1995karhunen} was related to reconstruction of human faces, and it has later been applied to fluid flow reconstruction \\cite{venturi2004gappy}. We will use the GPOD approach for comparison later in this study. \\\\ \n\nA similar approach is the technique of Compressed Sensing (CS) \\cite{donoho2006compressed}. As for the GPOD method, we want to solve a  linear system. However, in the CS-case this will be a under-determined linear system. That is we need some additional information about the system to be able to solve it, typically this can be a condition\/constraint related to the smoothness of the solution. The core difference between CS and GPOD is however the sparsity constraint. That is, instead of minimizing the L2-norm, we minimize the L1-norm. Minimizing the L1-norm favours sparse solutions, i.e. solutions with a small number of nonzero coefficients. \\\\ \n\nAnother reconstruction approach is Dynamical Mode Decomposition (DMD) \\cite{schmid2010dynamic}. Instead of using principal components in the spatial domain, DMD seek to find modes or representations that are associated with a specific frequency in the data, i.e. modes in the temporal domain. Again, the goal is to find a solution to an undetermined linear system and reconstruct based on the measurements, by minimizing the error of the observed values. \\\\\n\nDuring the last decade, data driven methods have become tremendously popular, partly because of the growth and availability of data, but also driven by new technology and improved hardware. To model a non-linear relationships with linear approximations is one of the fundamental limitation of the DMD, CS and GPOD. Recently we have seen development in methods where the artificial neural networks is informed with a physical law, the so called physic-informed neural networks (PINN) \\cite{raissi2019physics}. In PINNs the reconstruction is informed by a Partial Differential Equation (PDE) (e.g. Navier Stokes), thus the neural network can learn to fill the gap between measurements that are in compliance with the equation. This is what Rassi et. al. \\cite{raissi2018hidden} have shown for the benchmark examples such as flow around a 2D and 3D cylinder. Although PINNs are showing promising results, we have yet to see applications to complex systems such as atmospheric or oceanographic systems, where other aspect have to be accounted for, e.g. in large scale oceanic circulation models that are driven by forcing such as tides, bathymetry and river-influx. That being said, these problems may be resolved through PINNs in the future. Despite the promise of PINNs, they will not be a part of this study, as our approach is without any constraint related to the physical properties of the data. \\\\\n\nAnother non-linear data driven approaches for reconstruction of fluid flow are different variations of auto-encoders \\cite{ELMS2018, grover2019uncertainty}. An auto-encoder \\cite{Rumelhart86_autoencoder} is a special configuration of an artificial neural network that first encodes the data by gradually decreasing the size of the hidden layers. With this process, the data is represented in a lower dimensional space. A second neural network then takes the output of the encoder as input, and decodes the representation back to its original shape. These two neural networks together constitute an auto-encoder. Principal Component Analysis (PCA) \\cite{pearson1901_PCA} also represent the data in a different and more compact space. However, PCA reduce the dimension of the data by finding orthogonal basis functions or principal components through singular value decomposition. In fact, it has been showed with linear activation function, PCA and auto-encoders produces the same basis function \\cite{bourlard1988auto}. Probabilistic version of the auto-encoder are called Variational Auto-Encoders (VAEs) \\cite{kingma2013auto}. CVAEs \\cite{sohn2015learning} are conditional probabilistic auto-encoders, that is, the model is dependent on some additional information such that it is possible to create representations that are depend on this information. \\\\\n\nHere, we address the mentioned problem from a probabilistic point of view. Let $\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi: {\\mathcal P} \\to {\\mathbb R}^{2N}$ and $\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi: {\\mathcal Q} \\to {\\mathbb R}^{2M}$ be two multivariate random variables associated with the flow on ${\\mathcal P}$ and on ${\\mathcal Q}$, respectively. Then the data sets $\\ifmmode\\bm{X}\\else\\textbf{\\textit{X}}\\fi$ and $\\ifmmode\\bm{M}\\else\\textbf{\\textit{M}}\\fi$ consist of the realizations of $\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi$ and $\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi$, respectively. Using $\\ifmmode\\bm{X}\\else\\textbf{\\textit{X}}\\fi$ and $\\ifmmode\\bm{M}\\else\\textbf{\\textit{M}}\\fi,$ we intend to approximate the probability distribution $p(\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi|\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi).$ This would not only allow to predict $\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}$ given $\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)},$ but also to estimate an associated uncertainty. In this paper, we use a variational auto-encoder to approximate $p(\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi| \\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi)$. The method we use is a Bayesian Neural Network \\cite{MacKay92} approximated through variational inference \\cite{hoffman2013stochastic,blei2017variational}, that we have called \\textit{Semi-Conditional Variational Auto-encoder}, SCVAE. A detailed description of the SCVAE method for reconstruction and associated uncertainty quantification is given in \\cref{SCVAE_section}. \\\\\n\nHere we focus on fluid flow, being the main driving mechanism behind transport and dilution of tracers in marine waters. The world's oceans are  under tremendous stress \\citep{Halpern:2012hs}, UN has declared 2021-2030 as the ocean decade\\footnote{\\url{https:\/\/en.unesco.org\/ocean-decade}}, and an ecosystem based Marine Spatial Planning initiative has been launched by IOC \\citep{DominguezTejo:2016dt}. \\\\\n\nLocal and regional current conditions determines transport of tracers in the ocean \\cite{drange2001ocean,BARSTOW1983211}. Examples are accidental release of radioactive, biological or chemical substances from industrial complexes, e.g. organic waste from fish farms in Norwegian fjords \\citep{Ali:2011hd}, plastic \\cite{Law:2017}, or other contaminants that might have adverse effects on marine ecosystems \\citep{Hylland:2015gt}. \\\\ \n\nTo be able to predict the environmental impact of a release, i.e. concentrations as function of distance and direction from the source, requires reliable current conditions \\citep{Ali:2016go,Blackford:2020}. Subsequently, these transport predictions support design of marine environmental monitoring programs \\citep{Hvidevold:2015,Hvidevold:2016cx,Alendal:2017b, oleynik2020optimal}. The aim here is to model current conditions in a probabilistic manner using SCVAEs. This allows for predicting footprints in a Monte Carlo framework, providing simulated data for training networks used for, e.g., analysing environmental time series \\cite{gundersen2020binary}.\\\\ \n\nIn this study we will compare results  with the GPOD method \\cite{willcox2006unsteady}. We are aware that there recent methods (e.g. PINNS and traditional Auto-encoder) that may perform better on the specific data sets than the GPOD, however, the GPODs simplicity, versatility and not least its popularity \\cite{jo2019effective, mifsud2019fusing, callaham2019robust}, makes it a great method for comparison. \\\\\n\nThe reminder of this manuscript is outlined in the following: \\cref{A_Motivating_Example} presents a motivating example for the SCVAE-method in comparison with the GPOD-method. In \\cref{methods} we review both the VAE and CVAE method and present the SCVAE. Results of experiments on two different data sets are presented in \\cref{Experiment}. \\cref{discussion} summarize and discuss the method, experiments, drawbacks and benefits and potential extensions and further work.  \n\n\\section{A Motivating Example}\\label{A_Motivating_Example}\nHere we illustrate the performance of the proposed method vs the GPOD method in order to give a motivation for this study. We use simulations of a two dimensional viscous flow around a cylinder at the Raynolds number of $160,$ obtained from \\url{https:\/\/www.csc.kth.se\/~weinkauf\/notes\/cylinder2d.html}. The simulations were performed  by Weinkauf et. al. \\cite{weinkauf2010streak} with the Gerris Flow Solver software \\cite{gerrisflowsolver}. The data set consists of a horizontal $u$ and a vertical $v$ velocities on an uniform $400 \\times 50 \\times 1001$ grid of $[-0.5, 7.5] \\times [-0.5, 0.5] \\times [15, 23]$ spatial-temporal domain.\\footnote{The simulations are run from $t=0$ to $t=23$, but velocities are only extracted from $t=15$ to $t=23$} In particular, we have $400$ points in the horizontal, and $50$ points in the vertical  direction, and $1001$ points in time.  \n\\begin{figure}[h]\n\t\\centering\n\n\t\t\\includegraphics[width=0.99\\linewidth]{CW_original_data_u.png}\n    \t\\includegraphics[width=0.99\\linewidth]{CW_original_data_v.png}\n\t\t\\captionof{figure}{Typical data instance from the original 2D flow around a cylinder data set with $u$ and $v$ component presented at the upper and lower panel, respectively} \\label{Cw_data_motivating}\n\n\\end{figure}\nThe cylinder has the diameter of $0.125$ and is centered at the origin, see \\cref{Cw_data_motivating}. The left vertical boundary (inlet) has Dirichlet boundary condition $u=1$ and $v=0$. The homogeneous Neumann boundary condition is given at the right boundary (outlet), and the homogeneous Dirichlet conditions on the remaining boundaries. At the start of simulations, $t=0$, both velocities were equal to zero.  We plot the velocities at the time $t \\approx 19$ (time step $500$) in \\cref{Cw_data_motivating}. \\\\\n\nFor simplicity, in the experiment below we extract the data downstream from the cylinder, that is, from grid point $40$ to $200$ in the horizontal direction, and keep all grid points in vertical direction. Hence, $\\mathcal{P}$ contains $N = 8000$ points, $160$ points in the horizontal and $50$ in the vertical direction. The temporal resolution is kept as before, that is, the number of time steps in $\\mathcal{T}$ is $K=1001$. For validation purposes, the data set was split into a train, validation and test data set. The train and validation data sets were used for optimization of the model parameters. For both the SCVAE and the GPOD, the goal was to minimize the $L2$ error between the true and the modeled flow state.  \nThe restriction of the GPOD is that the number of components $r$ could be at most $2M.$ \nTo deal with this problem, and to account for the flow incompressibility, we added the  regularization term  $\\lambda \\|L_{div} x^{(i)}\\|,$ $\\lambda>0$, to the objective function, see \\cref{Appendix_C}. For the GPOD method, the parameters $r$ and\/or $\\lambda$ where optimized on the validation data set in order to have the smallest mean error.\nWe give more details about objective functions for the SCVAE in \\cref{SCVAE_section}. For now we mention that there are two versions, where one version uses an additional divergence regularization term similar to GPOD.\\\\\n\nIn \\cref{Error_boxplot_CW} we plot the mean  of the relative $L_2$ error calculated on the test data for both methods with and without the div-regularization. The results are presented for $3,$ $4,$ and $5$ measurement locations, that is, $M=3,4,5.$ For each of these three cases, we selected $20$ different configurations of $M.$. In particular, we created $20$ subgrids ${\\mathcal Q}$, each containing $5$ randomly sampled spatial grid points. Next we removed one and then two points from each of the $20$ subgrids ${\\mathcal Q},$ to create new subgrids of $4$ and $3$ measurements, respectively.\n\\begin{figure}[h!]\n    \\centering\n\t\\includegraphics[width=0.75\\linewidth]{Motivating_example_boxplot_3M_v1.png}\n\t\t\\captionof{figure}{The mean relative error for two reconstruction methods. \n\t\tThe orange and blue label correspond to the SCVAE with (div-on) and without (div-off) additional divergence regularization. The green and red labels correspond to the GPOD method. }\\label{Error_boxplot_CW}\n\\end{figure}\nAs it can be seen in \\cref{Error_boxplot_CW}, both methods perform well for the $5$ measurements case. The resulting relative errors have comparable mean and variance. When reducing the number of observations, the SCVAE method maintains low errors, while the GPOD error increases. The SCVAE seems to benefit from the additional regularization of minimizing the divergence, in terms of lower error and less variation in the error estimates. The effect is more profound with fewer measurements. \\\\\n\nThe key benefit of the SCVAE is that its predictions are optimal for the given measurement locations.  In a contrast, the POD based approaches, and in particular the GPOD, create a set of basis functions (principal components) based on the training data independently of the measurements. While this has an obvious computational advantage, the number of principle components for complex flows can be high and, as a result, many more measurements are needed, \\cite{willcox2006unsteady,manohar2018data,Proctor2014}. There are number of algorithms that aim to optimize to measurement locations to achieve the best performance of the POD based methods, see e.g., \\cite{jo2019effective,willcox2006unsteady,YILDIRIM2009160}. In practice, however, the locations are often fixed and another approaches are needed. The results in \\cref{Error_boxplot_CW} suggest that the SCVAE could be one of these approaches.\n\n\\section{Methods}\\label{methods}\nBefore we introduce the model used for reconstruction of flows, we give a brief introduction to VAEs and CVAEs. For a detailed introduction, see \\cite{vae_intro}. VAEs are neural network models that has been used for learning structured representations in a  wide variety of applications, e.g., image generation \\cite{gregor2015draw}, interpolation between sentences \\cite{bowman2015generating} and compressed sensing \\cite{grover2019uncertainty}. \n\n\\subsection{Preliminary}\nLet us assume that the data $\\ifmmode\\bm{X}\\else\\textbf{\\textit{X}}\\fi$ is generated by a random process that involves an unobserved continuous random variable $\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi.$ The process consists of two steps: (i) a value $\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi^{(i)}$ is sampled from a prior $p_{\\theta^*}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi);$ and (ii)  $\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}$ is generated from a conditional distribution  $p_{\\theta^*}(\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi|\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi).$ In the case of flow reconstruction, $\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi$ could be thought of as unknown boundary or initial conditions, tidal and wind forcing, etc. However, generally $\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi$ is just a convenient construct to represent $\\ifmmode\\bm{X}\\else\\textbf{\\textit{X}}\\fi,$ rather than a physically explained phenomena. Therefore it is for convenience assumed that $p_{\\theta^*}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi)$ and $p_{\\theta^*}(\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi|\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi)$ come from parametric families of distributions $p_{\\theta}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi)$ and $p_{\\theta}(\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi|\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi),$ and their density functions are differentiable almost everywhere w.r.t. both $\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi$ and $\\theta$. A probabilistic auto-encoder is neural network that is trained to represent its input $\\ifmmode\\bm{X}\\else\\textbf{\\textit{X}}\\fi$ as $p_\\theta(\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi)$ via \\textit{latent representation} $\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi \\sim p_{\\theta}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi),$ that is,\n\n\\begin{equation} \\label{eq:p_theta(x)}\np_{\\theta}(\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi) = \\int p_{\\theta}(\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi,\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi) d\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi = \\int p_{\\theta}(\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi|\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi)p_{\\theta}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi)d\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi.\n\\end{equation}\nAs $p_\\theta(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi)$ is unknown and observations $\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi^{(i)}$ are not accessible, we must use $\\ifmmode\\bm{X}\\else\\textbf{\\textit{X}}\\fi$ in order to generate $\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi \\sim p_\\theta(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi).$ That is, the network can be viewed as consisting of two parts: an \\textit{encoder} $p_\\theta(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi)$ and a \\textit{decoder} $p_\\theta(\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi|\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi).$ Typically the true posterior distribution $p_{\\theta}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi)$ is intractable but could be approximated with variational inference \\cite{hoffman2013stochastic,blei2017variational}.  That is, we define a so called recognition model $q_{\\phi}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi)$ with variational parameters $\\phi$, which aims to approximate $p_{\\theta}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi).$ The recognition model is often parameterized as a Gaussian. Thus, the problem of estimating $p_{\\theta}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi)$, is reduced to finding the best possible estimate for $\\phi$, effectively turning the problem into an optimization problem. \\\\\n\nAn auto-encoder that uses a recognition model is called Variational Auto-Encoder (VAE). In order to get good prediction we need to estimate the parameters $\\phi$ and $\\theta.$ The marginal likelihood  is equal to the sum over the marginal likelihoods of the individual samples, that is, $\\sum_{i=1}^K \\log p_{\\theta}(\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}).$ Therefore, we further on present estimates for an individual sample. The Kullback -Leibler divergence between two probability distributions  $q_{\\phi}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)})$ and $p_{\\theta}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)})$, defined as $$D_{KL}[q_{\\phi}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)})||p_{\\theta}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)})] = \\int q_{\\phi}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}) \\log\\left(\\frac{q_{\\phi}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)})}{p_{\\theta}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)})}\\right) d\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi,$$ can be interpreted as a measure of distinctiveness between these two distributions \\cite{kullback1951information}. It can be shown, see \\cite{vae_intro}, that\n\\begin{equation}\\label{eq:DKL_via_L}\n\\log p_{\\theta}(\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)})=D_{KL}[q_{\\phi}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)})||p_{\\theta}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)})] +\\mathcal{L}(\\theta,\\phi;\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}),\n\\end{equation}\nwhere\n$$\n\\mathcal{L}(\\theta,\\phi;\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}) = \n{\\mathbb E}_{q_{\\phi}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)})} \\left[ -\\log q_{\\phi}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)})+\\log p_{\\theta}(\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)},\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi) \\right].\n$$\nSince KL-divergence is non-negative, we have $\\log p_{\\theta}(\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}) \\geq\\mathcal{L}(\\theta,\\phi;\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)})$ and\n$\\mathcal{L}(\\theta,\\phi;\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)})$\nis called Evidence Lower Bound (ELBO) for the marginal likelihood $\\log p_{\\theta}(\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}).$\nThus, instead of maximizing the marginal probability, one can instead maximize its variational lower bound to which we also refer as an objective function. It can be further shown that the ELBO can be written as\n\\begin{equation}\n\t\\label{eq:VLB:2_no_beta}\n\t\\mathcal{L}(\\theta,\\phi;\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)})= {\\mathbb E}_{q_{\\phi}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)})} \\left[ \\log p_{\\theta}(\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}|\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi) \\right] - D_{KL}[q_{\\phi}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)})||p_{\\theta}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi)].\n\\end{equation}\nReformulating the traditional VAE framework as a constraint optimization problem, it is possible to obtain the $\\beta$-VAE \\cite{higgins2016beta} objective function if $p_\\theta(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi) =\\mathcal{N}({\\bf 0},{\\bm I}),$\n\\begin{equation}\n\t\\label{eq:VLB:2}\n\t\\mathcal{L}(\\theta,\\phi;\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)})= {\\mathbb E}_{q_{\\phi}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)})} \\left[ \\log p_{\\theta}(\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}|\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi) \\right] - \\beta D_{KL}[q_{\\phi}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)})||p_{\\theta}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi)],\n\\end{equation}\nwhere $\\beta>0.$ Here $\\beta$ is a regularisation coefficient that constrains the capacity of the latent representation $\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi$. The ${\\mathbb E}_{q_{\\phi}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)})} \\left[ \\log p_{\\theta}(\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}|\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi) \\right]$ can be interpreted as the reconstruction term, while the KL-term, $\\beta D_{KL}[q_{\\phi}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)})||p_{\\theta}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi)]$ as regularization term. \n\n\nConditional Variational Auto-encoders \\cite{sohn2015learning} (CVAE) are similar to VAEs, but differ by conditioning on an additional property of the data (e.g. a label or class), here denoted $\\ifmmode\\bm{c}\\else\\textbf{\\textit{c}}\\fi$. Conditioning both the recognition model and the true posteriori on both $\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}$ and $\\ifmmode\\bm{c}\\else\\textbf{\\textit{c}}\\fi$ results in the CVAE ELBO \n\\begin{align}\n    \\begin{split}\n        \\mathcal{L}(\\theta,\\phi;\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)},\\ifmmode\\bm{c}\\else\\textbf{\\textit{c}}\\fi)={\\mathbb E}_{q_{\\phi}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}, \\ifmmode\\bm{c}\\else\\textbf{\\textit{c}}\\fi)} \\left[ \\log p_{\\theta}(\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}|\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi,\\ifmmode\\bm{c}\\else\\textbf{\\textit{c}}\\fi) \\right] - D_{KL}[q(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)},\\ifmmode\\bm{c}\\else\\textbf{\\textit{c}}\\fi)||p_{\\theta}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{c}\\else\\textbf{\\textit{c}}\\fi)]. \\label{CVAE_objective}\n    \\end{split}\n\\end{align}\nIn the decoding phase, CVAE allows for conditional probabilistic reconstruction and permits sampling from the conditional distribution $p_{\\theta}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{c}\\else\\textbf{\\textit{c}}\\fi)$, which has been useful for generative modeling of data with known labels, see \\cite{sohn2015learning}. Here we investigate a special case of the CVAE when $\\ifmmode\\bm{c}\\else\\textbf{\\textit{c}}\\fi$ is a partial observation of $\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi.$ We call this Semi Conditional Variational Auto-encoder (SCVAE).\n\n\\subsection{Semi Conditional Variational Auto-encoder}\\label{SCVAE_section}\nThe SCVAE takes the input data $\\ifmmode\\bm{X}\\else\\textbf{\\textit{X}}\\fi$, conditioned on $\\ifmmode\\bm{M}\\else\\textbf{\\textit{M}}\\fi$ and approximates the probability distribution $p_{\\theta}(\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi|\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi,\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi).$ Then we can generate $\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}$, based on the observations $\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)}$ and latent representation $\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi$. As $\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)}=C \\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}$ where $C$ is a non-stochastic sampling matrix, we have \n$$p_{\\theta}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}, \\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)}) = p_{\\theta}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}), \\, \\mbox{ and } \\, q_{\\phi}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)},\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)})=q_{\\phi}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}).$$ Therefore, from \\cref{CVAE_objective} the ELBO for SCVAE is \n\\begin{equation}\n    \\begin{split}\n        \\log p_{\\theta}(\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}|\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)})\\geq \\mathcal{L}(\\theta,\\phi;\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)},\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)})=\n        &{\\mathbb E}_{q_{\\phi}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)})} \\left[ \\log p_{\\theta}(\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}|\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi, \\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)}) \\right] \\\\ & - D_{KL}[q_\\phi(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)})||p_{\\theta}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)})]\\label{eq:ELBO:SCVAE_no_beta}\n    \\end{split}\n\\end{equation}\nwhere $p_\\theta(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)}) = \\mathcal{N}({\\bf 0},{\\bf I}).$ Similarly as for the $\\beta$-VAE \\cite{higgins2016beta} we can obtain a relaxed version of \\cref{eq:ELBO:SCVAE_no_beta} by maximizing the parameters $\\{\\phi, \\theta\\}$ of the expected log-likelihood ${\\mathbb E}_{q_\\phi(\\cdot)} [\\log p_\\theta (\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}|\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)},\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi)])$ and treat it as an constrained optimization problem. That is, \n\\begin{align}\n    \\begin{split}\n    & \\max\\limits_{\\phi,\\theta} {\\mathbb E}_{q_\\phi(\\cdot)} [\\log p_\\theta (\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}|\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)},\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi)])  \\text{ subject to} \\\\ &\n    D_{KL}(q_\\phi(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)},\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)})||p_\\theta (\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)}) \\leq \\epsilon  \n    \\end{split}\\label{beta_SCVAE_opt_prob}\n\\end{align}\nwhere $\\epsilon>0$ is small. The subscript $q_\\phi(\\cdot)$ is short for $q_\\phi(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)},\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}).$ Since $\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)}$ is dependent on $\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}$ we have that $q_\\phi(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)},\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}) = q_\\phi(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}).$ \\cref{beta_SCVAE_opt_prob}  can expressed as a Lagrangian under the Karush\u2013Kuhn\u2013Tucker (KKT) conditions \\cite{kuhn2014nonlinear, karush1939minima}. Hence,\n\\begin{align}\n    \\begin{split}\n    \\mathcal{F}(\\theta, \\phi, \\beta, \\alpha, \\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}, \\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)}) & = \n    {\\mathbb E}_{q_\\phi(\\cdot)} [\\log p_\\theta (\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}|\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)},\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi)]) \\\\ & + \n    \\beta(D_{KL}(q_\\phi(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)})||p_\\theta (\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)})-\\epsilon)   \n    \\end{split}\\label{beta_SCVAE_constrained}\n\\end{align}\nAccording to the complementary slackness KKT condition $\\beta \\geq 0,$ we can rewrite \\cref{beta_SCVAE_constrained} as\n\\begin{align}\n    \\centering\n    \\begin{split}\n      \\mathcal{F}(\\theta, \\phi, \\beta, \\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}, \\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)}) \\geq  \\mathcal{L}(\\theta, \\phi, \\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}, \\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)}) & =  {\\mathbb E}_{q_\\phi(\\cdot)} [\\log p_\\theta (\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}|\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)},\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi)])  \\\\ & +\n     \\beta D_{KL}(q_\\phi(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)})||p_\\theta (\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)}).   \n    \\end{split}\\label{beta_SCVAE_constrained_2}\n\\end{align}\nObjective functions in \\cref{eq:ELBO:SCVAE_no_beta} and \\cref{beta_SCVAE_constrained_2}, and later \\cref{eq:ELBO:SCVAE:J}, show that if conditioning on a feature which is a known function of the original data, such as measurements, we do not need to account for them in the encoding phase.The measurements are then coupled with the encoded data in the decoder. We sketch the main components of the SCVAE in \\cref{DAE_neural_network_sketch}.\n\\begin{figure}[ht]\n    \\centering\n\t\t\\includegraphics[width=0.80\\linewidth]{Model_Sparse_autoencoder_new_notation_i.png}\n\t\t\\captionof{figure}{The figure shows a sketch of the model used to estimate \n\t\t$p_{\\theta}(\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi|\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)})$. During training both the observations $\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)}$ and the data $\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}$ will be used. After the model is trained, we can predict using only the decoder part of the neural network. The input to the decoder will then only be the observations and random samples from the latent space.}\\label{DAE_neural_network_sketch}\n\\end{figure}\nIn order to preserve some physical properties of the data $\\ifmmode\\bm{X}\\else\\textbf{\\textit{X}}\\fi,$ we can condition yet on another feature. Here we utilize the incompressibility property of the fluid, i.e., $\\ifmmode\\bm{d}\\else\\textbf{\\textit{d}}\\fi^{(i)}=L_{div} \\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)} \\approx 0,$ see \\cref{eq:L_div}. \\\\ \n\nWe intend to maximize a log-likelihood under an additional constrain $\\ifmmode\\bm{d}\\else\\textbf{\\textit{d}}\\fi^{(i)}$, compared to \\cref{beta_SCVAE_opt_prob}. That is\n\\begin{align}\n    \\begin{split}\n    & \\max\\limits_{\\phi,\\theta} {\\mathbb E}_{q_\\phi(\\cdot)} [\\log p_\\theta (\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}|\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)},\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi)])  \\text{ subject to} \\\\ &\n    D_{KL}(q_\\phi(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)})||p_\\theta (\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)},\\ifmmode\\bm{d}\\else\\textbf{\\textit{d}}\\fi^{(i)}) \\leq \\epsilon \\quad \\text{and} \\quad \\\\ &\n    -{\\mathbb E}_{q_\\phi(\\cdot)}[\\log p_\\theta(\\ifmmode\\bm{d}\\else\\textbf{\\textit{d}}\\fi^{(i)}|\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)},\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi)] \\leq \\delta \n    \\end{split}\\label{Constraint_optimization_prob_v3}\n\\end{align}\nwhere $\\epsilon, \\delta>0$ are small. \\cref{Constraint_optimization_prob_v3}  can expressed as a Lagrangian under the Karush\u2013Kuhn\u2013Tucker (KKT) conditions as before and as a consequence of the complementary slackness condition $\\lambda,\\beta \\geq 0,$ we can obtain the objective function \n\n\\begin{align}\n    \\centering\n    \\begin{split}\n      \\mathcal{F}(\\theta, \\phi, \\beta, \\alpha, \\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}, \\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)}, \\ifmmode\\bm{d}\\else\\textbf{\\textit{d}}\\fi^{(i)}) \\geq  \\mathcal{L}(\\theta, \\phi, \\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}, \\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)}, \\ifmmode\\bm{d}\\else\\textbf{\\textit{d}}\\fi^{(i)}) & =  {\\mathbb E}_{q_\\phi(\\cdot)} [\\log p_\\theta (\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}|\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)},\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi)])  \\\\ & +\n     \\lambda\\,{\\mathbb E}_{q_\\phi(\\cdot)} [\\log p_\\theta (\\ifmmode\\bm{d}\\else\\textbf{\\textit{d}}\\fi^{(i)}|\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)},\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi)])  \\\\ & -\n     \\beta D_{KL}(q_\\phi(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)})||p_\\theta (\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)},\\ifmmode\\bm{d}\\else\\textbf{\\textit{d}}\\fi^{(i)}),   \n    \\end{split}\\label{eq:ELBO:SCVAE:J}\n\\end{align}\nwhere $p(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)}, \\ifmmode\\bm{d}\\else\\textbf{\\textit{d}}\\fi^{(i)}) = \\mathcal{N}({\\bf 0},{\\bf I}).$ For convenience of notation we refer to the objective function \\cref{beta_SCVAE_constrained_2} as the case with $\\lambda=0$, and the objective function \\cref{eq:ELBO:SCVAE:J} as the case with $\\lambda > 0.$ Observe that under the Gaussian assumptions on the priors, \\cref{eq:ELBO:SCVAE:J} is equivalent to \\cref{beta_SCVAE_constrained_2} if $\\lambda=0.$ Thus, from now one we will refer to it as a special case of \\cref{eq:ELBO:SCVAE:J} and denote as $\\mathcal{L}_{0}.$ \\\\\n\nSimilarly to \\cite{kingma2013auto} we obtain $q_{\\phi}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)})= \\mathcal{N}({\\mu}^{(i)} \\mathbf{1},(\\sigma^{(i)})^2\\,\\mathbf{I})\n$, that is, $\\phi=\\{ \\mu, \\sigma\\}.$  This allows to express the KL-divergence terms in a closed form and avoid issues related to differentiability of the ELBOs.  Under these assumptions, the KL-divergence terms can be integrated analytically while the term\n${\\mathbb E}_{q_{\\phi}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)})} \\left[ \\log p_{\\theta}(\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}|\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi, \\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)}) \\right] $ and ${\\mathbb E}_{q_{\\phi}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)})} \\left[ \\log p_{\\theta}(\\ifmmode\\bm{d}\\else\\textbf{\\textit{d}}\\fi^{(i)}|\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi, \\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)}) \\right] $\nrequires estimation by sampling\n\\begin{equation}\\label{eq:E-estimate}\n    \\begin{array}{l}\n         {\\mathbb E}_{q_{\\phi}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)})} \\left[ \\log p_{\\theta}(\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}|\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi, \\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)}) \\right] \\approx \n        \\frac{1}{L}\\sum\\limits_{l=1}^{L} \\log  p_{\\theta}(\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}|\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi^{(i,l)},\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)}),\\\\\n        {\\mathbb E}_{q_{\\phi}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)})} \\left[ \\log p_{\\theta}(\\ifmmode\\bm{d}\\else\\textbf{\\textit{d}}\\fi^{(i)}|\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi, \\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)}) \\right] \\approx \n        \\frac{1}{L}\\sum\\limits_{l=1}^{L} \\log  p_{\\theta}(\\ifmmode\\bm{d}\\else\\textbf{\\textit{d}}\\fi^{(i)}|\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi^{(i,l)},\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)}),\\\\\n        \\mbox{where } \\, \\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi^{(i,l)} = g_{\\phi}(\\bm{\\epsilon}^{(i,l)}, \\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}), \\quad \n        \\bm{\\epsilon}^{l} \\sim p(\\bm{\\epsilon}).\n    \\end{array}\n\\end{equation}\nHere $\\bm{\\epsilon}^{l}$ is an auxiliary (noise) variable with independent marginal $p(\\bm{\\epsilon})$, and $g_{\\phi}(\\cdot)$ is a differentiable transformation of $\\bm{\\epsilon},$ parametrized by $\\phi,$  see for details \\cite{kingma2013auto}. We denote $\\mathcal{L}_\\lambda,$ $\\lambda \\geq 0$  \\cref{eq:ELBO:SCVAE:J} with the approximation above as $\\mathcal{\\widehat{L}_\\lambda},$ that is,\n\\begin{align}\n    \\begin{split}\n        & \\widehat{\\mathcal{L}}_\\lambda(\\theta, \\phi, \\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)},\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)},\\ifmmode\\bm{d}\\else\\textbf{\\textit{d}}\\fi^{(i)})  =   \\frac{1}{L}\\sum\\limits_{l=1}^{L} \\log  p_{\\theta}(\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}|\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi^{(i,l)},\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)}) \\\\\n         &+ \\lambda \\frac{1}{L}\\sum\\limits_{l=1}^{L} \\log p_{\\theta}(\\ifmmode\\bm{d}\\else\\textbf{\\textit{d}}\\fi^{(i)}|\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi^{(i,l)},\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)}) -\\beta D_{KL}[q_{\\phi}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)})||p_{\\theta}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)}, \\ifmmode\\bm{d}\\else\\textbf{\\textit{d}}\\fi^{(i)})].\n        \\label{Loss_function}\n    \\end{split}\n\\end{align}\nThe objective function $\\widehat{\\mathcal{L}}_\\lambda$ can be maximized by gradient descent. Since the gradient $\\nabla_{\\theta,\\phi}\\,\\widehat{\\mathcal{L}}_\\lambda$ cannot be calculated for large data sets, Stochastic Gradient Descent methods, see \\cite{kiefer1952stochastic, robbins1951stochastic} are typically used where \n\\begin{equation}\n    \\widehat{\\mathcal{L}}_\\lambda(\\theta, \\phi; \\ifmmode\\bm{X}\\else\\textbf{\\textit{X}}\\fi, \\ifmmode\\bm{M}\\else\\textbf{\\textit{M}}\\fi, \\bm{D}) \\approx \\widehat{\\mathcal{L}}^{R}(\\theta, \\phi; \\ifmmode\\bm{X}\\else\\textbf{\\textit{X}}\\fi^R, \\ifmmode\\bm{M}\\else\\textbf{\\textit{M}}\\fi^R, \\bm{D}^R) = \n    \\frac{K}{R}\\sum\\limits_{r=1}^{R} \\widehat{\\mathcal{L}}_\\lambda(\\theta, \\phi; \\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i_r)}, \\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i_r)}, \\ifmmode\\bm{d}\\else\\textbf{\\textit{d}}\\fi^{(i_r)}), \\quad \\lambda \\geq 0.\\label{obj_function_SCVAE}\n\\end{equation}\nHere  $\\ifmmode\\bm{X}\\else\\textbf{\\textit{X}}\\fi^{R}=\\left\\{\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i_r)}\\right\\}_{r=1}^{R},$  $R<K$ is a minibatch consisting of randomly sampled datapoints, $\\ifmmode\\bm{M}\\else\\textbf{\\textit{M}}\\fi^{R}=\\left\\{\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i_r)}\\right\\}_{r=1}^{R}$ and $\\bm{D}^{R}=\\left\\{\\ifmmode\\bm{d}\\else\\textbf{\\textit{d}}\\fi^{(i_r)}\\right\\}_{r=1}^{R}.$ After the network is optimized, a posterior predictive distribution $p_\\theta(\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi|\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi, \\ifmmode\\bm{d}\\else\\textbf{\\textit{d}}\\fi)$ can be approximated with a Monte Carlo estimator. \\\\\n\n\n\\subsubsection{Uncertainty Quantification}\\label{sec:posterior}\nLet  $\\hat{\\theta}$ and $\\hat{\\phi}$ be an estimation of generative and variational parameters, as described in \\cref{SCVAE_section}. Then the decoder can be used to predict the posterior as\n\\begin{align}\\label{eq:p_pred}\n    p_{\\hat{\\theta}}(\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi|\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^*, \\ifmmode\\bm{d}\\else\\textbf{\\textit{d}}\\fi^*) \\approx \\frac{1}{N_{MC}} \\sum_{j=1}^{N_{MC}} p_{\\hat{\\theta}}(\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi|\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi^{(j)},\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^*, \\ifmmode\\bm{d}\\else\\textbf{\\textit{d}}\\fi^*) \\xrightarrow[N_{MC} \\rightarrow \\infty]{} \\int\n    p_{\\hat{\\theta}}(\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi|\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi, \\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^*, \\ifmmode\\bm{d}\\else\\textbf{\\textit{d}}\\fi^*)p_{\\hat{\\theta}}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^*,\\ifmmode\\bm{d}\\else\\textbf{\\textit{d}}\\fi^*)d\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi.\n\\end{align}\nWhile sampling from the latent space has been viewed typically as an approach for generating new samples with similar properties, here we use it to estimate the prediction uncertainty of the trained model. From \\cref{eq:p_pred} we are able to estimate the mean prediction  $\\hat{\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi}^*$ and empirical covarience matrix $\\widehat{{\\bm \\Sigma}}$ using  a Monte Carlo estimator. We get\n\\begin{align} \n      \\widehat{\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi}^*=\n      \\frac{1}{N_{MC}} \\sum\\limits_{j=1}^{N_{MC}} \\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(j)}\\quad \\mbox{and} \\quad\n      \\widehat{{\\bm \\Sigma}} =  \\frac{1}{N_{MC}-1} \\sum\\limits_{j=1}^{N_{MC}}(\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(j)} -  \\widehat{\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi}^*)(\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(j)} -  \\widehat{\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi}^*)^T,\n        \\label{eq:mean_and_cov} \n               \n \\end{align}\nwhere $\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(j)}  \\sim p_{\\hat{\\theta}}(\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi|\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^*,\\ifmmode\\bm{d}\\else\\textbf{\\textit{d}}\\fi^*).$ The empirical standard deviation is then simply $\\widehat{\\bm \\sigma} = \\sqrt{diag(\\widehat{{\\bm \\Sigma}})}.$ To estimate the confidence region we assume that the predicted $p_{\\hat{\\theta}}(\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi|\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{*},\\ifmmode\\bm{d}\\else\\textbf{\\textit{d}}\\fi^*)$ is well approximated by a normal distribution $N({\\bm \\mu}, \\bm{\\Sigma}).$ Given that $\\widehat{\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi}^*$ and $\\widehat{{\\bm \\Sigma}}$ are approximations of $\\bm{\\mu}$ and $\\bm{\\Sigma},$ obtained from $N_{MC}$ samples as above, a confidence region estimate for a prediction $\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{*}$ can be given as\n\\begin{equation}\\label{eq:ConfRegion:1}\n    \\left\\{\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)} \\in {\\mathbb R}^{2N}: \\,(\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}-{\\widehat{\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi}^*})^T \\hat{\\bm \\Sigma}^{+} (\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}- \\widehat{\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi}^*)\\leq \\chi^2_k(p)\\right\\}\n\\end{equation}\nwhere $\\chi^2_k(p)$ is the quantile function for probability $p$ of the chi-squared distribution with $k=\\min\\{N_{MC},2N\\}$ degrees of freedom, and $\\widehat{\\bm \\Sigma}^{+}$ is the pseudoinverse of $\\widehat{\\bm \\Sigma}.$ Using the singular value decomposition, ${\\widehat{\\bm \\Sigma}}=\\bm{U S U}^T,$ the corresponding interval for $(\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^*)_n,$ $n=1,\\dots,2N,$ is\n\\begin{equation}\\label{eq:ConfRegion:2}\n   \\left[\n ({\\widehat{\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi}^*})_n-\\sqrt{\\chi^2_k(p)}\\,\\|{\\bm u}_n^T\\,{\\bm S}^{1\/2}\\|_2,\\quad\n ({\\widehat{\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi}^*})_n+\\sqrt{\\chi^2_k(p)}\\,\n \\|{\\bm u}_n^T\\,{\\bm S}^{1\/2}\\|_2\n  \\right]   \n\\end{equation}\nwhere ${\\bm u}_n^T$ is $n$th row of the matrix ${\\bm U}.$\n\n\n\\section{Experiments}\\label{Experiment}\nWe will present the SCVAE method on two different data sets. The first one is the 2D flow around a cylinder described in \\cref{A_Motivating_Example}, and the second is ocean currents on the seafloor created by the Bergen Ocean Model \\cite{berntsen2000users}. The data $\\ifmmode\\bm{X}\\else\\textbf{\\textit{X}}\\fi$ consists of the two dimensional velocities ${\\bm w} =(u, v).$ To illustrate the results we will plot $u$ and $v$ components of $\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)} \\in \\ifmmode\\bm{X}\\else\\textbf{\\textit{X}}\\fi,$ see \\cref{eq:x_i}. For validation of the models, the data $\\ifmmode\\bm{X}\\else\\textbf{\\textit{X}}\\fi$ is split into train, test and validation subsets, which are subscripted accordingly, if necessary. The data sets, spitting and preprocessing for each case are described in \\cref{Cylinder_wake_experiment,BOM_experiment}. \\\\\n\nWe use a schematically simple architecture to explore the SCVAE. The main ingredient of the encoder is convolutional neural network (CNN)  \\cite{lecun1998gradient, krizhevsky2012imagenet} and for the decoder we use transposed CNN-layers \\cite{noh2015learning}. The SCVAE has a slightly different architecture in each case, which we present in \\cref{Appendix_A}. \\\\\n\nThe SCVAE is trained to maximize the objective functions in \\cref{obj_function_SCVAE} with the backpropagation algorithm \\cite{lecun1989backpropagation} and the Adam algorithm \\cite{kingma2014adam}. We use an adaptive approach of weighing the reconstruction term with KL-divergence and\/or divergence terms \\cite{heydari2019softadapt}, that is, finding the regularization parameters $\\beta$ and $\\lambda$. Specifically, we calculate the proportion of each term contribution of the total value of the objective function, and scale the terms accordingly. This approach prevents posterior collapse. Posteriori collapse occurs if the KL-divergence term becomes too close to zero, resulting in a non probabilistic reconstruction. The approach of weighing the terms proportionally iteratively adjusts the weight of the KL-divergence term, $\\beta$, such that posterior collapse is avoided. For the result shown here we trained the SCVAEs  with an early stopping criteria of $50$ epochs, i.e., the optimization is stopped if we do not see any improvement after $50$ epochs, and returns the best model. We use a two-dimensional Gaussian  distribution for $p_\\theta(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)}, \\ifmmode\\bm{d}\\else\\textbf{\\textit{d}}\\fi^{(i)})$ in all the experiments.\\\\\n\nLet the test data $\\ifmmode\\bm{X}\\else\\textbf{\\textit{X}}\\fi_{test}$ consist of $n$ instances $\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)},$ $i=1,\\dots,n,$ and $\\widehat{\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi}^{(i)}$ denote a prediction of the true $\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}$ given $\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)}.$ In the case of the SCVAE, $\\widehat{\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi}^{(i)}$ is the mean prediction obtained as in \\cref{eq:mean_and_cov}. For the GPOD method, $\\widehat{\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi}^{(i)}$ is a deterministic output of the optimization problem, see \\cref{Appendix_C}. In order to compare the SCVAE results with the results of the GPOD method, we introduce the following measures; the mean of the relative error for the prediction\n\\begin{equation}\\label{L2_error}\n  \\mathcal{E} = \\frac{1}{n}\\sum\\limits_{i=1}^{n}\\frac{||\\widehat{\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi}^{(i)} - \\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}||_2}{||\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}||_2}\n\\end{equation}\nand the mean of the absolute error for the divergence \n\\begin{equation}\n        \\mathcal{E}_{div} = \\frac{1}{n}\\sum\\limits_{i=1}^{n}\\| L_{div} \\, \\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}\\|_2.\\label{divergence_error_1}\n\\end{equation}\n\n\\subsection{2D Flow Around a Cylinder}\\label{Cylinder_wake_experiment}\nHere we return to the example in \\cref{A_Motivating_Example}. Below we give some additional details of the data preprocessing and model implementation.\n\\subsubsection{Preprocessing}\nThe data is reduced as described in \\cref{A_Motivating_Example}. \nWe assess the SCVAE with a sequential split for train, test and validation. The sequantial split is defined as follows. The last $15 \\%$ of the data is used for test, the last $30\\%$ of the remaining data is used for validation, and the first $70\\%$ for training. To improve the conditioning of the optimization problem we scale the data as decribed in  \\cref{Appendix_D}. The errors $\\mathcal{E}$ (\\cref{L2_error}) and $\\mathcal{E}_{div} $ (\\cref{divergence_error_1}) are calculated after scaling the data back. The input to the SVAE $\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}$ was shaped as $(160 \\times 50 \\times 2)$ in order to apply convolutional layers. Here we use  $5,4,3$ and $2$ fixed spatial measurements, that is, four different subgrids ${\\mathcal Q}$ \n\\begin{align}\n    \\centering\n    \\begin{split}\n    {\\mathcal Q}_5 = & \\{(12,76),(47,8),(30,40),(153,34),(16,10)\\}, \\\\\n    {\\mathcal Q}_4 = & \\{(12,76),(47,8),(30,40),(153,34)\\},  \\\\\n    {\\mathcal Q}_3 = & \\{(12,76),(47,8),(30,40)\\}, \\\\\n    {\\mathcal Q}_2 = & \\{(12,76),(47,8)\\}.\n    \\end{split}\\label{CW_measurements}\n\\end{align}\nThe flow state at these specific locations constitute $\\ifmmode\\bm{M}\\else\\textbf{\\textit{M}}\\fi$.\n\n\\subsubsection{Model}\nA schematic description of the model is given in \\cref{Appendix_A}. \nThe first layer of the encoder is a zero-padding layer that expands the horizontal and vertical dimension by adding zeros on the boundaries. Here we add zero-padding of four in the horizontal and three in the vertical direction. The subsequent layers consists of two convolutional layers, where the first and second layer have $160$ and $200$ filters, respectively. We use a kernel size and strides of $2$ in both convolutional layers and ReLu activation functions. This design compresses the data into a $(42 \\times 14 \\times 200)$ shape. The compressed representation from the convolutional layers are flattened and are further compressed into a $64$ dimensional vector through a traditional dense layer. Two outputs layers are defined to represent the mean and log-variance of the latent representation $\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi$. The reparametrization trick is realized in a third layer, a so called  lambda layer, which takes the mean and log-variance as an input and generates $\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi$. The output of the encoder are the samples $\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi^{(i)}$ and the mean and the log-variance of $\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi$. \\\\\n\nThe decoder takes the latent representation $\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi^{(i)}$ and the measurements $\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)}$ as input. The input $\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)}$ is flattened and then concatenated with $\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi^{(i)}$. The next layer is a dense layer with shape $(42 \\times 14 \\times 200)$. Afterwards there are two transposed convolutional layers with filters of $200$ and $160$. The strides and the kernel size is the same as for the encoder. The final layer is a transposed convolutional layer, with same dimension as the input to the encoder, that is the dimension of $\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}$. A linear activation function is used for this output layer. The last layer of the model is a lambda layer that removes the zero-padding. In the next section we show statistics of the probabilistic reconstruction and compare with the GPOD method. \n\n\\subsubsection{Results}\nIn \\cref{Cylinder_wake_pred} we have plotted the true velocity field, the reconstructed velocities, the standard deviation of the velocities and the absolute error between the true and reconstructed velocity fields. The observations placement is shown as stars (black and white). The SCVAE with the objective function  $\\widehat{\\mathcal{L}}_0$, see \\cref{CVAE_objective}, was used for this prediction. To generate the posterior predictive distributions, \\cref{eq:p_pred},  we sample $100$ realizations from $\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi \\sim \\mathcal{N}(\\bm{0},\\bm{I})$ , which allows for calculation mean prediction and uncertainty estimates, see \\cref{eq:mean_and_cov}. \\\\\n\nWe emphasise here again that the SCVAE with $\\widehat{\\mathcal{L}}_0$ and with $\\widehat{\\mathcal{L}}_\\lambda,$ $\\lambda>0,$ are two different models. For the notation sake we here refer to $\\lambda=0$ when we mean the model with the objective function in \\cref{CVAE_objective}, and to $\\lambda>0$ when in \\cref{eq:ELBO:SCVAE:J}. The same holds for the GPOD method, see \\cref{Appendix_D}. When $\\lambda=0,$ the number of the principle components $r$ is less $2M.$ The number $r$ is chosen such that the prediction on the validation data has the smallest possible error on average. If $\\lambda>0,$ no restrictions on $r$ are imposed. In this case both  $\\lambda$ and $r$ are estimated from the validation data. \\\\\n\nThe general observation is that the SCVAE reconstruction fits the data well, with associated low uncertainty. \nThis can be explained by the periodicity in the data. In particular, the training and validation data sets represent the  test data well enough.\n\\begin{figure}[H]\n\t\\centering\n\n\t\t\\includegraphics[width=0.99\\linewidth]{CW_recon_3M_v1_CR.png}\n\t\\captionof{figure}{Left panels shows the u-velocities, and the right panel v-velocities. The results are based on a model trained with $\\lambda=0$ and $Q_3$ measurement locations. \\textbf{First panels:} The true solutions \\textbf{Second panels:} Reconstructed solution based on the SCVAE model \\textbf{Third panels:} Standard deviation of the predicted solution \\textbf{Fourth panels:} Absolute error between the true and predicted solution.}\\label{Cylinder_wake_pred}\n\n\\end{figure}\n\nIn \\cref{Cylinder_wake_time_series} we  have plotted four time series of the reconstructed test data at two specific grid points, together with the confidence regions constructed as in \\cref{eq:ConfRegion:2} with $p=0.95.$ The two upper panels represents the reconstruction at the grid point $(6, 31)$, and the lower at $(101,25)$ for $u$ and $v$ on the left and right side, respectively. The SCVAE reconstruction is significantly better than the GPOD, and close to the true solution for all time steps. \n\\begin{figure}[H]\n\t\\centering\n\n\t\t\\includegraphics[width=0.85\\linewidth]{CW_time_series_with_GPOD_CR.png}\n\t\t\\captionof{figure}{Velocities $u$ and $v$ at specific locations. The red line corresponds to the true values, blue to the SCVAE mean prediction, and orange to the GPOD reconstruction. Light blue shaded area represents the confidence region obtained in \\cref{eq:ConfRegion:2} with $p=0.95.$ The results are obtained from the model trained with $\\lambda=0$ and $Q_3$ measurement locations. \\textbf{Upper panels:} The time series and confidence region for $u$ (left) and $v$ (right) at  grid point $(6, 31).$ \\textbf{Lower panels:} The time series and confidence region for $u$ (left) and $v$ (right) at grid point $(101, 25).$ }\\label{Cylinder_wake_time_series}\n\n\\end{figure}\n\\begin{figure}[H]\n\t\\centering\n\n\t\t\\includegraphics[width=0.85\\linewidth]{CW_error_with_GPOD_CR.png}\n\t\t\\captionof{figure}{The difference between the true and predicted estimate for the SCVAE (blue) and for the GPOD (orange). The light blue shaded region represents the difference marginals, obtained from the confidence region in \\cref{BOM_time_series}. The estimates are based on a model trained with $\\lambda=0$ and $Q_3$ measurement locations. \\textbf{Upper panels:} The difference between the true and predicted estimate at grid point $(6,31)$ for $u$ (left) and $v$ (right), \\textbf{Lower panels:} The difference between the true and predicted estimate at point $(101,25)$ for $u$ (left) and $v$ (right).}\\label{Cylinder_wake_error}\n\n\\end{figure}\n\\cref{Cylinder_wake_error} shows the difference between the true values and the model prediction in time for the same two locations. This figure has to be seen in context with \\cref{Cylinder_wake_pred}. In \\cref{tab:comparison_CW}  we display the relative errors, \\cref{L2_error}, for the SCVAE and the GPOD method, both with and without divergence regularization, for $5, 4, 3,$ and $2$ measurement locations given in \\cref{CW_measurements}. \\\\\n\nThe results of the SCVAE depend on two stochastic inputs which are (i) randomness in the initialization of the prior weights and (ii) random mini batch sampling. We have trained the model with a each measurement configuration $10$ times, and chose the model that performs the best on the validation data set. Ideally we would run test cases where we used all the values as measurements,i.e., $\\ifmmode\\bm{M}\\else\\textbf{\\textit{M}}\\fi=\\ifmmode\\bm{X}\\else\\textbf{\\textit{X}}\\fi,$ and test how well the model would reconstruct in this case. This would then give us the lower bound of the best reconstruction that is possible for this specific architecture and hyper parameter settings. However, this scenario was not possible to test, due to limitations in memory in the GPU. Therefore we have used a large enough $M$ which still allowed us to run the model. In particular, we used every fifth and second pixel in the horizontal and vertical direction, which resulted in a total of $(32 \\times 25)$ measurement locations, or $M=800$. We believe that training the model with these settings, gave us a good indication of the lower bound of the reconstruction error. The error observed was of the magnitude of $10^{-3}$. \\\\\n\nThis lower bound has been reached for all measurement configurations \\cref{CW_measurements}. \nHowever, larger computational cost was needed to reach the lower bound for fewer measurement locations. \\cref{Epochs_per_measurement_CW} shows the number of epochs as a boxplot diagram. In comparison with GPOD, the SCVAE error is 10 times lower than the GPOD error, and this difference becomes larger with fewer measurements. Note that adding regularization did not have much effect on  the relative error. From the motivating example we observed that regularizing with $\\lambda>0$ is better in terms of a more consistent and low variable error estimation. Here we selected from the 10 trained models the one that performed best on the validation data set. This model selection approach shows that there are no significant differences between the two regularization techniques. The associated error in the divergence of the velocity fields is reported in  \\cref{tab:comparison_CW_div}. \n\\begin{table}[H]\n    \\centering\n    \\begin{tabular}{|c|c|c|c|c|c|} \n        \\hline\n         \\multirow{2}{*}{Method} & \\multirow{2}{*}{Regularization} & \\multicolumn{4}{|c|}{Measurement Locations} \\\\  \\cline{3-6} \n          & & 5 & 4 & 3 & 2  \\\\  \\hline\n         \\multirow{2}{*}{SCVAE} & $\\lambda = 0$\n         & 0.30e-02 & 0.33e-02 & 0.26e-02 & 0.28e-02    \\\\ \\cline{2-6}\n         & $\\lambda > 0$  & 0.31e-02 & 0.32e-02 & 0.30e-02 & 0.28e-02 \\\\ \\cline{3-6} \\hline\n         \\multirow{2}{*}{GPOD} & $\\lambda = 0$  & 2.35e-02& 2.49e-02  \n           & 3.38e-02& 17.38e-02\\\\ \\cline{2-6}\n         & $\\lambda > 0$ &   2.12e-02 & 2.33e-02& 3.15e-02& 16.38e-02    \\\\ \\hline\n    \\end{tabular}\n    \\caption{The mean relative error $\\mathcal{E}$ (\\cref{L2_error}) for the  SCVAE prediction  and  the GPOD prediction with or without div-regularization, and different number of measurements.}\n    \\label{tab:comparison_CW}\n\\end{table}\n\\begin{table}[H]\n    \\centering\n    \\begin{tabular}{|c|c|c|c|c|c|} \n        \\hline\n         \\multirow{2}{*}{Method} & \\multirow{2}{*}{Regularization} & \\multicolumn{4}{|c|}{Measurement Locations} \\\\  \\cline{3-6} \n          & & 5 & 4 & 3 & 2  \\\\  \\hline\n         \n         \\multirow{2}{*}{SCVAE} & $\\lambda = 0$\n          & 0.1439 & 0.1580 & 0.1383 & 0.1432    \\\\ \\cline{2-6}\n         & $\\lambda > 0$ & 0.1533 & 0.1408 & 0.1468 & 0.1410    \\\\ \\cline{3-6} \\hline\n         \\multirow{2}{*}{GPOD} & $\\lambda = 0$ & 0.1052 & 0.1047 & 0.0943 & 0.08866    \\\\ \\cline{2-6}\n         & $\\lambda > 0$  & 0.1039 & 0.1051 & 0.0966 & 0.0669   \\\\ \\hline\n    \\end{tabular}\n    \\caption{Comparison of the divergence error $\\mathcal{E}_{div}$ as calculated in \\cref{divergence_error_1} for the different methods and regularization techniques. The true divergence error on the entire test data set is $0.1058$}\n    \\label{tab:comparison_CW_div}\n\\end{table}\n\\begin{figure}[H]\n    \\centering\n \t    \\includegraphics[width=0.75\\linewidth]{Epochs_per_Measurement.png}\n \t    \\captionof{figure}{Number of epochs trained depending on the number of measurements. For each measurement configuration and regularization technique the model is run $10$ times. The variation of number of epochs for for each measurement locations is due to different priors of the weights and random mini-batch sampling.}\\label{Epochs_per_measurement_CW}\n\\end{figure}\n\n\\subsection{Current data from Bergen ocean model}\\label{BOM_experiment}\nWe tested the SCVAE on simulations from the Bergen Ocean Model (BOM) \\cite{berntsen2000users}. BOM is a three-dimensional terrain-following nonhydrostatic ocean model with capabilities of resolving mesoscale to large-scale processes. Here we use velocities simulated by Ali. et. al \\cite{Ali:2016go}. The simulations where conducted on the entire North Sea with 800 meter horizontal and vertical grid resolution and 41 layers for the period from 1st to 15th of January 2012. Forcing of the model consist of wind, atmospheric pressure, harmonic tides, rivers, and initial fields for salinity and temperature. For details of the setup of the model, forcing and the simulations we refer to \\cite{Ali:2016go}. \\\\\n\nHere, the horizontal and vertical velocities of an excerpt of 25.6 $\\times$ 25.6 km$^2$ at the bottom layer centered at the Sleipner CO2 injection site ($58.36^\\circ N, \\, 1.91^\\circ E$) is used as data set for reconstruction. In \\cref{Data_set_time_series_BOM} we have plotted the mean and extreme values of $u$ and $v$ for each time $t$ in ${\\mathcal T}$. \n\\begin{figure}[H]\n\t\\centering\n\n\t\t\\includegraphics[width=0.99\\linewidth]{Data_CW_U.png}\n        \\includegraphics[width=0.99\\linewidth]{Data_CW_V.png}\n\t\t\\captionof{figure}{The light-blue line represent the maximum, the orange the minimum and the green mean value of $u$ and $v$ for each time $t$ in ${\\mathcal T}.$ The horizontal lines indicate the sequential data split. }\\label{Data_set_time_series_BOM}\n\n\\end{figure}\n\\subsubsection{Preprocessing}\nWe extract $32 \\times 32$ central grid from the bottom layer velocity data. Hence, $\\mathcal{P}$ contains $N = 1024$ points, $32$ points in the horizontal and $32$ in the vertical direction. The temporal resolution is originally $105000$ and the time between each time step is $1$ minute. We downsample the temporal dimension of the original data uniformly such that the number of time steps in $\\mathcal{T}$ is $K=8500.$ We train and validate the SCVAE with two different data splits: randomized and sequential in time. For the sequential split we have used the last $15 \\%$ for the test, the last $30\\%$ of the remaining data is used for validation, and the fist $70\\%$ for training. In \\cref{Data_set_time_series_BOM}, the red and blue vertical lines indicate the data split for this case. For the random split, the instances $\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}$ are drawn randomly from $\\ifmmode\\bm{X}\\else\\textbf{\\textit{X}}\\fi$ with the same percentage. The data was scaled as described in \\cref{Appendix_D}.  The input $\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}$ to the SCVAE was shaped as $(32 \\times 32 \\times 2)$ in order to apply convolutional layers. We use  $9,5$ and $3$ fixed spatial measurement locations. In particular, the subgrid ${\\mathcal Q}$ is given as \n\\begin{align}\\label{BOM_measurements}\n    \\centering\n    \\begin{split}\n    {\\mathcal Q}_9 = & \\{(6,6),(6,17),(6,27),(17,17),(17,27),(17,6),(27,6), (27,17),(27,27) \\}, \\\\\n    {\\mathcal Q}_5 = & \\{(6,6),(17,17),(27,27),(6,27),(27,6) \\}, \\\\\n    {\\mathcal Q}_3 = & \\{ (6,27),(17,17),(27,6) \\}. \n    \\end{split}\n\\end{align}\nAs before, the values of $u$ and $v$ at these specific locations constitute the measurements $\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)} \\in \\ifmmode\\bm{M}\\else\\textbf{\\textit{M}}\\fi$.\n\n\\subsubsection{Model} \nA schematic description of the model is given in \\cref{Appendix_A.3,Appendix_A.4}. \nThe first two layers of the encoder are convolutional layers with $64$ and $128$ filters with strides and kernel size of $2$ and ReLu activation functions. This compresses the data into a shape of $(8 \\times 8 \\times 128)$. The next layers are flattening and dense layers, where the latter have $16$ filters and ReLu activation. The subsequent layers defines the mean and log-variance of the latent representation $\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi$, which is input to a lambda layer for realization of the reparametrization trick. The encoder outputs the samples $\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi^{(i)}$ and the mean and the log-variance of $\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi$.\\\\\n\nInput to the decoder is the output $\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi^{(i)}$ of the encoder and the measurement $\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)}.$ To concatenate the inputs, $\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)}$ is flattened. After concatenation of $\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi^{(i)}$ and $\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)}$, the next layer is a dense layer with shape $(8 \\times 8 \\times 128)$ and ReLu activation. This allows for use of transposed convolutional layers to obtain the original shape of the data. Hence, the following layers are two transposed convolutional layers with $64$ and $128$ filters, strides and kernel size of $2$ and ReLu activation's. The final layer is a transposed convolutional with linear activation functions and filter size  of shape $(32 \\times 32 \\times 2),$ i.e., the same shape as $\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}$. \n\n\\subsubsection{Results}\\label{BOM_results}\nWe illustrate the obtained posterior predictive distribution in terms the predictive mean and standard deviation for the prediction at a specific time. The SCVAE is compared with the GPOD method, both with $\\lambda >0$ and $\\lambda = 0$ for measurement locations given in \\cref{BOM_measurements} for random and sequential split cases. To generate the posterior predictive distributions, \\cref{eq:p_pred},  we sample $200$ realizations from $\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi \\sim \\mathcal{N}(\\bm{0},\\bm{I})$ , which allows for calculation mean prediction and uncertainty\nestimates, see \\cref{eq:mean_and_cov}. \\cref{UV_prediction} shows the results of the prediction at time step $1185$ for both the $u$ and $v$ component and associated uncertainty estimates for a trained model with $\\lambda=0$ and $Q_3$ measurement locations (see \\cref{BOM_measurements}).  \n\n\\begin{figure}[H]\n    \\centering\n\t\t\\includegraphics[width=0.75\\linewidth]{Sleipner_prediction_std_abs_v1.png}\n\t\t\\captionof{figure}{Presentation of statistics for\n\t\tthe reconstruction of the $u$ and $v$ component of the velocity for sample $1185$ in the test data set based on the trained model with $\\lambda=0$ and $Q_3$ measurement locations. \\textbf{Left panels:} From top to bottom; True velocity in $u$, predicted mean velocity field of $u$, the standard deviation of the prediction for $u$ and the absolute error of $u.$ \\textbf{Lower panels:} Similar as describe for the upper panels, but for $v$ }\\label{UV_prediction}\n\\end{figure} \nIn \\cref{BOM_time_series} we plot the true solution and the predicted mean velocity \\cref{eq:mean_and_cov} with the associated uncertainty, see \\cref{eq:ConfRegion:2}, for two grid points. We plot only the first $600$ time steps for readability. The first grid point is $(26,6)$ and $(4,1).$ One location is approximately $5.1$ km from the nearest observation, and another one is about $16.1$ km away.\n\\begin{figure}[H]\n        \\centering\n\t\t\\includegraphics[width=0.85\\linewidth]{BOM_plots_time_series_CR.png}\n\t\t\\captionof{figure}{Velocities $u$ and $v$ at specific locations based on the trained model with $\\lambda=0$ and $Q_3$ measurement locations. The red line corresponds to the true values, blue to the SCVAE mean prediction, and orange to the GPOD reconstruction. Light blue shaded area represents the confidence region obtained in \\cref{eq:ConfRegion:2} with $p=0.95.$.  \\textbf{Upper panels:} The time series and confidence region for $u$ (left) and $v$ (right) at  grid point $(26,6)$, approximately $5.1$ km from nearest observation. \\textbf{Lower panels:} The time series and confidence region for $u$ (left) and $v$ (right) at grid point $(4,1)$ approximately $16.1$ km from nearest observation.}\\label{BOM_time_series}\n\\end{figure} \n\\cref{BOM_error} has to be viewed in context with \\cref{BOM_time_series} and show the difference between the true and the predicted solutions with associated difference marginal in time for the two locations as in \\cref{BOM_time_series}. \n\\begin{figure}[H]\n\t\\centering\n\t\\includegraphics[width=0.85\\linewidth]{BOM_error_plot_CR.png}\n\t\\captionof{figure}{The difference between the true and predicted estimate for the SCVAE (blue) and for the GPOD (orange) based on the $\\lambda=0$ model and $Q_3$ measurement locations. The light blue shaded region represents the difference marginals, obtained from the confidence region in \\cref{BOM_time_series}. \\textbf{Upper panels:} The difference between the true and predicted estimate at grid point $(26,6)$ for $u$ (left) and $v$ (right), \\textbf{Lower panels:} The difference between the true and predicted estimate at point $(4,1)$ for $u$ (left) and $v$ (right).}\\label{BOM_error}\n\\end{figure}\nIntegrating over the latent space generates a posterior distribution of the reconstruction, as described in \\cref{sec:posterior}. It is also possible to use the latent space to generate new statistically sound versions of $u$ and $v$. This is presented in \\cref{z_space_sampling_v} where it is sampled uniformly over the 2 dimensional latent space $\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi \\sim \\mathcal{N}(\\bm{0},\\bm{I})$ and the result shows how different variations can be created with the SCVAE model, given only the sparse measurements.    \n\\begin{figure}[H]\n    \\centering\n\t\t\\includegraphics[width=0.70\\linewidth]{Sleipner_generating_data_v1.png}\n\t\t\\captionof{figure}{The left panels shows 9 different generated velocity-field-snapshots for the $\\ifmmode\\bm{u}\\else\\textbf{\\textit{u}}\\fi$ and $\\ifmmode\\bm{v}\\else\\textbf{\\textit{v}}\\fi$ component for test sample number $1185$. The predictions are generated from the model with $\\lambda=0$ and $Q_3$ measurement locations. We sample uniformly over the latent space and predicts with the decoder, given the measurements.}\\label{z_space_sampling_v}\n\\end{figure}\nThese sampled velocities could be used for ensemble simulations when estimating uncertainty in a passive tracer transport, see e.g., \\cite{oleynik2020optimal}. \n\nThe SCVAE results are are compared with results of the GPOD method, see \\cref{tab:comparison_BOM} and \\cref{tab:comparison_BOM_div}. The tables show the errors as calculated in \\cref{L2_error} and \\cref{divergence_error_1} of the test data set for both sequential and random split.\nFor the sequential splitting, the SCVAE is better for $3$ measurement locations, while the GPOD method performs better for $9$ and $5$ locations.  From  \\cref{Data_set_time_series_BOM}, we observe that test data set seems to arise from a different process than the train and validation data (especially for $v$). Thus, the SCVAE generalize worse than a simpler model such as the GPOD, \\cite{model_selection}. For the $3$ location case, the number of components in the GPOD is not enough to compete with the SCVAE. \\\\\n\nWith random split on the train, test and validation data, we see that the SCVAE is significantly better than the GPOD. The training data and measurements represent the test data and test measurements better with random splitting. This highlights the importance of large data sets that cover as many outcomes as possible. Demanding that $\\lambda > 0$ in \\cref{obj_function_SCVAE} do not improve the result. The SCVAE-models with $\\lambda = 0$ learns that the reconstructed representations should have low divergence without explicitly demanding it during optimization. However, as discussed in the 2D flow around cylinder experiment, demanding $\\lambda>0$ seems to improve the conditioning of the optimization problem and give more consistent results.  \n\\begin{table}[ht]\n    \\centering\n    \\begin{tabular}{|c|c|c|c|c|c|} \n        \\hline\n         \\multirow{2}{*}{Split} & \\multirow{2}{*}{Regularization} & \\multirow{2}{*}{Method} & \\multicolumn{3}{|c|}{Measurement Locations} \\\\  \\cline{4-6} \n          & & & 9 & 5 & 3 \\\\  \\hline\n         \n         \\multirow{4}{*}{Random} & \\multirow{2}{*}{$\\lambda=0$}\n         & SCVAE & 0.1379 & 0.2097 & 0.2928   \\\\ \\cline{3-6}\n         & & GPOD & 0.3300 & 0.3822 & 0.4349  \\\\ \\cline{2-6}\n         & \\multirow{2}{*}{$\\lambda>0$}\n         & SCVAE & 0.1403 & 0.2025 & 0.3016   \\\\ \\cline{3-6}\n         & & GPOD & 0.2971 & 0.3579 & 0.4039  \\\\ \\cline{3-6} \\hline\n         \n         \\multirow{4}{*}{\\makecell{Time \\\\ Dependent}} & \\multirow{2}{*}{$\\lambda=0$}\n         & SCVAE & 0.3493 & 0.3913 & 0.4155  \\\\ \\cline{3-6}\n         & & GPOD & 0.3767 & 0.4031 & 0.4678  \\\\ \\cline{2-6}\n         & \\multirow{2}{*}{$\\lambda>0$}\n         & SCVAE & 0.3527 & 0.3889 & 0.4141  \\\\ \\cline{3-6}\n         & & GPOD &0.3362 & 0.3695 & 0.4462  \\\\ \\hline\n    \\end{tabular}\n    \\caption{Errors as calculated in \\cref{L2_error} for the different methods, regularization techniques ($\\lambda=0$ or $\\lambda > 0$), split regimes and measurements}\n    \\label{tab:comparison_BOM}\n\\end{table}\n\\begin{table}[ht]\n    \\centering\n    \\begin{tabular}{|c|c|c|c|c|c|} \n        \\hline\n         \\multirow{2}{*}{Split} & \\multirow{2}{*}{Regularization} & \\multirow{2}{*}{Method} & \\multicolumn{3}{|c|}{Measurement Locations} \\\\  \\cline{4-6} \n          & & & 9 & 5 & 3  \\\\  \\hline\n         \n         \\multirow{4}{*}{Random} & \\multirow{2}{*}{$\\lambda=0$}\n         & SCVAE & 3.75e-05 & 3.62e-05 & 3.42e-05   \\\\ \\cline{3-6}\n         & & GPOD & 6.51e-05 & 5.88e-05 & 5.02e-05   \\\\ \\cline{2-6}\n         & \\multirow{2}{*}{$\\lambda>0$}\n         & SCVAE &  3.60e-05 & 3.60e-05 & 3.13e-05  \\\\ \\cline{3-6}\n         & & GPOD & 6.23e-05 & 4.77e-05 & 4.14e-05  \\\\ \\cline{3-6} \\hline\n         \n         \\multirow{4}{*}{\\makecell{Time \\\\ Dependent}} & \\multirow{2}{*}{$\\lambda=0$}\n         & SCVAE & 2.02e-05 & 1.80e-05 & 1.69e-05   \\\\ \\cline{3-6}\n         & & GPOD & 5.09e-05 & 4.03e-05 & 4.15e-05  \\\\ \\cline{2-6}\n         & \\multirow{2}{*}{$\\lambda>0$}\n         & SCVAE & 2.05e-05 & 1.99e-05 & 1.85e-05  \\\\ \\cline{3-6}\n         & & GPOD & 4.39e-05 & 3.65e-05 & 2.92e-05  \\\\ \\hline\n    \\end{tabular}\n    \\caption{Divergence errors as calculated in \\cref{divergence_error_1} for the different methods, regularization techniques ($\\lambda=0$ or $\\lambda > 0$), split regimes and measurements. The true divergence of the test data is of order  $10^{-4}.$ }\n    \\label{tab:comparison_BOM_div}\n\\end{table}\n\\begin{figure}[H]\n    \\centering\n \t    \\includegraphics[width=0.75\\linewidth]{BOM_epochs_random_split.png}\n \t    \\captionof{figure}{The figure shows number of epochs and number of measurement locations. For each measurement configuration and regularization technique the model is optimized 10 times. The variation in the number of epochs for each measurement and regularization technique is due to different priors of the weights and mini-batch sampling.}\\label{Epochs_per_measurement}\n\\end{figure}\n\n\\section{Discussion}\\label{discussion}\nWe have presented the SCVAE method for efficient data reconstruction based on sparse observations. The derived objective functions for the network optimization  show that the encoding is independent of measurements. This allows for a simpler model structure with fewer model parameters than a CVAE and results in an optimization procedure that requires less computations.  \\\\\n\nWe have shown that the SCVAE is suitable for reconstruction of fluid flow. The method is showcased on two different data sets, velocity data from simulations of 2D flow around a cylinder, and bottom currents from the BOM. The fact that the fluids studied in the experiments are incompressible served as a motivation for adding an extra term to the objective function, see \\cref{obj_function_SCVAE} with $\\lambda>0$. \\\\\n\nOur investigation of additional regularization showed that the mean reconstruction error over all models was lower with $\\lambda>0$ compared to the model where $\\lambda=0$, but the best reconstruction error was similar for $\\lambda=0$ and $\\lambda>0$. In \\cref{Experiment} we optimized 10 models for every experiment, and chose the model that performed best on the validation data sets. With this approach we did not observe significant differences between optimizing with  $\\lambda=0$ and $\\lambda>0$. However, the reconstruction became less sensitive to the stochasticity involved in optimization (minibatch selection, network weights priors) when the regularization was used, see \\cref{A_Motivating_Example}. \\\\ \n\nThe SCVAE is a probabilistic model, which allows to make predictions, estimate their uncertainty, see \\cref{sec:posterior},  and draw multiple samples from the predictive distribution, see \\cref{z_space_sampling_v}. The last two properties make the SCVAE a useful method especially when the predictions are used in another application, i.e., ensemble simulation of tracer transport. Motivated by \\cite{YILDIRIM2009160}, we compared the SCVAE predictions with the predictions of a modified GPOD method, see \\cref{Appendix_C}. \\\\\n\nUnlike the GPOD-method, a benefit with the SCVAE-method is that it scales well to larger data sets. Another aspect and as the experiments in \\cref{Experiment} suggest, the GPOD seems more sensitive to the number of measurement locations than the SCVAE. On the other hand, the experiments suggested that GPOD is better than SCVAE with a larger number of measurement locations if the training data and the test data are too different, see BOM experiment with sequential splitting \\cref{BOM_results}. Essentially the SCVAE overfit to the training data, and as a result performing poorly on the test data set. This fact shows the importance of training the the SCVAE on large data sets, which covers as many potential flow patterns as possible. Further, the results show that the GPOD is more sensitive to the measurement location choice than the SCVAE, see \\cref{A_Motivating_Example}, and the GPOD-method is not expected to preform well on a complex flow with very few fixed measurement locations. \\\\\n\nVAEs has been used for generating data in e.g. computer vision \\cite{kingma2013auto}, and auto-encoders is a natural to use in reconstruction tasks \\cite{ELMS2018}. Many reconstruction approaches, including the GPOD approach, first create a basis, then use the basis and minimize the error of the observations \\cite{willcox2006unsteady, bui2004aerodynamic}. This makes the GPOD suitable for fast optimization of measurement locations that minimize the reconstruction error. On the other hand,  the SCVAE optimizes the basis function given the measurements, i.e. they are known and fixed. This makes it challenging to use the framework for optimizing sensor layout. But if the measurement locations are fixed and large amounts of training data are available, the SCVAE outperforms the GPOD for reconstruction. SCVAE optimize the latent representation and the neural network model parameters, variational and generative parameters, given the measurements. This ensures that the reconstruction is adapted to the specific configuration of measurements. \\\\\n\nA limitation of our experiments is that we used only $100$ and $200$ samples and constructed the confidence region under further simplifying assumptions. The uncertainty estimate could be improved by increasing the sample size and better model for the confidence region. \\\\\n\nNatural applications for the SCVAE are related to environmental data, where we often have sparse measurements. It is for example possible to optimize sensor layout to best possible detect unintentional discharges in the marine environment by using a simple transport model \\cite{oleynik2020optimal}. Oleynik. et al. used deterministic flow fields to transport the contaminant and thus obtain a footprint of the leakage. SCVAE can be used to improve that method and efficiently generate probabilistic footprints of a discharges. This may be important as input to design, environmental risk assessments, and emergency preparedness plans. \n\nWe have highlighted the SCVAE through the reconstruction of currents and flow field reconstruction, however, the SCVAE method is not limited to fluid flow problems. For instance, the same principles could be used in computer vision to generate new picture based on sparse pixel representations or in time series reconstruction. \\\\\n\nA natural extension of the SCVAE is to set it up as a partially hidden Markov model. That is to predict the current state $p_\\theta(\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi_t|\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi_t, \\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi_{t-1}),$ given the measurements and the reconstruction from the previous time step. This could potentially improve the reconstruction further.\n\n\\section*{Acknowledgements}\nThis work is part of the project ACTOM, funded through the ACT programme (Accelerating CCS Technologies, Horizon2020 Project No 294766). Financial contributions made from; The Research Council of Norway, (RCN), Norway, Netherlands Enterprise Agency (RVO), Netherlands, Department for Business, Energy \\& Industrial Strategy (BEIS) together with extra funding from NERC and EPSRC research councils, United Kingdom, US-Department of Energy (US-DOE), USA. Kristian Gundersen has been supported by the Research Council of Norway, through the CLIMIT program (project 254711, BayMode) and the European Union Horizon 2020 research and innovation program under grant agreement 654462, STEMM-CCS. The authors would like to acknowledge NVIDIA Corporation for providing their GPUs in the academic GPU Grant Program.\n\n\\clearpage\n\n\\bibliographystyle{unsrt}\n\\clearpage\n\n\\section{Introduction}\nReconstruction of non-linear dynamic processes based on sparse observations is an important and difficult problem. The problem traditionally requires knowledge of the governing equations or processes to be able to generalize from the the sparse observations to a wider area around, in-between and beyond the measurements. Alternatively it is possible to learn the underlying processes or equations based on data itself, so called data driven methods. In geophysics and environmental monitoring measurements is often only available at sparse locations. For instance, within the field of meteorology, atmospheric pressures, temperatures and wind are only measured at limited number of stations. To produce accurate and general weather predictions, requires methods that both forecast in the future, but also reconstruct where no data is available. Within oceanography one faces the same problem, that in-situ information about the ocean dynamics is only available at sparse locations such as buoys or sub-sea sensors. \\\\\n\nBoth the weather and ocean currents can be approximated with models that are governed by physical laws, e.g. the Navier-Stokes Equation. However, to get accurate reliable reconstructions and forecasts it is of crucial importance to incorporate observations. \\\\ \n\nReconstruction and inference based on sparse observations is important in many applications both in engineering and physical science \\cite{brunton2015closed, kong2018application, bolton2019applications, venturi2004gappy,callaham2018robust, manohar2018data}. Bolton et. al. \\cite{bolton2019applications}  used convolutional neural networks to hindcast ocean models, and in \\cite{yeo2019data} K. Yeo reconstructs time series of nonlinear dynamics from sparse observation.  Oikonomo et. al. \\cite{oikonomou2018novel} proposed a method for filling data gaps in groundwater level observations and Kong. et. al \\cite{kong2018application} used reconstruction techniques to modeling the characteristics of cartridge valves.  \\\\\n\nThe above mentioned applications are just some of the many examples of reconstruction of a dynamic process based on limited information. Here we focus on reconstruction of flow. This problem can be formulated as follows. Let $\\bm w \\in {\\mathbb R}^d,$ $d \\in {\\mathbb N},$ represent a state of the flow, for example velocity, pressure, temperature, etc. Here, we will focus on incompressible unsteady flows and ${\\bm w}=(u,v)\\in {\\mathbb R}^2$ where $u$ and $v$ are the horizontal and vertical velocities, respectively. The velocities ${\\bm w}$ are typically obtained from computational fluid dynamic simulations on a meshed spatial domain $\\mathcal{P}$ at discrete times $\\mathcal{T} = \\{t_1,...,t_K\\} \\subset {\\mathbb R}$. \\\\\n\nLet $ \\mathcal{P}=\\{p_1,...,p_N\\}$ consist of $N$ grid points $p_n,$ $n=1,...,N.$  \nThen the state of the flow $\\bm w$ evaluated on ${\\mathcal P}$ at a time $t_i \\in {\\mathcal T}$ can be represented as a vector \n$\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)} \\in {\\mathbb R}^{2N},$ \n\\begin{equation}\n\\label{eq:x_i}\n\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}=(u(p_1,t_i),...,u(p_N,t_i), v(p_1,t_i),...,v(p_N,t_i))^T.\n\\end{equation}\nThe collection of $\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)},$ $i=1,\\dots,K,$ constitutes the data set $\\ifmmode\\bm{X}\\else\\textbf{\\textit{X}}\\fi.$ In order to account for incompressibility, we introduce a discrete divergence operator $L_{div}$, which is given by a $N \\times 2N$ matrix associated with a finite difference scheme, and\n\\begin{equation}\n\\label{eq:L_div}\n(L_{div} \\, \\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi)_{k} \\approx (\\nabla \\cdot w)(p_k)=0.\n\\end{equation}\n\nFurther, we assume that the state can be measured only at specific points in ${\\mathcal P},$ that is, at ${\\mathcal Q}=\\{q_1,...,q_M\\} \\subset {\\mathcal P}$ where $M$ is typically much less than $N.$  Hence, there is $\\ifmmode\\bm{M}\\else\\textbf{\\textit{M}}\\fi=\\{\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)} \\in {\\mathbb R}^{2M}: \\, \\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)}=C\\,\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}, \\, \\forall \\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)} \\in \\ifmmode\\bm{X}\\else\\textbf{\\textit{X}}\\fi \\},$ where ${\\bm C} \\in {\\mathbb R}^{2M \\times 2N}$ is a sampling matrix. More specifically, $\\bm C$ is a two block matrix \n\n$$\n{\\bm C}=\\begin{pmatrix}\n{\\bm C}_{1\/2}& O\\\\\nO&{\\bm C}_{1\/2}\\\\\n\\end{pmatrix}, \\quad \n({\\bm C}_{1\/2})_{ij}= \\left\\{\n\\begin{array}{ll}\n1,& \\mbox{if } \\, q_i =p_j\\\\\n0,& \\mbox{otherwise}\n\\end{array} \\right. , \\quad i=1,...,N \\quad j=1,...,M,\n$$\nand ${\\bm O}\\in {\\mathbb R}^{M\\times N}$ is a zero matrix. The problem of reconstructing fluid flow $\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)} \\in \\ifmmode\\bm{X}\\else\\textbf{\\textit{X}}\\fi$ from $\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)}\\in \\ifmmode\\bm{M}\\else\\textbf{\\textit{M}}\\fi$ is presented as a schematic plot in \\cref{Sketch_map}.\n\\begin{figure}[h]\n\t\\centering\n\n\t\t\\includegraphics[width=0.70\\linewidth]{Subset_M_of_D_v2_new_notation_i.png}\n\t\t\\captionof{figure}{Sketch of reconstruction of $\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}$ from $\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)}$. The dots on the right side represent the grid ${\\mathcal P}$, and those on  the left side represent the measurement locations ${\\mathcal Q}.$}\\label{Sketch_map}\n\n\\end{figure}\nThere have been a wide range of methods for solving the problem, e.g. \\cite{sirovich1987turbulence, everson1995karhunen, donoho2006compressed, schmid2010dynamic, ELMS2018, raissi2019physics}. In particular, use of proper orthogonal decomposition (POD) \\cite{sirovich1987turbulence} techniques has been popular. \\\\ \n\nPOD \\cite{sirovich1987turbulence} is a traditional dimensional reduction technique where based on a data set, a number of basis functions are constructed. The key idea is that a linear combination of the basis functions can reconstruct the original data within some error margin, efficiently reducing the dimension of the problem. In a modified version of the POD, the Gappy POD (GPOD) \\citep{everson1995karhunen}, the aim is to fill the gap in-between sparse measurements. Given a POD basis one can minimize the $L_2$-error of the measurements and find a linear combination of the POD-basis that complements between the measurements. If the basis is not know, a iterative scheme can be formulated to optimize the basis based on the measurements. The original application of GPOD \\citep{everson1995karhunen} was related to reconstruction of human faces, and it has later been applied to fluid flow reconstruction \\cite{venturi2004gappy}. We will use the GPOD approach for comparison later in this study. \\\\ \n\nA similar approach is the technique of Compressed Sensing (CS) \\cite{donoho2006compressed}. As for the GPOD method, we want to solve a  linear system. However, in the CS-case this will be a under-determined linear system. That is we need some additional information about the system to be able to solve it, typically this can be a condition\/constraint related to the smoothness of the solution. The core difference between CS and GPOD is however the sparsity constraint. That is, instead of minimizing the L2-norm, we minimize the L1-norm. Minimizing the L1-norm favours sparse solutions, i.e. solutions with a small number of nonzero coefficients. \\\\ \n\nAnother reconstruction approach is Dynamical Mode Decomposition (DMD) \\cite{schmid2010dynamic}. Instead of using principal components in the spatial domain, DMD seek to find modes or representations that are associated with a specific frequency in the data, i.e. modes in the temporal domain. Again, the goal is to find a solution to an undetermined linear system and reconstruct based on the measurements, by minimizing the error of the observed values. \\\\\n\nDuring the last decade, data driven methods have become tremendously popular, partly because of the growth and availability of data, but also driven by new technology and improved hardware. To model a non-linear relationships with linear approximations is one of the fundamental limitation of the DMD, CS and GPOD. Recently we have seen development in methods where the artificial neural networks is informed with a physical law, the so called physic-informed neural networks (PINN) \\cite{raissi2019physics}. In PINNs the reconstruction is informed by a Partial Differential Equation (PDE) (e.g. Navier Stokes), thus the neural network can learn to fill the gap between measurements that are in compliance with the equation. This is what Rassi et. al. \\cite{raissi2018hidden} have shown for the benchmark examples such as flow around a 2D and 3D cylinder. Although PINNs are showing promising results, we have yet to see applications to complex systems such as atmospheric or oceanographic systems, where other aspect have to be accounted for, e.g. in large scale oceanic circulation models that are driven by forcing such as tides, bathymetry and river-influx. That being said, these problems may be resolved through PINNs in the future. Despite the promise of PINNs, they will not be a part of this study, as our approach is without any constraint related to the physical properties of the data. \\\\\n\nAnother non-linear data driven approaches for reconstruction of fluid flow are different variations of auto-encoders \\cite{ELMS2018, grover2019uncertainty}. An auto-encoder \\cite{Rumelhart86_autoencoder} is a special configuration of an artificial neural network that first encodes the data by gradually decreasing the size of the hidden layers. With this process, the data is represented in a lower dimensional space. A second neural network then takes the output of the encoder as input, and decodes the representation back to its original shape. These two neural networks together constitute an auto-encoder. Principal Component Analysis (PCA) \\cite{pearson1901_PCA} also represent the data in a different and more compact space. However, PCA reduce the dimension of the data by finding orthogonal basis functions or principal components through singular value decomposition. In fact, it has been showed with linear activation function, PCA and auto-encoders produces the same basis function \\cite{bourlard1988auto}. Probabilistic version of the auto-encoder are called Variational Auto-Encoders (VAEs) \\cite{kingma2013auto}. CVAEs \\cite{sohn2015learning} are conditional probabilistic auto-encoders, that is, the model is dependent on some additional information such that it is possible to create representations that are depend on this information. \\\\\n\nHere, we address the mentioned problem from a probabilistic point of view. Let $\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi: {\\mathcal P} \\to {\\mathbb R}^{2N}$ and $\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi: {\\mathcal Q} \\to {\\mathbb R}^{2M}$ be two multivariate random variables associated with the flow on ${\\mathcal P}$ and on ${\\mathcal Q}$, respectively. Then the data sets $\\ifmmode\\bm{X}\\else\\textbf{\\textit{X}}\\fi$ and $\\ifmmode\\bm{M}\\else\\textbf{\\textit{M}}\\fi$ consist of the realizations of $\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi$ and $\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi$, respectively. Using $\\ifmmode\\bm{X}\\else\\textbf{\\textit{X}}\\fi$ and $\\ifmmode\\bm{M}\\else\\textbf{\\textit{M}}\\fi,$ we intend to approximate the probability distribution $p(\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi|\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi).$ This would not only allow to predict $\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}$ given $\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)},$ but also to estimate an associated uncertainty. In this paper, we use a variational auto-encoder to approximate $p(\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi| \\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi)$. The method we use is a Bayesian Neural Network \\cite{MacKay92} approximated through variational inference \\cite{hoffman2013stochastic,blei2017variational}, that we have called \\textit{Semi-Conditional Variational Auto-encoder}, SCVAE. A detailed description of the SCVAE method for reconstruction and associated uncertainty quantification is given in \\cref{SCVAE_section}. \\\\\n\nHere we focus on fluid flow, being the main driving mechanism behind transport and dilution of tracers in marine waters. The world's oceans are  under tremendous stress \\citep{Halpern:2012hs}, UN has declared 2021-2030 as the ocean decade\\footnote{\\url{https:\/\/en.unesco.org\/ocean-decade}}, and an ecosystem based Marine Spatial Planning initiative has been launched by IOC \\citep{DominguezTejo:2016dt}. \\\\\n\nLocal and regional current conditions determines transport of tracers in the ocean \\cite{drange2001ocean,BARSTOW1983211}. Examples are accidental release of radioactive, biological or chemical substances from industrial complexes, e.g. organic waste from fish farms in Norwegian fjords \\citep{Ali:2011hd}, plastic \\cite{Law:2017}, or other contaminants that might have adverse effects on marine ecosystems \\citep{Hylland:2015gt}. \\\\ \n\nTo be able to predict the environmental impact of a release, i.e. concentrations as function of distance and direction from the source, requires reliable current conditions \\citep{Ali:2016go,Blackford:2020}. Subsequently, these transport predictions support design of marine environmental monitoring programs \\citep{Hvidevold:2015,Hvidevold:2016cx,Alendal:2017b, oleynik2020optimal}. The aim here is to model current conditions in a probabilistic manner using SCVAEs. This allows for predicting footprints in a Monte Carlo framework, providing simulated data for training networks used for, e.g., analysing environmental time series \\cite{gundersen2020binary}.\\\\ \n\nIn this study we will compare results  with the GPOD method \\cite{willcox2006unsteady}. We are aware that there recent methods (e.g. PINNS and traditional Auto-encoder) that may perform better on the specific data sets than the GPOD, however, the GPODs simplicity, versatility and not least its popularity \\cite{jo2019effective, mifsud2019fusing, callaham2019robust}, makes it a great method for comparison. \\\\\n\nThe reminder of this manuscript is outlined in the following: \\cref{A_Motivating_Example} presents a motivating example for the SCVAE-method in comparison with the GPOD-method. In \\cref{methods} we review both the VAE and CVAE method and present the SCVAE. Results of experiments on two different data sets are presented in \\cref{Experiment}. \\cref{discussion} summarize and discuss the method, experiments, drawbacks and benefits and potential extensions and further work.  \n\n\\section{A Motivating Example}\\label{A_Motivating_Example}\nHere we illustrate the performance of the proposed method vs the GPOD method in order to give a motivation for this study. We use simulations of a two dimensional viscous flow around a cylinder at the Raynolds number of $160,$ obtained from \\url{https:\/\/www.csc.kth.se\/~weinkauf\/notes\/cylinder2d.html}. The simulations were performed  by Weinkauf et. al. \\cite{weinkauf2010streak} with the Gerris Flow Solver software \\cite{gerrisflowsolver}. The data set consists of a horizontal $u$ and a vertical $v$ velocities on an uniform $400 \\times 50 \\times 1001$ grid of $[-0.5, 7.5] \\times [-0.5, 0.5] \\times [15, 23]$ spatial-temporal domain.\\footnote{The simulations are run from $t=0$ to $t=23$, but velocities are only extracted from $t=15$ to $t=23$} In particular, we have $400$ points in the horizontal, and $50$ points in the vertical  direction, and $1001$ points in time.  \n\\begin{figure}[h]\n\t\\centering\n\n\t\t\\includegraphics[width=0.99\\linewidth]{CW_original_data_u.png}\n    \t\\includegraphics[width=0.99\\linewidth]{CW_original_data_v.png}\n\t\t\\captionof{figure}{Typical data instance from the original 2D flow around a cylinder data set with $u$ and $v$ component presented at the upper and lower panel, respectively} \\label{Cw_data_motivating}\n\n\\end{figure}\nThe cylinder has the diameter of $0.125$ and is centered at the origin, see \\cref{Cw_data_motivating}. The left vertical boundary (inlet) has Dirichlet boundary condition $u=1$ and $v=0$. The homogeneous Neumann boundary condition is given at the right boundary (outlet), and the homogeneous Dirichlet conditions on the remaining boundaries. At the start of simulations, $t=0$, both velocities were equal to zero.  We plot the velocities at the time $t \\approx 19$ (time step $500$) in \\cref{Cw_data_motivating}. \\\\\n\nFor simplicity, in the experiment below we extract the data downstream from the cylinder, that is, from grid point $40$ to $200$ in the horizontal direction, and keep all grid points in vertical direction. Hence, $\\mathcal{P}$ contains $N = 8000$ points, $160$ points in the horizontal and $50$ in the vertical direction. The temporal resolution is kept as before, that is, the number of time steps in $\\mathcal{T}$ is $K=1001$. For validation purposes, the data set was split into a train, validation and test data set. The train and validation data sets were used for optimization of the model parameters. For both the SCVAE and the GPOD, the goal was to minimize the $L2$ error between the true and the modeled flow state.  \nThe restriction of the GPOD is that the number of components $r$ could be at most $2M.$ \nTo deal with this problem, and to account for the flow incompressibility, we added the  regularization term  $\\lambda \\|L_{div} x^{(i)}\\|,$ $\\lambda>0$, to the objective function, see \\cref{Appendix_C}. For the GPOD method, the parameters $r$ and\/or $\\lambda$ where optimized on the validation data set in order to have the smallest mean error.\nWe give more details about objective functions for the SCVAE in \\cref{SCVAE_section}. For now we mention that there are two versions, where one version uses an additional divergence regularization term similar to GPOD.\\\\\n\nIn \\cref{Error_boxplot_CW} we plot the mean  of the relative $L_2$ error calculated on the test data for both methods with and without the div-regularization. The results are presented for $3,$ $4,$ and $5$ measurement locations, that is, $M=3,4,5.$ For each of these three cases, we selected $20$ different configurations of $M.$. In particular, we created $20$ subgrids ${\\mathcal Q}$, each containing $5$ randomly sampled spatial grid points. Next we removed one and then two points from each of the $20$ subgrids ${\\mathcal Q},$ to create new subgrids of $4$ and $3$ measurements, respectively.\n\\begin{figure}[h!]\n    \\centering\n\t\\includegraphics[width=0.75\\linewidth]{Motivating_example_boxplot_3M_v1.png}\n\t\t\\captionof{figure}{The mean relative error for two reconstruction methods. \n\t\tThe orange and blue label correspond to the SCVAE with (div-on) and without (div-off) additional divergence regularization. The green and red labels correspond to the GPOD method. }\\label{Error_boxplot_CW}\n\\end{figure}\nAs it can be seen in \\cref{Error_boxplot_CW}, both methods perform well for the $5$ measurements case. The resulting relative errors have comparable mean and variance. When reducing the number of observations, the SCVAE method maintains low errors, while the GPOD error increases. The SCVAE seems to benefit from the additional regularization of minimizing the divergence, in terms of lower error and less variation in the error estimates. The effect is more profound with fewer measurements. \\\\\n\nThe key benefit of the SCVAE is that its predictions are optimal for the given measurement locations.  In a contrast, the POD based approaches, and in particular the GPOD, create a set of basis functions (principal components) based on the training data independently of the measurements. While this has an obvious computational advantage, the number of principle components for complex flows can be high and, as a result, many more measurements are needed, \\cite{willcox2006unsteady,manohar2018data,Proctor2014}. There are number of algorithms that aim to optimize to measurement locations to achieve the best performance of the POD based methods, see e.g., \\cite{jo2019effective,willcox2006unsteady,YILDIRIM2009160}. In practice, however, the locations are often fixed and another approaches are needed. The results in \\cref{Error_boxplot_CW} suggest that the SCVAE could be one of these approaches.\n\n\\section{Methods}\\label{methods}\nBefore we introduce the model used for reconstruction of flows, we give a brief introduction to VAEs and CVAEs. For a detailed introduction, see \\cite{vae_intro}. VAEs are neural network models that has been used for learning structured representations in a  wide variety of applications, e.g., image generation \\cite{gregor2015draw}, interpolation between sentences \\cite{bowman2015generating} and compressed sensing \\cite{grover2019uncertainty}. \n\n\\subsection{Preliminary}\nLet us assume that the data $\\ifmmode\\bm{X}\\else\\textbf{\\textit{X}}\\fi$ is generated by a random process that involves an unobserved continuous random variable $\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi.$ The process consists of two steps: (i) a value $\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi^{(i)}$ is sampled from a prior $p_{\\theta^*}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi);$ and (ii)  $\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}$ is generated from a conditional distribution  $p_{\\theta^*}(\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi|\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi).$ In the case of flow reconstruction, $\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi$ could be thought of as unknown boundary or initial conditions, tidal and wind forcing, etc. However, generally $\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi$ is just a convenient construct to represent $\\ifmmode\\bm{X}\\else\\textbf{\\textit{X}}\\fi,$ rather than a physically explained phenomena. Therefore it is for convenience assumed that $p_{\\theta^*}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi)$ and $p_{\\theta^*}(\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi|\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi)$ come from parametric families of distributions $p_{\\theta}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi)$ and $p_{\\theta}(\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi|\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi),$ and their density functions are differentiable almost everywhere w.r.t. both $\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi$ and $\\theta$. A probabilistic auto-encoder is neural network that is trained to represent its input $\\ifmmode\\bm{X}\\else\\textbf{\\textit{X}}\\fi$ as $p_\\theta(\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi)$ via \\textit{latent representation} $\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi \\sim p_{\\theta}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi),$ that is,\n\n\\begin{equation} \\label{eq:p_theta(x)}\np_{\\theta}(\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi) = \\int p_{\\theta}(\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi,\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi) d\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi = \\int p_{\\theta}(\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi|\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi)p_{\\theta}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi)d\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi.\n\\end{equation}\nAs $p_\\theta(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi)$ is unknown and observations $\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi^{(i)}$ are not accessible, we must use $\\ifmmode\\bm{X}\\else\\textbf{\\textit{X}}\\fi$ in order to generate $\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi \\sim p_\\theta(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi).$ That is, the network can be viewed as consisting of two parts: an \\textit{encoder} $p_\\theta(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi)$ and a \\textit{decoder} $p_\\theta(\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi|\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi).$ Typically the true posterior distribution $p_{\\theta}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi)$ is intractable but could be approximated with variational inference \\cite{hoffman2013stochastic,blei2017variational}.  That is, we define a so called recognition model $q_{\\phi}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi)$ with variational parameters $\\phi$, which aims to approximate $p_{\\theta}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi).$ The recognition model is often parameterized as a Gaussian. Thus, the problem of estimating $p_{\\theta}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi)$, is reduced to finding the best possible estimate for $\\phi$, effectively turning the problem into an optimization problem. \\\\\n\nAn auto-encoder that uses a recognition model is called Variational Auto-Encoder (VAE). In order to get good prediction we need to estimate the parameters $\\phi$ and $\\theta.$ The marginal likelihood  is equal to the sum over the marginal likelihoods of the individual samples, that is, $\\sum_{i=1}^K \\log p_{\\theta}(\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}).$ Therefore, we further on present estimates for an individual sample. The Kullback -Leibler divergence between two probability distributions  $q_{\\phi}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)})$ and $p_{\\theta}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)})$, defined as $$D_{KL}[q_{\\phi}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)})||p_{\\theta}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)})] = \\int q_{\\phi}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}) \\log\\left(\\frac{q_{\\phi}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)})}{p_{\\theta}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)})}\\right) d\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi,$$ can be interpreted as a measure of distinctiveness between these two distributions \\cite{kullback1951information}. It can be shown, see \\cite{vae_intro}, that\n\\begin{equation}\\label{eq:DKL_via_L}\n\\log p_{\\theta}(\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)})=D_{KL}[q_{\\phi}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)})||p_{\\theta}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)})] +\\mathcal{L}(\\theta,\\phi;\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}),\n\\end{equation}\nwhere\n$$\n\\mathcal{L}(\\theta,\\phi;\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}) = \n{\\mathbb E}_{q_{\\phi}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)})} \\left[ -\\log q_{\\phi}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)})+\\log p_{\\theta}(\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)},\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi) \\right].\n$$\nSince KL-divergence is non-negative, we have $\\log p_{\\theta}(\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}) \\geq\\mathcal{L}(\\theta,\\phi;\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)})$ and\n$\\mathcal{L}(\\theta,\\phi;\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)})$\nis called Evidence Lower Bound (ELBO) for the marginal likelihood $\\log p_{\\theta}(\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}).$\nThus, instead of maximizing the marginal probability, one can instead maximize its variational lower bound to which we also refer as an objective function. It can be further shown that the ELBO can be written as\n\\begin{equation}\n\t\\label{eq:VLB:2_no_beta}\n\t\\mathcal{L}(\\theta,\\phi;\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)})= {\\mathbb E}_{q_{\\phi}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)})} \\left[ \\log p_{\\theta}(\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}|\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi) \\right] - D_{KL}[q_{\\phi}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)})||p_{\\theta}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi)].\n\\end{equation}\nReformulating the traditional VAE framework as a constraint optimization problem, it is possible to obtain the $\\beta$-VAE \\cite{higgins2016beta} objective function if $p_\\theta(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi) =\\mathcal{N}({\\bf 0},{\\bm I}),$\n\\begin{equation}\n\t\\label{eq:VLB:2}\n\t\\mathcal{L}(\\theta,\\phi;\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)})= {\\mathbb E}_{q_{\\phi}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)})} \\left[ \\log p_{\\theta}(\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}|\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi) \\right] - \\beta D_{KL}[q_{\\phi}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)})||p_{\\theta}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi)],\n\\end{equation}\nwhere $\\beta>0.$ Here $\\beta$ is a regularisation coefficient that constrains the capacity of the latent representation $\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi$. The ${\\mathbb E}_{q_{\\phi}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)})} \\left[ \\log p_{\\theta}(\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}|\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi) \\right]$ can be interpreted as the reconstruction term, while the KL-term, $\\beta D_{KL}[q_{\\phi}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)})||p_{\\theta}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi)]$ as regularization term. \n\n\nConditional Variational Auto-encoders \\cite{sohn2015learning} (CVAE) are similar to VAEs, but differ by conditioning on an additional property of the data (e.g. a label or class), here denoted $\\ifmmode\\bm{c}\\else\\textbf{\\textit{c}}\\fi$. Conditioning both the recognition model and the true posteriori on both $\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}$ and $\\ifmmode\\bm{c}\\else\\textbf{\\textit{c}}\\fi$ results in the CVAE ELBO \n\\begin{align}\n    \\begin{split}\n        \\mathcal{L}(\\theta,\\phi;\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)},\\ifmmode\\bm{c}\\else\\textbf{\\textit{c}}\\fi)={\\mathbb E}_{q_{\\phi}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}, \\ifmmode\\bm{c}\\else\\textbf{\\textit{c}}\\fi)} \\left[ \\log p_{\\theta}(\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}|\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi,\\ifmmode\\bm{c}\\else\\textbf{\\textit{c}}\\fi) \\right] - D_{KL}[q(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)},\\ifmmode\\bm{c}\\else\\textbf{\\textit{c}}\\fi)||p_{\\theta}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{c}\\else\\textbf{\\textit{c}}\\fi)]. \\label{CVAE_objective}\n    \\end{split}\n\\end{align}\nIn the decoding phase, CVAE allows for conditional probabilistic reconstruction and permits sampling from the conditional distribution $p_{\\theta}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{c}\\else\\textbf{\\textit{c}}\\fi)$, which has been useful for generative modeling of data with known labels, see \\cite{sohn2015learning}. Here we investigate a special case of the CVAE when $\\ifmmode\\bm{c}\\else\\textbf{\\textit{c}}\\fi$ is a partial observation of $\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi.$ We call this Semi Conditional Variational Auto-encoder (SCVAE).\n\n\\subsection{Semi Conditional Variational Auto-encoder}\\label{SCVAE_section}\nThe SCVAE takes the input data $\\ifmmode\\bm{X}\\else\\textbf{\\textit{X}}\\fi$, conditioned on $\\ifmmode\\bm{M}\\else\\textbf{\\textit{M}}\\fi$ and approximates the probability distribution $p_{\\theta}(\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi|\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi,\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi).$ Then we can generate $\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}$, based on the observations $\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)}$ and latent representation $\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi$. As $\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)}=C \\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}$ where $C$ is a non-stochastic sampling matrix, we have \n$$p_{\\theta}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}, \\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)}) = p_{\\theta}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}), \\, \\mbox{ and } \\, q_{\\phi}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)},\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)})=q_{\\phi}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}).$$ Therefore, from \\cref{CVAE_objective} the ELBO for SCVAE is \n\\begin{equation}\n    \\begin{split}\n        \\log p_{\\theta}(\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}|\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)})\\geq \\mathcal{L}(\\theta,\\phi;\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)},\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)})=\n        &{\\mathbb E}_{q_{\\phi}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)})} \\left[ \\log p_{\\theta}(\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}|\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi, \\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)}) \\right] \\\\ & - D_{KL}[q_\\phi(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)})||p_{\\theta}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)})]\\label{eq:ELBO:SCVAE_no_beta}\n    \\end{split}\n\\end{equation}\nwhere $p_\\theta(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)}) = \\mathcal{N}({\\bf 0},{\\bf I}).$ Similarly as for the $\\beta$-VAE \\cite{higgins2016beta} we can obtain a relaxed version of \\cref{eq:ELBO:SCVAE_no_beta} by maximizing the parameters $\\{\\phi, \\theta\\}$ of the expected log-likelihood ${\\mathbb E}_{q_\\phi(\\cdot)} [\\log p_\\theta (\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}|\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)},\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi)])$ and treat it as an constrained optimization problem. That is, \n\\begin{align}\n    \\begin{split}\n    & \\max\\limits_{\\phi,\\theta} {\\mathbb E}_{q_\\phi(\\cdot)} [\\log p_\\theta (\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}|\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)},\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi)])  \\text{ subject to} \\\\ &\n    D_{KL}(q_\\phi(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)},\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)})||p_\\theta (\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)}) \\leq \\epsilon  \n    \\end{split}\\label{beta_SCVAE_opt_prob}\n\\end{align}\nwhere $\\epsilon>0$ is small. The subscript $q_\\phi(\\cdot)$ is short for $q_\\phi(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)},\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}).$ Since $\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)}$ is dependent on $\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}$ we have that $q_\\phi(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)},\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}) = q_\\phi(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}).$ \\cref{beta_SCVAE_opt_prob}  can expressed as a Lagrangian under the Karush\u2013Kuhn\u2013Tucker (KKT) conditions \\cite{kuhn2014nonlinear, karush1939minima}. Hence,\n\\begin{align}\n    \\begin{split}\n    \\mathcal{F}(\\theta, \\phi, \\beta, \\alpha, \\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}, \\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)}) & = \n    {\\mathbb E}_{q_\\phi(\\cdot)} [\\log p_\\theta (\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}|\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)},\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi)]) \\\\ & + \n    \\beta(D_{KL}(q_\\phi(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)})||p_\\theta (\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)})-\\epsilon)   \n    \\end{split}\\label{beta_SCVAE_constrained}\n\\end{align}\nAccording to the complementary slackness KKT condition $\\beta \\geq 0,$ we can rewrite \\cref{beta_SCVAE_constrained} as\n\\begin{align}\n    \\centering\n    \\begin{split}\n      \\mathcal{F}(\\theta, \\phi, \\beta, \\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}, \\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)}) \\geq  \\mathcal{L}(\\theta, \\phi, \\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}, \\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)}) & =  {\\mathbb E}_{q_\\phi(\\cdot)} [\\log p_\\theta (\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}|\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)},\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi)])  \\\\ & +\n     \\beta D_{KL}(q_\\phi(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)})||p_\\theta (\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)}).   \n    \\end{split}\\label{beta_SCVAE_constrained_2}\n\\end{align}\nObjective functions in \\cref{eq:ELBO:SCVAE_no_beta} and \\cref{beta_SCVAE_constrained_2}, and later \\cref{eq:ELBO:SCVAE:J}, show that if conditioning on a feature which is a known function of the original data, such as measurements, we do not need to account for them in the encoding phase.The measurements are then coupled with the encoded data in the decoder. We sketch the main components of the SCVAE in \\cref{DAE_neural_network_sketch}.\n\\begin{figure}[ht]\n    \\centering\n\t\t\\includegraphics[width=0.80\\linewidth]{Model_Sparse_autoencoder_new_notation_i.png}\n\t\t\\captionof{figure}{The figure shows a sketch of the model used to estimate \n\t\t$p_{\\theta}(\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi|\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)})$. During training both the observations $\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)}$ and the data $\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}$ will be used. After the model is trained, we can predict using only the decoder part of the neural network. The input to the decoder will then only be the observations and random samples from the latent space.}\\label{DAE_neural_network_sketch}\n\\end{figure}\nIn order to preserve some physical properties of the data $\\ifmmode\\bm{X}\\else\\textbf{\\textit{X}}\\fi,$ we can condition yet on another feature. Here we utilize the incompressibility property of the fluid, i.e., $\\ifmmode\\bm{d}\\else\\textbf{\\textit{d}}\\fi^{(i)}=L_{div} \\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)} \\approx 0,$ see \\cref{eq:L_div}. \\\\ \n\nWe intend to maximize a log-likelihood under an additional constrain $\\ifmmode\\bm{d}\\else\\textbf{\\textit{d}}\\fi^{(i)}$, compared to \\cref{beta_SCVAE_opt_prob}. That is\n\\begin{align}\n    \\begin{split}\n    & \\max\\limits_{\\phi,\\theta} {\\mathbb E}_{q_\\phi(\\cdot)} [\\log p_\\theta (\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}|\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)},\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi)])  \\text{ subject to} \\\\ &\n    D_{KL}(q_\\phi(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)})||p_\\theta (\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)},\\ifmmode\\bm{d}\\else\\textbf{\\textit{d}}\\fi^{(i)}) \\leq \\epsilon \\quad \\text{and} \\quad \\\\ &\n    -{\\mathbb E}_{q_\\phi(\\cdot)}[\\log p_\\theta(\\ifmmode\\bm{d}\\else\\textbf{\\textit{d}}\\fi^{(i)}|\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)},\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi)] \\leq \\delta \n    \\end{split}\\label{Constraint_optimization_prob_v3}\n\\end{align}\nwhere $\\epsilon, \\delta>0$ are small. \\cref{Constraint_optimization_prob_v3}  can expressed as a Lagrangian under the Karush\u2013Kuhn\u2013Tucker (KKT) conditions as before and as a consequence of the complementary slackness condition $\\lambda,\\beta \\geq 0,$ we can obtain the objective function \n\n\\begin{align}\n    \\centering\n    \\begin{split}\n      \\mathcal{F}(\\theta, \\phi, \\beta, \\alpha, \\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}, \\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)}, \\ifmmode\\bm{d}\\else\\textbf{\\textit{d}}\\fi^{(i)}) \\geq  \\mathcal{L}(\\theta, \\phi, \\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}, \\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)}, \\ifmmode\\bm{d}\\else\\textbf{\\textit{d}}\\fi^{(i)}) & =  {\\mathbb E}_{q_\\phi(\\cdot)} [\\log p_\\theta (\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}|\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)},\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi)])  \\\\ & +\n     \\lambda\\,{\\mathbb E}_{q_\\phi(\\cdot)} [\\log p_\\theta (\\ifmmode\\bm{d}\\else\\textbf{\\textit{d}}\\fi^{(i)}|\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)},\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi)])  \\\\ & -\n     \\beta D_{KL}(q_\\phi(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)})||p_\\theta (\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)},\\ifmmode\\bm{d}\\else\\textbf{\\textit{d}}\\fi^{(i)}),   \n    \\end{split}\\label{eq:ELBO:SCVAE:J}\n\\end{align}\nwhere $p(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)}, \\ifmmode\\bm{d}\\else\\textbf{\\textit{d}}\\fi^{(i)}) = \\mathcal{N}({\\bf 0},{\\bf I}).$ For convenience of notation we refer to the objective function \\cref{beta_SCVAE_constrained_2} as the case with $\\lambda=0$, and the objective function \\cref{eq:ELBO:SCVAE:J} as the case with $\\lambda > 0.$ Observe that under the Gaussian assumptions on the priors, \\cref{eq:ELBO:SCVAE:J} is equivalent to \\cref{beta_SCVAE_constrained_2} if $\\lambda=0.$ Thus, from now one we will refer to it as a special case of \\cref{eq:ELBO:SCVAE:J} and denote as $\\mathcal{L}_{0}.$ \\\\\n\nSimilarly to \\cite{kingma2013auto} we obtain $q_{\\phi}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)})= \\mathcal{N}({\\mu}^{(i)} \\mathbf{1},(\\sigma^{(i)})^2\\,\\mathbf{I})\n$, that is, $\\phi=\\{ \\mu, \\sigma\\}.$  This allows to express the KL-divergence terms in a closed form and avoid issues related to differentiability of the ELBOs.  Under these assumptions, the KL-divergence terms can be integrated analytically while the term\n${\\mathbb E}_{q_{\\phi}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)})} \\left[ \\log p_{\\theta}(\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}|\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi, \\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)}) \\right] $ and ${\\mathbb E}_{q_{\\phi}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)})} \\left[ \\log p_{\\theta}(\\ifmmode\\bm{d}\\else\\textbf{\\textit{d}}\\fi^{(i)}|\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi, \\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)}) \\right] $\nrequires estimation by sampling\n\\begin{equation}\\label{eq:E-estimate}\n    \\begin{array}{l}\n         {\\mathbb E}_{q_{\\phi}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)})} \\left[ \\log p_{\\theta}(\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}|\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi, \\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)}) \\right] \\approx \n        \\frac{1}{L}\\sum\\limits_{l=1}^{L} \\log  p_{\\theta}(\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}|\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi^{(i,l)},\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)}),\\\\\n        {\\mathbb E}_{q_{\\phi}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)})} \\left[ \\log p_{\\theta}(\\ifmmode\\bm{d}\\else\\textbf{\\textit{d}}\\fi^{(i)}|\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi, \\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)}) \\right] \\approx \n        \\frac{1}{L}\\sum\\limits_{l=1}^{L} \\log  p_{\\theta}(\\ifmmode\\bm{d}\\else\\textbf{\\textit{d}}\\fi^{(i)}|\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi^{(i,l)},\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)}),\\\\\n        \\mbox{where } \\, \\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi^{(i,l)} = g_{\\phi}(\\bm{\\epsilon}^{(i,l)}, \\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}), \\quad \n        \\bm{\\epsilon}^{l} \\sim p(\\bm{\\epsilon}).\n    \\end{array}\n\\end{equation}\nHere $\\bm{\\epsilon}^{l}$ is an auxiliary (noise) variable with independent marginal $p(\\bm{\\epsilon})$, and $g_{\\phi}(\\cdot)$ is a differentiable transformation of $\\bm{\\epsilon},$ parametrized by $\\phi,$  see for details \\cite{kingma2013auto}. We denote $\\mathcal{L}_\\lambda,$ $\\lambda \\geq 0$  \\cref{eq:ELBO:SCVAE:J} with the approximation above as $\\mathcal{\\widehat{L}_\\lambda},$ that is,\n\\begin{align}\n    \\begin{split}\n        & \\widehat{\\mathcal{L}}_\\lambda(\\theta, \\phi, \\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)},\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)},\\ifmmode\\bm{d}\\else\\textbf{\\textit{d}}\\fi^{(i)})  =   \\frac{1}{L}\\sum\\limits_{l=1}^{L} \\log  p_{\\theta}(\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}|\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi^{(i,l)},\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)}) \\\\\n         &+ \\lambda \\frac{1}{L}\\sum\\limits_{l=1}^{L} \\log p_{\\theta}(\\ifmmode\\bm{d}\\else\\textbf{\\textit{d}}\\fi^{(i)}|\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi^{(i,l)},\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)}) -\\beta D_{KL}[q_{\\phi}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)})||p_{\\theta}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)}, \\ifmmode\\bm{d}\\else\\textbf{\\textit{d}}\\fi^{(i)})].\n        \\label{Loss_function}\n    \\end{split}\n\\end{align}\nThe objective function $\\widehat{\\mathcal{L}}_\\lambda$ can be maximized by gradient descent. Since the gradient $\\nabla_{\\theta,\\phi}\\,\\widehat{\\mathcal{L}}_\\lambda$ cannot be calculated for large data sets, Stochastic Gradient Descent methods, see \\cite{kiefer1952stochastic, robbins1951stochastic} are typically used where \n\\begin{equation}\n    \\widehat{\\mathcal{L}}_\\lambda(\\theta, \\phi; \\ifmmode\\bm{X}\\else\\textbf{\\textit{X}}\\fi, \\ifmmode\\bm{M}\\else\\textbf{\\textit{M}}\\fi, \\bm{D}) \\approx \\widehat{\\mathcal{L}}^{R}(\\theta, \\phi; \\ifmmode\\bm{X}\\else\\textbf{\\textit{X}}\\fi^R, \\ifmmode\\bm{M}\\else\\textbf{\\textit{M}}\\fi^R, \\bm{D}^R) = \n    \\frac{K}{R}\\sum\\limits_{r=1}^{R} \\widehat{\\mathcal{L}}_\\lambda(\\theta, \\phi; \\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i_r)}, \\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i_r)}, \\ifmmode\\bm{d}\\else\\textbf{\\textit{d}}\\fi^{(i_r)}), \\quad \\lambda \\geq 0.\\label{obj_function_SCVAE}\n\\end{equation}\nHere  $\\ifmmode\\bm{X}\\else\\textbf{\\textit{X}}\\fi^{R}=\\left\\{\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i_r)}\\right\\}_{r=1}^{R},$  $R<K$ is a minibatch consisting of randomly sampled datapoints, $\\ifmmode\\bm{M}\\else\\textbf{\\textit{M}}\\fi^{R}=\\left\\{\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i_r)}\\right\\}_{r=1}^{R}$ and $\\bm{D}^{R}=\\left\\{\\ifmmode\\bm{d}\\else\\textbf{\\textit{d}}\\fi^{(i_r)}\\right\\}_{r=1}^{R}.$ After the network is optimized, a posterior predictive distribution $p_\\theta(\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi|\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi, \\ifmmode\\bm{d}\\else\\textbf{\\textit{d}}\\fi)$ can be approximated with a Monte Carlo estimator. \\\\\n\n\n\\subsubsection{Uncertainty Quantification}\\label{sec:posterior}\nLet  $\\hat{\\theta}$ and $\\hat{\\phi}$ be an estimation of generative and variational parameters, as described in \\cref{SCVAE_section}. Then the decoder can be used to predict the posterior as\n\\begin{align}\\label{eq:p_pred}\n    p_{\\hat{\\theta}}(\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi|\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^*, \\ifmmode\\bm{d}\\else\\textbf{\\textit{d}}\\fi^*) \\approx \\frac{1}{N_{MC}} \\sum_{j=1}^{N_{MC}} p_{\\hat{\\theta}}(\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi|\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi^{(j)},\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^*, \\ifmmode\\bm{d}\\else\\textbf{\\textit{d}}\\fi^*) \\xrightarrow[N_{MC} \\rightarrow \\infty]{} \\int\n    p_{\\hat{\\theta}}(\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi|\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi, \\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^*, \\ifmmode\\bm{d}\\else\\textbf{\\textit{d}}\\fi^*)p_{\\hat{\\theta}}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^*,\\ifmmode\\bm{d}\\else\\textbf{\\textit{d}}\\fi^*)d\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi.\n\\end{align}\nWhile sampling from the latent space has been viewed typically as an approach for generating new samples with similar properties, here we use it to estimate the prediction uncertainty of the trained model. From \\cref{eq:p_pred} we are able to estimate the mean prediction  $\\hat{\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi}^*$ and empirical covarience matrix $\\widehat{{\\bm \\Sigma}}$ using  a Monte Carlo estimator. We get\n\\begin{align} \n      \\widehat{\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi}^*=\n      \\frac{1}{N_{MC}} \\sum\\limits_{j=1}^{N_{MC}} \\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(j)}\\quad \\mbox{and} \\quad\n      \\widehat{{\\bm \\Sigma}} =  \\frac{1}{N_{MC}-1} \\sum\\limits_{j=1}^{N_{MC}}(\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(j)} -  \\widehat{\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi}^*)(\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(j)} -  \\widehat{\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi}^*)^T,\n        \\label{eq:mean_and_cov} \n               \n \\end{align}\nwhere $\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(j)}  \\sim p_{\\hat{\\theta}}(\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi|\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^*,\\ifmmode\\bm{d}\\else\\textbf{\\textit{d}}\\fi^*).$ The empirical standard deviation is then simply $\\widehat{\\bm \\sigma} = \\sqrt{diag(\\widehat{{\\bm \\Sigma}})}.$ To estimate the confidence region we assume that the predicted $p_{\\hat{\\theta}}(\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi|\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{*},\\ifmmode\\bm{d}\\else\\textbf{\\textit{d}}\\fi^*)$ is well approximated by a normal distribution $N({\\bm \\mu}, \\bm{\\Sigma}).$ Given that $\\widehat{\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi}^*$ and $\\widehat{{\\bm \\Sigma}}$ are approximations of $\\bm{\\mu}$ and $\\bm{\\Sigma},$ obtained from $N_{MC}$ samples as above, a confidence region estimate for a prediction $\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{*}$ can be given as\n\\begin{equation}\\label{eq:ConfRegion:1}\n    \\left\\{\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)} \\in {\\mathbb R}^{2N}: \\,(\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}-{\\widehat{\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi}^*})^T \\hat{\\bm \\Sigma}^{+} (\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}- \\widehat{\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi}^*)\\leq \\chi^2_k(p)\\right\\}\n\\end{equation}\nwhere $\\chi^2_k(p)$ is the quantile function for probability $p$ of the chi-squared distribution with $k=\\min\\{N_{MC},2N\\}$ degrees of freedom, and $\\widehat{\\bm \\Sigma}^{+}$ is the pseudoinverse of $\\widehat{\\bm \\Sigma}.$ Using the singular value decomposition, ${\\widehat{\\bm \\Sigma}}=\\bm{U S U}^T,$ the corresponding interval for $(\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^*)_n,$ $n=1,\\dots,2N,$ is\n\\begin{equation}\\label{eq:ConfRegion:2}\n   \\left[\n ({\\widehat{\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi}^*})_n-\\sqrt{\\chi^2_k(p)}\\,\\|{\\bm u}_n^T\\,{\\bm S}^{1\/2}\\|_2,\\quad\n ({\\widehat{\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi}^*})_n+\\sqrt{\\chi^2_k(p)}\\,\n \\|{\\bm u}_n^T\\,{\\bm S}^{1\/2}\\|_2\n  \\right]   \n\\end{equation}\nwhere ${\\bm u}_n^T$ is $n$th row of the matrix ${\\bm U}.$\n\n\n\\section{Experiments}\\label{Experiment}\nWe will present the SCVAE method on two different data sets. The first one is the 2D flow around a cylinder described in \\cref{A_Motivating_Example}, and the second is ocean currents on the seafloor created by the Bergen Ocean Model \\cite{berntsen2000users}. The data $\\ifmmode\\bm{X}\\else\\textbf{\\textit{X}}\\fi$ consists of the two dimensional velocities ${\\bm w} =(u, v).$ To illustrate the results we will plot $u$ and $v$ components of $\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)} \\in \\ifmmode\\bm{X}\\else\\textbf{\\textit{X}}\\fi,$ see \\cref{eq:x_i}. For validation of the models, the data $\\ifmmode\\bm{X}\\else\\textbf{\\textit{X}}\\fi$ is split into train, test and validation subsets, which are subscripted accordingly, if necessary. The data sets, spitting and preprocessing for each case are described in \\cref{Cylinder_wake_experiment,BOM_experiment}. \\\\\n\nWe use a schematically simple architecture to explore the SCVAE. The main ingredient of the encoder is convolutional neural network (CNN)  \\cite{lecun1998gradient, krizhevsky2012imagenet} and for the decoder we use transposed CNN-layers \\cite{noh2015learning}. The SCVAE has a slightly different architecture in each case, which we present in \\cref{Appendix_A}. \\\\\n\nThe SCVAE is trained to maximize the objective functions in \\cref{obj_function_SCVAE} with the backpropagation algorithm \\cite{lecun1989backpropagation} and the Adam algorithm \\cite{kingma2014adam}. We use an adaptive approach of weighing the reconstruction term with KL-divergence and\/or divergence terms \\cite{heydari2019softadapt}, that is, finding the regularization parameters $\\beta$ and $\\lambda$. Specifically, we calculate the proportion of each term contribution of the total value of the objective function, and scale the terms accordingly. This approach prevents posterior collapse. Posteriori collapse occurs if the KL-divergence term becomes too close to zero, resulting in a non probabilistic reconstruction. The approach of weighing the terms proportionally iteratively adjusts the weight of the KL-divergence term, $\\beta$, such that posterior collapse is avoided. For the result shown here we trained the SCVAEs  with an early stopping criteria of $50$ epochs, i.e., the optimization is stopped if we do not see any improvement after $50$ epochs, and returns the best model. We use a two-dimensional Gaussian  distribution for $p_\\theta(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)}, \\ifmmode\\bm{d}\\else\\textbf{\\textit{d}}\\fi^{(i)})$ in all the experiments.\\\\\n\nLet the test data $\\ifmmode\\bm{X}\\else\\textbf{\\textit{X}}\\fi_{test}$ consist of $n$ instances $\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)},$ $i=1,\\dots,n,$ and $\\widehat{\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi}^{(i)}$ denote a prediction of the true $\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}$ given $\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)}.$ In the case of the SCVAE, $\\widehat{\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi}^{(i)}$ is the mean prediction obtained as in \\cref{eq:mean_and_cov}. For the GPOD method, $\\widehat{\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi}^{(i)}$ is a deterministic output of the optimization problem, see \\cref{Appendix_C}. In order to compare the SCVAE results with the results of the GPOD method, we introduce the following measures; the mean of the relative error for the prediction\n\\begin{equation}\\label{L2_error}\n  \\mathcal{E} = \\frac{1}{n}\\sum\\limits_{i=1}^{n}\\frac{||\\widehat{\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi}^{(i)} - \\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}||_2}{||\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}||_2}\n\\end{equation}\nand the mean of the absolute error for the divergence \n\\begin{equation}\n        \\mathcal{E}_{div} = \\frac{1}{n}\\sum\\limits_{i=1}^{n}\\| L_{div} \\, \\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}\\|_2.\\label{divergence_error_1}\n\\end{equation}\n\n\\subsection{2D Flow Around a Cylinder}\\label{Cylinder_wake_experiment}\nHere we return to the example in \\cref{A_Motivating_Example}. Below we give some additional details of the data preprocessing and model implementation.\n\\subsubsection{Preprocessing}\nThe data is reduced as described in \\cref{A_Motivating_Example}. \nWe assess the SCVAE with a sequential split for train, test and validation. The sequantial split is defined as follows. The last $15 \\%$ of the data is used for test, the last $30\\%$ of the remaining data is used for validation, and the first $70\\%$ for training. To improve the conditioning of the optimization problem we scale the data as decribed in  \\cref{Appendix_D}. The errors $\\mathcal{E}$ (\\cref{L2_error}) and $\\mathcal{E}_{div} $ (\\cref{divergence_error_1}) are calculated after scaling the data back. The input to the SVAE $\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}$ was shaped as $(160 \\times 50 \\times 2)$ in order to apply convolutional layers. Here we use  $5,4,3$ and $2$ fixed spatial measurements, that is, four different subgrids ${\\mathcal Q}$ \n\\begin{align}\n    \\centering\n    \\begin{split}\n    {\\mathcal Q}_5 = & \\{(12,76),(47,8),(30,40),(153,34),(16,10)\\}, \\\\\n    {\\mathcal Q}_4 = & \\{(12,76),(47,8),(30,40),(153,34)\\},  \\\\\n    {\\mathcal Q}_3 = & \\{(12,76),(47,8),(30,40)\\}, \\\\\n    {\\mathcal Q}_2 = & \\{(12,76),(47,8)\\}.\n    \\end{split}\\label{CW_measurements}\n\\end{align}\nThe flow state at these specific locations constitute $\\ifmmode\\bm{M}\\else\\textbf{\\textit{M}}\\fi$.\n\n\\subsubsection{Model}\nA schematic description of the model is given in \\cref{Appendix_A}. \nThe first layer of the encoder is a zero-padding layer that expands the horizontal and vertical dimension by adding zeros on the boundaries. Here we add zero-padding of four in the horizontal and three in the vertical direction. The subsequent layers consists of two convolutional layers, where the first and second layer have $160$ and $200$ filters, respectively. We use a kernel size and strides of $2$ in both convolutional layers and ReLu activation functions. This design compresses the data into a $(42 \\times 14 \\times 200)$ shape. The compressed representation from the convolutional layers are flattened and are further compressed into a $64$ dimensional vector through a traditional dense layer. Two outputs layers are defined to represent the mean and log-variance of the latent representation $\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi$. The reparametrization trick is realized in a third layer, a so called  lambda layer, which takes the mean and log-variance as an input and generates $\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi$. The output of the encoder are the samples $\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi^{(i)}$ and the mean and the log-variance of $\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi$. \\\\\n\nThe decoder takes the latent representation $\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi^{(i)}$ and the measurements $\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)}$ as input. The input $\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)}$ is flattened and then concatenated with $\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi^{(i)}$. The next layer is a dense layer with shape $(42 \\times 14 \\times 200)$. Afterwards there are two transposed convolutional layers with filters of $200$ and $160$. The strides and the kernel size is the same as for the encoder. The final layer is a transposed convolutional layer, with same dimension as the input to the encoder, that is the dimension of $\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}$. A linear activation function is used for this output layer. The last layer of the model is a lambda layer that removes the zero-padding. In the next section we show statistics of the probabilistic reconstruction and compare with the GPOD method. \n\n\\subsubsection{Results}\nIn \\cref{Cylinder_wake_pred} we have plotted the true velocity field, the reconstructed velocities, the standard deviation of the velocities and the absolute error between the true and reconstructed velocity fields. The observations placement is shown as stars (black and white). The SCVAE with the objective function  $\\widehat{\\mathcal{L}}_0$, see \\cref{CVAE_objective}, was used for this prediction. To generate the posterior predictive distributions, \\cref{eq:p_pred},  we sample $100$ realizations from $\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi \\sim \\mathcal{N}(\\bm{0},\\bm{I})$ , which allows for calculation mean prediction and uncertainty estimates, see \\cref{eq:mean_and_cov}. \\\\\n\nWe emphasise here again that the SCVAE with $\\widehat{\\mathcal{L}}_0$ and with $\\widehat{\\mathcal{L}}_\\lambda,$ $\\lambda>0,$ are two different models. For the notation sake we here refer to $\\lambda=0$ when we mean the model with the objective function in \\cref{CVAE_objective}, and to $\\lambda>0$ when in \\cref{eq:ELBO:SCVAE:J}. The same holds for the GPOD method, see \\cref{Appendix_D}. When $\\lambda=0,$ the number of the principle components $r$ is less $2M.$ The number $r$ is chosen such that the prediction on the validation data has the smallest possible error on average. If $\\lambda>0,$ no restrictions on $r$ are imposed. In this case both  $\\lambda$ and $r$ are estimated from the validation data. \\\\\n\nThe general observation is that the SCVAE reconstruction fits the data well, with associated low uncertainty. \nThis can be explained by the periodicity in the data. In particular, the training and validation data sets represent the  test data well enough.\n\\begin{figure}[H]\n\t\\centering\n\n\t\t\\includegraphics[width=0.99\\linewidth]{CW_recon_3M_v1_CR.png}\n\t\\captionof{figure}{Left panels shows the u-velocities, and the right panel v-velocities. The results are based on a model trained with $\\lambda=0$ and $Q_3$ measurement locations. \\textbf{First panels:} The true solutions \\textbf{Second panels:} Reconstructed solution based on the SCVAE model \\textbf{Third panels:} Standard deviation of the predicted solution \\textbf{Fourth panels:} Absolute error between the true and predicted solution.}\\label{Cylinder_wake_pred}\n\n\\end{figure}\n\nIn \\cref{Cylinder_wake_time_series} we  have plotted four time series of the reconstructed test data at two specific grid points, together with the confidence regions constructed as in \\cref{eq:ConfRegion:2} with $p=0.95.$ The two upper panels represents the reconstruction at the grid point $(6, 31)$, and the lower at $(101,25)$ for $u$ and $v$ on the left and right side, respectively. The SCVAE reconstruction is significantly better than the GPOD, and close to the true solution for all time steps. \n\\begin{figure}[H]\n\t\\centering\n\n\t\t\\includegraphics[width=0.85\\linewidth]{CW_time_series_with_GPOD_CR.png}\n\t\t\\captionof{figure}{Velocities $u$ and $v$ at specific locations. The red line corresponds to the true values, blue to the SCVAE mean prediction, and orange to the GPOD reconstruction. Light blue shaded area represents the confidence region obtained in \\cref{eq:ConfRegion:2} with $p=0.95.$ The results are obtained from the model trained with $\\lambda=0$ and $Q_3$ measurement locations. \\textbf{Upper panels:} The time series and confidence region for $u$ (left) and $v$ (right) at  grid point $(6, 31).$ \\textbf{Lower panels:} The time series and confidence region for $u$ (left) and $v$ (right) at grid point $(101, 25).$ }\\label{Cylinder_wake_time_series}\n\n\\end{figure}\n\\begin{figure}[H]\n\t\\centering\n\n\t\t\\includegraphics[width=0.85\\linewidth]{CW_error_with_GPOD_CR.png}\n\t\t\\captionof{figure}{The difference between the true and predicted estimate for the SCVAE (blue) and for the GPOD (orange). The light blue shaded region represents the difference marginals, obtained from the confidence region in \\cref{BOM_time_series}. The estimates are based on a model trained with $\\lambda=0$ and $Q_3$ measurement locations. \\textbf{Upper panels:} The difference between the true and predicted estimate at grid point $(6,31)$ for $u$ (left) and $v$ (right), \\textbf{Lower panels:} The difference between the true and predicted estimate at point $(101,25)$ for $u$ (left) and $v$ (right).}\\label{Cylinder_wake_error}\n\n\\end{figure}\n\\cref{Cylinder_wake_error} shows the difference between the true values and the model prediction in time for the same two locations. This figure has to be seen in context with \\cref{Cylinder_wake_pred}. In \\cref{tab:comparison_CW}  we display the relative errors, \\cref{L2_error}, for the SCVAE and the GPOD method, both with and without divergence regularization, for $5, 4, 3,$ and $2$ measurement locations given in \\cref{CW_measurements}. \\\\\n\nThe results of the SCVAE depend on two stochastic inputs which are (i) randomness in the initialization of the prior weights and (ii) random mini batch sampling. We have trained the model with a each measurement configuration $10$ times, and chose the model that performs the best on the validation data set. Ideally we would run test cases where we used all the values as measurements,i.e., $\\ifmmode\\bm{M}\\else\\textbf{\\textit{M}}\\fi=\\ifmmode\\bm{X}\\else\\textbf{\\textit{X}}\\fi,$ and test how well the model would reconstruct in this case. This would then give us the lower bound of the best reconstruction that is possible for this specific architecture and hyper parameter settings. However, this scenario was not possible to test, due to limitations in memory in the GPU. Therefore we have used a large enough $M$ which still allowed us to run the model. In particular, we used every fifth and second pixel in the horizontal and vertical direction, which resulted in a total of $(32 \\times 25)$ measurement locations, or $M=800$. We believe that training the model with these settings, gave us a good indication of the lower bound of the reconstruction error. The error observed was of the magnitude of $10^{-3}$. \\\\\n\nThis lower bound has been reached for all measurement configurations \\cref{CW_measurements}. \nHowever, larger computational cost was needed to reach the lower bound for fewer measurement locations. \\cref{Epochs_per_measurement_CW} shows the number of epochs as a boxplot diagram. In comparison with GPOD, the SCVAE error is 10 times lower than the GPOD error, and this difference becomes larger with fewer measurements. Note that adding regularization did not have much effect on  the relative error. From the motivating example we observed that regularizing with $\\lambda>0$ is better in terms of a more consistent and low variable error estimation. Here we selected from the 10 trained models the one that performed best on the validation data set. This model selection approach shows that there are no significant differences between the two regularization techniques. The associated error in the divergence of the velocity fields is reported in  \\cref{tab:comparison_CW_div}. \n\\begin{table}[H]\n    \\centering\n    \\begin{tabular}{|c|c|c|c|c|c|} \n        \\hline\n         \\multirow{2}{*}{Method} & \\multirow{2}{*}{Regularization} & \\multicolumn{4}{|c|}{Measurement Locations} \\\\  \\cline{3-6} \n          & & 5 & 4 & 3 & 2  \\\\  \\hline\n         \\multirow{2}{*}{SCVAE} & $\\lambda = 0$\n         & 0.30e-02 & 0.33e-02 & 0.26e-02 & 0.28e-02    \\\\ \\cline{2-6}\n         & $\\lambda > 0$  & 0.31e-02 & 0.32e-02 & 0.30e-02 & 0.28e-02 \\\\ \\cline{3-6} \\hline\n         \\multirow{2}{*}{GPOD} & $\\lambda = 0$  & 2.35e-02& 2.49e-02  \n           & 3.38e-02& 17.38e-02\\\\ \\cline{2-6}\n         & $\\lambda > 0$ &   2.12e-02 & 2.33e-02& 3.15e-02& 16.38e-02    \\\\ \\hline\n    \\end{tabular}\n    \\caption{The mean relative error $\\mathcal{E}$ (\\cref{L2_error}) for the  SCVAE prediction  and  the GPOD prediction with or without div-regularization, and different number of measurements.}\n    \\label{tab:comparison_CW}\n\\end{table}\n\\begin{table}[H]\n    \\centering\n    \\begin{tabular}{|c|c|c|c|c|c|} \n        \\hline\n         \\multirow{2}{*}{Method} & \\multirow{2}{*}{Regularization} & \\multicolumn{4}{|c|}{Measurement Locations} \\\\  \\cline{3-6} \n          & & 5 & 4 & 3 & 2  \\\\  \\hline\n         \n         \\multirow{2}{*}{SCVAE} & $\\lambda = 0$\n          & 0.1439 & 0.1580 & 0.1383 & 0.1432    \\\\ \\cline{2-6}\n         & $\\lambda > 0$ & 0.1533 & 0.1408 & 0.1468 & 0.1410    \\\\ \\cline{3-6} \\hline\n         \\multirow{2}{*}{GPOD} & $\\lambda = 0$ & 0.1052 & 0.1047 & 0.0943 & 0.08866    \\\\ \\cline{2-6}\n         & $\\lambda > 0$  & 0.1039 & 0.1051 & 0.0966 & 0.0669   \\\\ \\hline\n    \\end{tabular}\n    \\caption{Comparison of the divergence error $\\mathcal{E}_{div}$ as calculated in \\cref{divergence_error_1} for the different methods and regularization techniques. The true divergence error on the entire test data set is $0.1058$}\n    \\label{tab:comparison_CW_div}\n\\end{table}\n\\begin{figure}[H]\n    \\centering\n \t    \\includegraphics[width=0.75\\linewidth]{Epochs_per_Measurement.png}\n \t    \\captionof{figure}{Number of epochs trained depending on the number of measurements. For each measurement configuration and regularization technique the model is run $10$ times. The variation of number of epochs for for each measurement locations is due to different priors of the weights and random mini-batch sampling.}\\label{Epochs_per_measurement_CW}\n\\end{figure}\n\n\\subsection{Current data from Bergen ocean model}\\label{BOM_experiment}\nWe tested the SCVAE on simulations from the Bergen Ocean Model (BOM) \\cite{berntsen2000users}. BOM is a three-dimensional terrain-following nonhydrostatic ocean model with capabilities of resolving mesoscale to large-scale processes. Here we use velocities simulated by Ali. et. al \\cite{Ali:2016go}. The simulations where conducted on the entire North Sea with 800 meter horizontal and vertical grid resolution and 41 layers for the period from 1st to 15th of January 2012. Forcing of the model consist of wind, atmospheric pressure, harmonic tides, rivers, and initial fields for salinity and temperature. For details of the setup of the model, forcing and the simulations we refer to \\cite{Ali:2016go}. \\\\\n\nHere, the horizontal and vertical velocities of an excerpt of 25.6 $\\times$ 25.6 km$^2$ at the bottom layer centered at the Sleipner CO2 injection site ($58.36^\\circ N, \\, 1.91^\\circ E$) is used as data set for reconstruction. In \\cref{Data_set_time_series_BOM} we have plotted the mean and extreme values of $u$ and $v$ for each time $t$ in ${\\mathcal T}$. \n\\begin{figure}[H]\n\t\\centering\n\n\t\t\\includegraphics[width=0.99\\linewidth]{Data_CW_U.png}\n        \\includegraphics[width=0.99\\linewidth]{Data_CW_V.png}\n\t\t\\captionof{figure}{The light-blue line represent the maximum, the orange the minimum and the green mean value of $u$ and $v$ for each time $t$ in ${\\mathcal T}.$ The horizontal lines indicate the sequential data split. }\\label{Data_set_time_series_BOM}\n\n\\end{figure}\n\\subsubsection{Preprocessing}\nWe extract $32 \\times 32$ central grid from the bottom layer velocity data. Hence, $\\mathcal{P}$ contains $N = 1024$ points, $32$ points in the horizontal and $32$ in the vertical direction. The temporal resolution is originally $105000$ and the time between each time step is $1$ minute. We downsample the temporal dimension of the original data uniformly such that the number of time steps in $\\mathcal{T}$ is $K=8500.$ We train and validate the SCVAE with two different data splits: randomized and sequential in time. For the sequential split we have used the last $15 \\%$ for the test, the last $30\\%$ of the remaining data is used for validation, and the fist $70\\%$ for training. In \\cref{Data_set_time_series_BOM}, the red and blue vertical lines indicate the data split for this case. For the random split, the instances $\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}$ are drawn randomly from $\\ifmmode\\bm{X}\\else\\textbf{\\textit{X}}\\fi$ with the same percentage. The data was scaled as described in \\cref{Appendix_D}.  The input $\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}$ to the SCVAE was shaped as $(32 \\times 32 \\times 2)$ in order to apply convolutional layers. We use  $9,5$ and $3$ fixed spatial measurement locations. In particular, the subgrid ${\\mathcal Q}$ is given as \n\\begin{align}\\label{BOM_measurements}\n    \\centering\n    \\begin{split}\n    {\\mathcal Q}_9 = & \\{(6,6),(6,17),(6,27),(17,17),(17,27),(17,6),(27,6), (27,17),(27,27) \\}, \\\\\n    {\\mathcal Q}_5 = & \\{(6,6),(17,17),(27,27),(6,27),(27,6) \\}, \\\\\n    {\\mathcal Q}_3 = & \\{ (6,27),(17,17),(27,6) \\}. \n    \\end{split}\n\\end{align}\nAs before, the values of $u$ and $v$ at these specific locations constitute the measurements $\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)} \\in \\ifmmode\\bm{M}\\else\\textbf{\\textit{M}}\\fi$.\n\n\\subsubsection{Model} \nA schematic description of the model is given in \\cref{Appendix_A.3,Appendix_A.4}. \nThe first two layers of the encoder are convolutional layers with $64$ and $128$ filters with strides and kernel size of $2$ and ReLu activation functions. This compresses the data into a shape of $(8 \\times 8 \\times 128)$. The next layers are flattening and dense layers, where the latter have $16$ filters and ReLu activation. The subsequent layers defines the mean and log-variance of the latent representation $\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi$, which is input to a lambda layer for realization of the reparametrization trick. The encoder outputs the samples $\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi^{(i)}$ and the mean and the log-variance of $\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi$.\\\\\n\nInput to the decoder is the output $\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi^{(i)}$ of the encoder and the measurement $\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)}.$ To concatenate the inputs, $\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)}$ is flattened. After concatenation of $\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi^{(i)}$ and $\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)}$, the next layer is a dense layer with shape $(8 \\times 8 \\times 128)$ and ReLu activation. This allows for use of transposed convolutional layers to obtain the original shape of the data. Hence, the following layers are two transposed convolutional layers with $64$ and $128$ filters, strides and kernel size of $2$ and ReLu activation's. The final layer is a transposed convolutional with linear activation functions and filter size  of shape $(32 \\times 32 \\times 2),$ i.e., the same shape as $\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}$. \n\n\\subsubsection{Results}\\label{BOM_results}\nWe illustrate the obtained posterior predictive distribution in terms the predictive mean and standard deviation for the prediction at a specific time. The SCVAE is compared with the GPOD method, both with $\\lambda >0$ and $\\lambda = 0$ for measurement locations given in \\cref{BOM_measurements} for random and sequential split cases. To generate the posterior predictive distributions, \\cref{eq:p_pred},  we sample $200$ realizations from $\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi \\sim \\mathcal{N}(\\bm{0},\\bm{I})$ , which allows for calculation mean prediction and uncertainty\nestimates, see \\cref{eq:mean_and_cov}. \\cref{UV_prediction} shows the results of the prediction at time step $1185$ for both the $u$ and $v$ component and associated uncertainty estimates for a trained model with $\\lambda=0$ and $Q_3$ measurement locations (see \\cref{BOM_measurements}).  \n\n\\begin{figure}[H]\n    \\centering\n\t\t\\includegraphics[width=0.75\\linewidth]{Sleipner_prediction_std_abs_v1.png}\n\t\t\\captionof{figure}{Presentation of statistics for\n\t\tthe reconstruction of the $u$ and $v$ component of the velocity for sample $1185$ in the test data set based on the trained model with $\\lambda=0$ and $Q_3$ measurement locations. \\textbf{Left panels:} From top to bottom; True velocity in $u$, predicted mean velocity field of $u$, the standard deviation of the prediction for $u$ and the absolute error of $u.$ \\textbf{Lower panels:} Similar as describe for the upper panels, but for $v$ }\\label{UV_prediction}\n\\end{figure} \nIn \\cref{BOM_time_series} we plot the true solution and the predicted mean velocity \\cref{eq:mean_and_cov} with the associated uncertainty, see \\cref{eq:ConfRegion:2}, for two grid points. We plot only the first $600$ time steps for readability. The first grid point is $(26,6)$ and $(4,1).$ One location is approximately $5.1$ km from the nearest observation, and another one is about $16.1$ km away.\n\\begin{figure}[H]\n        \\centering\n\t\t\\includegraphics[width=0.85\\linewidth]{BOM_plots_time_series_CR.png}\n\t\t\\captionof{figure}{Velocities $u$ and $v$ at specific locations based on the trained model with $\\lambda=0$ and $Q_3$ measurement locations. The red line corresponds to the true values, blue to the SCVAE mean prediction, and orange to the GPOD reconstruction. Light blue shaded area represents the confidence region obtained in \\cref{eq:ConfRegion:2} with $p=0.95.$.  \\textbf{Upper panels:} The time series and confidence region for $u$ (left) and $v$ (right) at  grid point $(26,6)$, approximately $5.1$ km from nearest observation. \\textbf{Lower panels:} The time series and confidence region for $u$ (left) and $v$ (right) at grid point $(4,1)$ approximately $16.1$ km from nearest observation.}\\label{BOM_time_series}\n\\end{figure} \n\\cref{BOM_error} has to be viewed in context with \\cref{BOM_time_series} and show the difference between the true and the predicted solutions with associated difference marginal in time for the two locations as in \\cref{BOM_time_series}. \n\\begin{figure}[H]\n\t\\centering\n\t\\includegraphics[width=0.85\\linewidth]{BOM_error_plot_CR.png}\n\t\\captionof{figure}{The difference between the true and predicted estimate for the SCVAE (blue) and for the GPOD (orange) based on the $\\lambda=0$ model and $Q_3$ measurement locations. The light blue shaded region represents the difference marginals, obtained from the confidence region in \\cref{BOM_time_series}. \\textbf{Upper panels:} The difference between the true and predicted estimate at grid point $(26,6)$ for $u$ (left) and $v$ (right), \\textbf{Lower panels:} The difference between the true and predicted estimate at point $(4,1)$ for $u$ (left) and $v$ (right).}\\label{BOM_error}\n\\end{figure}\nIntegrating over the latent space generates a posterior distribution of the reconstruction, as described in \\cref{sec:posterior}. It is also possible to use the latent space to generate new statistically sound versions of $u$ and $v$. This is presented in \\cref{z_space_sampling_v} where it is sampled uniformly over the 2 dimensional latent space $\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi \\sim \\mathcal{N}(\\bm{0},\\bm{I})$ and the result shows how different variations can be created with the SCVAE model, given only the sparse measurements.    \n\\begin{figure}[H]\n    \\centering\n\t\t\\includegraphics[width=0.70\\linewidth]{Sleipner_generating_data_v1.png}\n\t\t\\captionof{figure}{The left panels shows 9 different generated velocity-field-snapshots for the $\\ifmmode\\bm{u}\\else\\textbf{\\textit{u}}\\fi$ and $\\ifmmode\\bm{v}\\else\\textbf{\\textit{v}}\\fi$ component for test sample number $1185$. The predictions are generated from the model with $\\lambda=0$ and $Q_3$ measurement locations. We sample uniformly over the latent space and predicts with the decoder, given the measurements.}\\label{z_space_sampling_v}\n\\end{figure}\nThese sampled velocities could be used for ensemble simulations when estimating uncertainty in a passive tracer transport, see e.g., \\cite{oleynik2020optimal}. \n\nThe SCVAE results are are compared with results of the GPOD method, see \\cref{tab:comparison_BOM} and \\cref{tab:comparison_BOM_div}. The tables show the errors as calculated in \\cref{L2_error} and \\cref{divergence_error_1} of the test data set for both sequential and random split.\nFor the sequential splitting, the SCVAE is better for $3$ measurement locations, while the GPOD method performs better for $9$ and $5$ locations.  From  \\cref{Data_set_time_series_BOM}, we observe that test data set seems to arise from a different process than the train and validation data (especially for $v$). Thus, the SCVAE generalize worse than a simpler model such as the GPOD, \\cite{model_selection}. For the $3$ location case, the number of components in the GPOD is not enough to compete with the SCVAE. \\\\\n\nWith random split on the train, test and validation data, we see that the SCVAE is significantly better than the GPOD. The training data and measurements represent the test data and test measurements better with random splitting. This highlights the importance of large data sets that cover as many outcomes as possible. Demanding that $\\lambda > 0$ in \\cref{obj_function_SCVAE} do not improve the result. The SCVAE-models with $\\lambda = 0$ learns that the reconstructed representations should have low divergence without explicitly demanding it during optimization. However, as discussed in the 2D flow around cylinder experiment, demanding $\\lambda>0$ seems to improve the conditioning of the optimization problem and give more consistent results.  \n\\begin{table}[ht]\n    \\centering\n    \\begin{tabular}{|c|c|c|c|c|c|} \n        \\hline\n         \\multirow{2}{*}{Split} & \\multirow{2}{*}{Regularization} & \\multirow{2}{*}{Method} & \\multicolumn{3}{|c|}{Measurement Locations} \\\\  \\cline{4-6} \n          & & & 9 & 5 & 3 \\\\  \\hline\n         \n         \\multirow{4}{*}{Random} & \\multirow{2}{*}{$\\lambda=0$}\n         & SCVAE & 0.1379 & 0.2097 & 0.2928   \\\\ \\cline{3-6}\n         & & GPOD & 0.3300 & 0.3822 & 0.4349  \\\\ \\cline{2-6}\n         & \\multirow{2}{*}{$\\lambda>0$}\n         & SCVAE & 0.1403 & 0.2025 & 0.3016   \\\\ \\cline{3-6}\n         & & GPOD & 0.2971 & 0.3579 & 0.4039  \\\\ \\cline{3-6} \\hline\n         \n         \\multirow{4}{*}{\\makecell{Time \\\\ Dependent}} & \\multirow{2}{*}{$\\lambda=0$}\n         & SCVAE & 0.3493 & 0.3913 & 0.4155  \\\\ \\cline{3-6}\n         & & GPOD & 0.3767 & 0.4031 & 0.4678  \\\\ \\cline{2-6}\n         & \\multirow{2}{*}{$\\lambda>0$}\n         & SCVAE & 0.3527 & 0.3889 & 0.4141  \\\\ \\cline{3-6}\n         & & GPOD &0.3362 & 0.3695 & 0.4462  \\\\ \\hline\n    \\end{tabular}\n    \\caption{Errors as calculated in \\cref{L2_error} for the different methods, regularization techniques ($\\lambda=0$ or $\\lambda > 0$), split regimes and measurements}\n    \\label{tab:comparison_BOM}\n\\end{table}\n\\begin{table}[ht]\n    \\centering\n    \\begin{tabular}{|c|c|c|c|c|c|} \n        \\hline\n         \\multirow{2}{*}{Split} & \\multirow{2}{*}{Regularization} & \\multirow{2}{*}{Method} & \\multicolumn{3}{|c|}{Measurement Locations} \\\\  \\cline{4-6} \n          & & & 9 & 5 & 3  \\\\  \\hline\n         \n         \\multirow{4}{*}{Random} & \\multirow{2}{*}{$\\lambda=0$}\n         & SCVAE & 3.75e-05 & 3.62e-05 & 3.42e-05   \\\\ \\cline{3-6}\n         & & GPOD & 6.51e-05 & 5.88e-05 & 5.02e-05   \\\\ \\cline{2-6}\n         & \\multirow{2}{*}{$\\lambda>0$}\n         & SCVAE &  3.60e-05 & 3.60e-05 & 3.13e-05  \\\\ \\cline{3-6}\n         & & GPOD & 6.23e-05 & 4.77e-05 & 4.14e-05  \\\\ \\cline{3-6} \\hline\n         \n         \\multirow{4}{*}{\\makecell{Time \\\\ Dependent}} & \\multirow{2}{*}{$\\lambda=0$}\n         & SCVAE & 2.02e-05 & 1.80e-05 & 1.69e-05   \\\\ \\cline{3-6}\n         & & GPOD & 5.09e-05 & 4.03e-05 & 4.15e-05  \\\\ \\cline{2-6}\n         & \\multirow{2}{*}{$\\lambda>0$}\n         & SCVAE & 2.05e-05 & 1.99e-05 & 1.85e-05  \\\\ \\cline{3-6}\n         & & GPOD & 4.39e-05 & 3.65e-05 & 2.92e-05  \\\\ \\hline\n    \\end{tabular}\n    \\caption{Divergence errors as calculated in \\cref{divergence_error_1} for the different methods, regularization techniques ($\\lambda=0$ or $\\lambda > 0$), split regimes and measurements. The true divergence of the test data is of order  $10^{-4}.$ }\n    \\label{tab:comparison_BOM_div}\n\\end{table}\n\\begin{figure}[H]\n    \\centering\n \t    \\includegraphics[width=0.75\\linewidth]{BOM_epochs_random_split.png}\n \t    \\captionof{figure}{The figure shows number of epochs and number of measurement locations. For each measurement configuration and regularization technique the model is optimized 10 times. The variation in the number of epochs for each measurement and regularization technique is due to different priors of the weights and mini-batch sampling.}\\label{Epochs_per_measurement}\n\\end{figure}\n\n\\section{Discussion}\\label{discussion}\nWe have presented the SCVAE method for efficient data reconstruction based on sparse observations. The derived objective functions for the network optimization  show that the encoding is independent of measurements. This allows for a simpler model structure with fewer model parameters than a CVAE and results in an optimization procedure that requires less computations.  \\\\\n\nWe have shown that the SCVAE is suitable for reconstruction of fluid flow. The method is showcased on two different data sets, velocity data from simulations of 2D flow around a cylinder, and bottom currents from the BOM. The fact that the fluids studied in the experiments are incompressible served as a motivation for adding an extra term to the objective function, see \\cref{obj_function_SCVAE} with $\\lambda>0$. \\\\\n\nOur investigation of additional regularization showed that the mean reconstruction error over all models was lower with $\\lambda>0$ compared to the model where $\\lambda=0$, but the best reconstruction error was similar for $\\lambda=0$ and $\\lambda>0$. In \\cref{Experiment} we optimized 10 models for every experiment, and chose the model that performed best on the validation data sets. With this approach we did not observe significant differences between optimizing with  $\\lambda=0$ and $\\lambda>0$. However, the reconstruction became less sensitive to the stochasticity involved in optimization (minibatch selection, network weights priors) when the regularization was used, see \\cref{A_Motivating_Example}. \\\\ \n\nThe SCVAE is a probabilistic model, which allows to make predictions, estimate their uncertainty, see \\cref{sec:posterior},  and draw multiple samples from the predictive distribution, see \\cref{z_space_sampling_v}. The last two properties make the SCVAE a useful method especially when the predictions are used in another application, i.e., ensemble simulation of tracer transport. Motivated by \\cite{YILDIRIM2009160}, we compared the SCVAE predictions with the predictions of a modified GPOD method, see \\cref{Appendix_C}. \\\\\n\nUnlike the GPOD-method, a benefit with the SCVAE-method is that it scales well to larger data sets. Another aspect and as the experiments in \\cref{Experiment} suggest, the GPOD seems more sensitive to the number of measurement locations than the SCVAE. On the other hand, the experiments suggested that GPOD is better than SCVAE with a larger number of measurement locations if the training data and the test data are too different, see BOM experiment with sequential splitting \\cref{BOM_results}. Essentially the SCVAE overfit to the training data, and as a result performing poorly on the test data set. This fact shows the importance of training the the SCVAE on large data sets, which covers as many potential flow patterns as possible. Further, the results show that the GPOD is more sensitive to the measurement location choice than the SCVAE, see \\cref{A_Motivating_Example}, and the GPOD-method is not expected to preform well on a complex flow with very few fixed measurement locations. \\\\\n\nVAEs has been used for generating data in e.g. computer vision \\cite{kingma2013auto}, and auto-encoders is a natural to use in reconstruction tasks \\cite{ELMS2018}. Many reconstruction approaches, including the GPOD approach, first create a basis, then use the basis and minimize the error of the observations \\cite{willcox2006unsteady, bui2004aerodynamic}. This makes the GPOD suitable for fast optimization of measurement locations that minimize the reconstruction error. On the other hand,  the SCVAE optimizes the basis function given the measurements, i.e. they are known and fixed. This makes it challenging to use the framework for optimizing sensor layout. But if the measurement locations are fixed and large amounts of training data are available, the SCVAE outperforms the GPOD for reconstruction. SCVAE optimize the latent representation and the neural network model parameters, variational and generative parameters, given the measurements. This ensures that the reconstruction is adapted to the specific configuration of measurements. \\\\\n\nA limitation of our experiments is that we used only $100$ and $200$ samples and constructed the confidence region under further simplifying assumptions. The uncertainty estimate could be improved by increasing the sample size and better model for the confidence region. \\\\\n\nNatural applications for the SCVAE are related to environmental data, where we often have sparse measurements. It is for example possible to optimize sensor layout to best possible detect unintentional discharges in the marine environment by using a simple transport model \\cite{oleynik2020optimal}. Oleynik. et al. used deterministic flow fields to transport the contaminant and thus obtain a footprint of the leakage. SCVAE can be used to improve that method and efficiently generate probabilistic footprints of a discharges. This may be important as input to design, environmental risk assessments, and emergency preparedness plans. \n\nWe have highlighted the SCVAE through the reconstruction of currents and flow field reconstruction, however, the SCVAE method is not limited to fluid flow problems. For instance, the same principles could be used in computer vision to generate new picture based on sparse pixel representations or in time series reconstruction. \\\\\n\nA natural extension of the SCVAE is to set it up as a partially hidden Markov model. That is to predict the current state $p_\\theta(\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi_t|\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi_t, \\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi_{t-1}),$ given the measurements and the reconstruction from the previous time step. This could potentially improve the reconstruction further.\n\n\\section*{Acknowledgements}\nThis work is part of the project ACTOM, funded through the ACT programme (Accelerating CCS Technologies, Horizon2020 Project No 294766). Financial contributions made from; The Research Council of Norway, (RCN), Norway, Netherlands Enterprise Agency (RVO), Netherlands, Department for Business, Energy \\& Industrial Strategy (BEIS) together with extra funding from NERC and EPSRC research councils, United Kingdom, US-Department of Energy (US-DOE), USA. Kristian Gundersen has been supported by the Research Council of Norway, through the CLIMIT program (project 254711, BayMode) and the European Union Horizon 2020 research and innovation program under grant agreement 654462, STEMM-CCS. The authors would like to acknowledge NVIDIA Corporation for providing their GPUs in the academic GPU Grant Program.\n\n\\clearpage\n\n\\bibliographystyle{unsrt}\n\\clearpage\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\n\n\\subsection{Training SNNs}\n\n\n\\begin{figure}[t!]\n  \\centering\n  \\includegraphics[width=0.8\\linewidth]{figures\/overview.pdf}\n  \\caption{Overview of the knowledge distillation method for SNNs.}\n  \\label{fig:overview}\n\\end{figure}\n\n\\subsection{Analyzing conventional knowledge distillation applied to SNNs}\n\\label{conventional_KD_analysis}\nAs already mentioned in the previous section, knowledge distillation has been conventionally applied to SNN models based on the method proposed in \\cite{hinton2015distilling}.\nThe overall process of knowledge distillation in SNNs is visualized in Fig.~\\ref{fig:overview}.\nIn this method, the loss term is formulated as\n\\begin{equation}\n\\label{eq:total_loss}\n    L = \\alpha L_{\\mathrm{CE}} + (1-\\alpha) L_{\\mathrm{KD}} \\textrm{,}\n\\end{equation}\nwhere $L_{\\mathrm{CE}}$ is the original cross entropy loss, $L_{\\mathrm{KD}}$ is additional distillation loss, and the two loss terms are balanced by a weighting hyperparameter $\\alpha$.\nMoreover, $L_{\\mathrm{KD}}$ can be computed as\n\\begin{equation}\n\\label{eq:KD_loss_equal_T}\n    L_{\\mathrm{KD}} = \\tau^{2}D_{\\mathrm{KL}}(\\sigma(z^{\\mathrm{s}};T=\\tau)||\\sigma(z^{\\mathrm{t}};T=\\tau)) \\textrm{,}\n\\end{equation}\nwhere Kullback Leibler divergence is adopted, and temperature $T$ with $\\tau > 1$ is used to soften the output logits of the student and the teacher, notated $z^s$ and $z^t$ respectively.\nThe softened output logits are normalized by a softmax function $\\sigma(\\cdot)$, which can be described by\n\\begin{equation}\n\\label{eq:softmax}\n    \\sigma(z_k;T=\\tau)=\\frac{e^{z_k\/\\tau}}{\\sum_{i}e^{z_i\/\\tau}} \\textrm{.}\n\\end{equation}\n\nThe performance of knowledge distillation depends on the choices of the two hyperparameters, $T$ and $\\alpha$.\nIn past studies, $T$ and $\\alpha$ have been commonly decided in a heuristic manner, usually following the popular choices \\cite{cho2019efficacy}, to derive only the best inference accuracy.~\\cite{kushawaha2020distilling,takuya2021training}\nHowever, when applying knowledge distillation to SNNs, both accuracy and energy efficiency should be considered together as important metrics when evaluating the performance of the models.\nAccordingly, in this paper we first investigate the performance of knowledge distillation in SNN models in terms of inference accuracy and energy efficiency by changing the values of $T$ and $\\alpha$.\nFor energy efficiency, we measure the number of spikes occurring inside the network, which serves as an effective indicator for estimating energy efficiency.\nA baseline SNN model trained from scratch is also evaluated in the same way for fair comparison.\n\nFrom the aforementioned analysis, we find that the hyperparameter sets that lead to student SNN models with high accuracy also result in the increase in the number of spikes, which will be described in detail in Section~\\ref{result:conventional_KD_analysis}.\nThis can act as a major drawback when trying to apply knowledge distillation to SNNs, as the number of spikes is strongly related to the energy efficiency of the model.\nTo address this issue, we conduct a second analysis on the weight and output distributions to determine the cause for this increase.\nThe related results can be found in Section~\\ref{result:why_spikes_increase}.\n\n\\subsection{Knowledge distillation with heterogeneous temperature}\n\\label{KD_proposed}\nWe come to the conclusion that the temperature is the key factor, which has been conventionally applied to both the teacher and the student outputs with the same value without doubt, as in Eq.\\ref{eq:KD_loss_equal_T}.\nHowever, we find that in SNNs, $T$ applied to the teacher output and $T$ applied to the student output play different roles in the distilling process.\nWith this in mind, we propose a novel knowledge distillation method that significantly reduces the number of spikes while boosting the inference accuracy of the student SNN models when compared with the conventional method.\n\nWe construct a new loss term by introducing heterogeneous temperatures for the outputs of student and teacher models.\nThe reformed KD loss $L_{\\mathrm{KD}}$ is as follows:\n\\begin{equation}\n\\label{eq:KD_loss_hetero_T}\n    L_{\\mathrm{KD}} = T_{\\mathrm{s}}T_{\\mathrm{t}}D_{\\mathrm{KL}}(\\sigma(z^{\\mathrm{s}};T=T_{\\mathrm{s}})||\\sigma(z^{\\mathrm{t}};T=T_{\\mathrm{t}})) \\textrm{,}\n\\end{equation}\nwhere $T_{\\mathrm{s}}$ and $T_{\\mathrm{t}}$ are the temperatures applied to the student SNN model and the teacher ANN model, respectively.\nWith this simple change in the loss term, by only tweaking the existing temperature parameter, we can effectively regulate the number of spikes in student SNN models while maintaining their accuracy.\nDetailed results are covered in the next section.\n\n\n\n\n\n\n\n\n\\section{Introduction}\n\\label{intro}\n\\input{1-introduction}\n\n\n\\section{Background}\n\\label{background}\n\n\\subsection{Spiking neural networks}\n\\label{bg:SNN}\n\\input{2.1-spiking_neural_networks}\n\n\\subsection{Knowledge distillation}\n\\label{bg:KD}\n\\input{2.2-knowledge_distillation}\n\n\n\\section{Methodology}\n\\label{method}\n\\input{3-methodology}\n\n\n\\section{Results}\n\\label{result}\n\n\\subsection{Experimental settings}\n\\label{result:settings}\n\\input{4.1-experimental_settings}\n\n\\subsection{Analyzing the performance of conventional knowledge distillation}\n\\label{result:conventional_KD_analysis}\n\\input{4.2-conventional_KD_analysis}\n\n\\subsection{Why does the number of spikes increase?}\n\\label{result:why_spikes_increase}\n\\input{4.3-why_spikes_increase}\n\n\\subsection{Analyzing the performance of proposed knowledge distillation}\n\\label{result:proposed_KD_analysis}\n\\input{4.4-proposed_KD_analysis}\n\n\\subsection{How does the number of spikes decrease?}\n\\label{result:how_spikes_decrease}\n\\input{4.5-how_spikes_decrease}\n\n\\subsection{Comparison with other methods}\n\\label{result:comparison}\n\\input{4.6-comparison_with_other_methods}\n\n\n\\section{Conclusion}\n\\label{conclusion}\n\\input{5-conclusion}\n\n\n\\bibliographystyle{abbrvnat}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \\label{sec:intro} Recent research on route planning in transportation networks~\\cite{bdgmpsww-rptn-14} has produced several speedup techniques varying in preprocessing time, space, query performance, and simplicity. Overall, queries on road networks are several orders of magnitude faster than on public transit~\\cite{bdgmpsww-rptn-14}. Our aim is to reduce this gap.\n\nThere are many natural query types in public transit. An \\emph{earliest arrival} query seeks a journey that arrives at a target stop \\targetStop as early as possible, given a source stop \\sourceStop and a departure time~(\\eg, ``now''). A \\emph{multicriteria} query also considers the number of transfers when traveling from \\sourceStop\\ to \\targetStop. A \\emph{profile} query reports all quickest journeys between two stops within a time range.\n\nThese problems can be approached by variants of Dijkstra's algorithm~\\cite{d-ntpcg-59} applied to a graph modeling the public transit network, with various techniques to handle time-dependency~\\cite{pswz-emtip-08}. In particular, the \\emph{time-expanded}~(TE) graph encodes time in the vertices, creating a vertex for every \\emph{event} (\\eg, a train departure or arrival at a stop at a specific time). Newer approaches, like CSA~\\cite{dpsw-isftr-13} and RAPTOR~\\cite{dpw-rbptr-14}, work directly on the timetable. Speedup techniques~\\cite{bdgmpsww-rptn-14} such as Transfer Patterns~\\cite{bceghrv-frvlp-10,bs-fbspt-14}, Timetable Contraction Hierarchies~\\cite{g-ctnrt-10}, and ACSA~\\cite{sw-csa-13} use preprocessing to create auxiliary data that is then used to accelerate queries.\n\nFor aperiodic timetables, the TE model yields a \\emph{directed acyclic graph}~(DAG), and several public transit query problems translate to reachability problems. Although these can be solved by simple graph searches, this is too slow for our application. Different methodologies exist to enable faster reachability computation~\\cite{chwf-tflat-13,jw-sfsro-13,ms-prcha-14,sabw-ferra-13,yaiy-fsrqg-13,ycz-grail-10,zlwx-rqldg-14}. In particular, the \\emph{2-hop labeling}~\\cite{chkz-rdqhl-03} scheme associates with each vertex two labels (forward and backward); reachability (or shortest-path distance) can be determined by intersecting the source's forward label and the target's backward label. On continental road networks, 2-hop labeling distance queries take less than a microsecond~\\cite{adgw-hhlsp-12}.\n\nIn this work, we adapt 2-hop labeling to public transit networks, improving query performance by orders of magnitude over previous methods, while keeping preprocessing time practical. Starting from the time-expanded graph model~(Section~\\ref{sec:basic}), we extend the labeling scheme by carefully exploiting properties of public transit networks (Section~\\ref{sec:leverage}). Besides earliest arrival and profile queries, we address multicriteria and location-to-location queries, as well as reporting the full journey description quickly~(Section~\\ref{sec:practical}). We validate our Public Transit Labeling~(PTL) algorithm by careful experimental evaluation on large metropolitan and national transit networks (Section~\\ref{sec:exp}), achieving queries within microseconds.\n\n\\section{Preliminaries} \\label{sec:prelim}\n\nLet~$G = (V,A)$ be a (weighted) \\emph{directed graph}, where~$V$ is the set of vertices and~$A$ the set of arcs. An arc between two vertices $\\avertex,\\bvertex \\in V$ is denoted by~$(\\avertex,\\bvertex)$. A \\emph{path} is a sequence of adjacent vertices. A vertex~\\bvertex is \\emph{reachable} from a vertex~\\avertex if there is a path from~\\avertex to \\bvertex. A \\emph{DAG} is a graph that is both directed and acyclic.\n\nWe consider \\emph{aperiodic} timetables, consisting of sets of stops~\\stopSet, events~\\eventSet, trips~\\tripSet, and footpaths~\\footSet. \\emph{Stops} are distinct locations where one can board a transit vehicle~(such as bus stops or subway platforms). \\emph{Events} are the scheduled departures and arrivals of vehicles. Each event~$\\event \\in \\eventSet$ has an associated stop~$\\eventStop(\\event)$ and time~$\\eventTime(\\event)$. Let $\\stopEventList{\\astop} = \\{\\event_0(\\astop),\\ldots,\\event_{k_\\astop}(\\astop)\\}$ be the list (ordered by time) of events at a stop~\\astop. We set~$\\eventTime(\\event_i(\\astop)) = -\\infty$ for $i < 0$, and~$\\eventTime(\\event_i(\\astop)) = \\infty$ for $i > k_\\astop$. For simplicity, we may drop the index of an event (as in~$\\event(\\astop) \\in \\stopEventList{\\astop}$) or its stop (as in $\\event \\in \\eventSet$). A \\emph{trip} is a sequence of events served by the same vehicle. A pair of a consecutive departure and arrival events of a trip is a \\emph{connection}. \\emph{Footpaths} model transfers between nearby stops, each with a predetermined walking duration.\n\nA journey planning algorithm outputs a set of \\emph{journeys}. A journey is a sequence of trips (each with a pair of pick-up and drop-off stops) and footpaths in the order of travel. Journeys can be measured according to several criteria, such as arrival time or number of transfers. A journey~\\ajourney \\emph{dominates} a journey~\\bjourney if and only if~\\ajourney is no worse in any criterion than~\\bjourney. If~$\\ajourney$ and~$\\bjourney$ are equal in all criteria, we break ties arbitrarily. A set of non-dominated journeys is called a \\emph{Pareto set}. Multicriteria Pareto optimization is NP-hard in general, but practical for natural criteria in public transit networks~\\cite{dpw-rbptr-14,dpsw-isftr-13,mw-op-06,pswz-emtip-08}. A journey is \\emph{tight} if there is no other journey between the same source and target that dominates it in terms of departure and arrival time, \\eg, that departs later and arrives earlier.\n\nGiven a timetable, stops~$\\sourceStop$ and~$\\targetStop$, and a departure time~$\\depTime$, the~\\emph{$(\\sourceStop,\\targetStop,\\depTime)$-earliest arrival}~(EA) problem asks for an~$\\sourceStop$--$\\targetStop$ journey that arrives at~\\targetStop as early as possible and departs at~\\sourceStop no earlier than~$\\depTime$. The~\\emph{$(\\sourceStop,\\targetStop)$-profile} problem asks for a Pareto set of all tight journeys between~$\\sourceStop$ and~$\\targetStop$ over the entire timetable period. Finally, the~\\emph{$(\\sourceStop,\\targetStop,\\depTime)$-multicriteria}~(MC) problem asks for a Pareto set of journeys departing at~$\\sourceStop$ no earlier than~$\\depTime$ and minimizing the criteria arrival time and number of transfers. We focus on computing the \\emph{values} of the associated optimization criteria of the journeys~(\\ie, departure time, arrival times, number of transfers), which is enough for many applications. Section~\\ref{sec:practical} discusses how the full journey description can be obtained with little overhead.\n\nOur algorithms are based on the 2-hop~labeling scheme for directed graphs~\\cite{chkz-rdqhl-03}. It associates with every vertex~\\bvertex a \\emph{forward label}~$\\flabel(\\bvertex)$ and a \\emph{backward label}~$\\rlabel(\\bvertex)$. In a \\emph{reachability labeling}, labels are subsets of~$V$, and vertices $\\avertex \\in \\flabel(\\bvertex) \\cup \\rlabel(\\bvertex)$ are \\emph{hubs} of \\bvertex. Every hub in $\\flabel(\\bvertex)$ must be reachable from $\\bvertex$, which in turn must be reachable by every hub in $\\rlabel(\\bvertex)$. In addition, labels must obey the \\emph{cover property}: for any pair of vertices~\\avertex and~\\bvertex, the intersection~$\\flabel(\\avertex) \\cap \\rlabel(\\bvertex)$ must contain at least one hub on a~\\avertex--\\bvertex~path~(if it exists). It follows from this definition that~$\\flabel(\\avertex) \\cap \\rlabel(\\bvertex) \\neq \\emptyset$ if and only if~\\bvertex is reachable from~\\avertex.\n\nIn a \\emph{shortest path labeling}, each hub~$\\avertex \\in \\flabel(\\bvertex)$ also keeps the associated distance~$\\dist(\\avertex, \\bvertex)$, or~$\\dist(\\bvertex,\\avertex)$ for backward labels, and the cover property requires $\\flabel(\\avertex) \\cap \\rlabel(\\bvertex)$ to contain at least one hub on a \\emph{shortest} \\mbox{\\avertex--\\bvertex~path}. If labels are kept sorted by hub ID, a \\emph{distance label query} efficiently computes $\\dist(\\avertex,\\bvertex)$ by a coordinated linear sweep over~$\\flabel(\\avertex)$ and $\\rlabel(\\bvertex)$, finding the hub~$\\cvertex \\in \\flabel(\\avertex) \\cap \\rlabel(\\bvertex)$ that minimizes $\\dist(\\avertex,\\cvertex) + \\dist(\\cvertex,\\bvertex)$. In contrast, a \\emph{reachability label query} can stop as soon as any matching hub is found.\n\nIn general, smaller labels lead to less space and faster queries. Many algorithms to compute labelings have been proposed~\\cite{adgw-hhlsp-12,aiy-f-13,chwf-tflat-13,jw-sfsro-13,yaiy-fsrqg-13,zlwx-rqldg-14}, often for restricted graph classes. We leverage (as a black box) the recent RXL algorithm~\\cite{dgpw-rdqmn-14}, which efficiently computes small shortest path labelings for a variety of graph classes at scale. It is a sampling-based greedy algorithm that builds labels one hub at a time, with priority to vertices that cover as many relevant paths as possible.\n\nDifferent approaches for transforming a timetable into a graph exist~(see~\\cite{pswz-emtip-08} for an overview). In this work, we focus on the \\emph{time-expanded model}. Since it uses scalar arc costs, it is a natural choice for adapting the labeling approach. In contrast, the \\emph{time-dependent model}~(another popular approach) associates functions with the arcs, which makes adaption more difficult.\n\n\\section{Basic Approach} \\label{sec:basic}\n\nWe build the time-expanded graph from the timetable as follows. We group all departure and arrival events by the stop where they occur. We sort all events at a stop by time, merging events that happen at the same stop and time. We then add a vertex for each unique event, a \\emph{waiting arc} between two consecutive events of the same stop, and a \\emph{connection arc} for each connection (between the corresponding departure and arrival event). The cost of arc $(\\avertex,\\bvertex)$ is $\\eventTime(\\bvertex) - \\eventTime(\\avertex)$, \\ie, the time difference of the corresponding events. To account for footpaths between two stops $a$ and $b$, we add, from each vertex at stop $a$, a \\emph{foot arc} to the first reachable vertex at~$b$ (based on walking time), and vice versa. As events and vertices are tightly coupled in this model, we use the terms interchangeably.\n\nAny label generation scheme~(we use RXL~\\cite{dgpw-rdqmn-14}) on the time-expanded graph creates two~(forward and backward) \\emph{event labels} for every vertex~(event), enabling \\emph{event-to-event queries}. For our application \\emph{reachability} labels~\\cite{yaiy-fsrqg-13}, which only store hubs~(without distances), suffice. First, since all arcs point to the future, time-expanded graphs are DAGs. Second, if an event \\event\\ is reachable from another event $\\event'$ (\\ie, $\\flabel(\\event') \\cap \\rlabel(\\event) \\neq \\emptyset$), we can compute the time to get from $\\event'$\\ to \\event\\ as $\\eventTime(\\event) - \\eventTime(\\event')$. In fact, \\emph{all} paths between two events have equal cost.\n\nIn practice, however, event-to-event queries are of limited use, as they require users to specify both departure \\emph{and} arrival times, one of which is usually unknown. Therefore, we discuss earliest arrival and profile queries, which \\emph{optimize} arrival time and are thus more meaningful. See Section~\\ref{sec:practical} for multicriteria queries.\n\n\\paragraph{Earliest Arrival Queries.} Given event labels, we answer an ($\\sourceStop,\\targetStop,\\depTime$)-EA query as follows. We first find the earliest event $\\event_i(\\sourceStop) \\in \\stopEventList{\\sourceStop}$ at the source stop~\\sourceStop that suits the departure time, \\ie, with $\\eventTime(\\event_i(\\sourceStop)) \\geq \\depTime$ and \\mbox{$\\eventTime(\\event_{i-1}(\\sourceStop)) < \\depTime$}. Next, we search at the target stop~\\targetStop for the earliest event $\\event_j(\\targetStop) \\in \\stopEventList{\\targetStop}$ that is reachable from~$\\event_i(\\sourceStop)$ by testing whether~$\\flabel(\\event_i(\\sourceStop)) \\cap \\rlabel(\\event_j(\\targetStop)) \\neq \\emptyset$ and \\mbox{$\\flabel(\\event_i(\\sourceStop)) \\cap \\rlabel(\\event_{j-1}(\\targetStop)) = \\emptyset$}. Then, $\\eventTime(\\event_j(\\targetStop))$ is the earliest arrival time. One could find $\\event_j(\\targetStop)$ using linear search (which is simple and cache-friendly), but binary search is faster in theory and in practice. To accelerate queries, we \\emph{prune} (skip) all events~$\\event(\\targetStop)$ with~$\\eventTime(\\event(\\targetStop)) < \\depTime$, since~$\\flabel(\\event_i(\\sourceStop)) \\cap \\rlabel(\\event(\\targetStop)) = \\emptyset$ always holds in such cases. Moreover, to avoid evaluating $\\flabel(\\event_i(\\sourceStop))$ multiple times, we use \\emph{hash-based queries}~\\cite{dgpw-rdqmn-14}: we first build a hash set of the hubs in $\\flabel(\\event_i(\\sourceStop))$, then check the reachability for an event $\\event(\\targetStop)$ by probing the hash with hubs $h \\in \\rlabel(\\event(\\targetStop))$.\n\n\\paragraph{Profile Queries.} To answer an ($\\sourceStop,\\targetStop$)-profile query, we perform a coordinated sweep over the events at $\\sourceStop$ and $\\targetStop$. For the current event~$\\event_i(\\sourceStop) \\in \\stopEventList{\\sourceStop}$ at the source stop~(initialized to the earliest event~$\\event_0(\\sourceStop) \\in \\stopEventList{\\sourceStop}$), we find the first event~\\mbox{$\\event_j(\\targetStop) \\in \\stopEventList{\\targetStop}$} at the target stop that is reachable, \\ie, such that \\mbox{$\\flabel(\\event_i(\\sourceStop)) \\cap \\rlabel(\\event_j(\\targetStop)) \\neq \\emptyset$} and~$\\flabel(\\event_i(\\sourceStop)) \\cap \\rlabel(\\event_{j-1}(\\targetStop)) = \\emptyset$. This gives us the earliest arrival time~$\\eventTime(\\event_j(\\targetStop))$. To identify the latest departure time from \\sourceStop\\ for that earliest arrival event (and thus have a tight journey), we increase $i$ until~$\\flabel(\\event_{i}(\\sourceStop)) \\cap \\rlabel(\\event_j(\\targetStop)) = \\emptyset$, then add~$(\\eventTime(\\event_{i-1}(\\sourceStop)),\\eventTime(\\event_j(\\targetStop)))$ to the profile. We repeat the process starting from the events~$\\event_{i}(\\sourceStop)$ and~$\\event_{j+1}(\\targetStop)$. Since we increase either~$i$ or~$j$ after each intersection test, the worst-case time to find all tight journeys is linear in the number of events~(at~$\\sourceStop$ and~$\\targetStop$) multiplied by the size of their largest label.\n\n\\section{Leveraging Public Transit} \\label{sec:leverage} Our approach can be refined to exploit features specific to public transit networks. As described so far, our labeling scheme maintains reachability information for \\emph{all pairs} of events (by covering all paths of the time-expanded graph, breaking ties arbitrarily). However, in public transit networks we actually are only interested in \\emph{certain paths}. In particular, the labeling does \\emph{not} need to cover any path ending at a departure event (or beginning at an arrival event). We can thus discard forward labels from arrival events and backward labels from departure events.\n\n\\paragraph{Trimmed Event Labels.} Moreover, we can disregard paths representing dominated journeys that depart earlier and arrive later than others (\\ie, journeys that are not tight, \\cf~Section~\\ref{sec:prelim}). Consider all departure events of a stop. If a certain hub is reachable from event~$\\event_i(\\sourceStop)$, then it is also reachable from $\\event_0(\\sourceStop), \\ldots, \\event_{i-1}(\\sourceStop)$, and is thus potentially added to the forward labels of all these earlier events. In fact, experiments show that on average the same hub is added to 1.8--5.0 events per stop~(depending on the network). We therefore compute \\emph{trimmed event labels} by discarding all but the latest occurrence of each hub from the forward labels. Similarly, we only keep the earliest occurrence of each hub in the backward labels. (Preliminary experiments have shown that we obtain very similar label sizes with a much slower algorithm that greedily covers tight journeys explicitly~\\cite{adgw-hhlsp-12,dgpw-rdqmn-14}.)\n\nUnfortunately, we can no longer just apply the query algorithms from Section~\\ref{sec:basic} with trimmed event labels: if the selected departure event at~$\\sourceStop$ does not correspond to a tight journey toward~$\\targetStop$, the algorithm will not find a solution~(though one might exist). One could circumvent this issue by also running the algorithm from subsequent departure events at~$\\sourceStop$, which however may lead to quadratic query complexity in the worst case~(for both EA and profile queries).\n\n\\paragraph{Stop Labels.} We solve this problem by working with \\emph{stop labels}: For each stop~\\astop, we merge all forward event labels $\\flabel(\\event_0(\\astop)), \\ldots, \\flabel(\\event_k(\\astop))$ into a forward stop label \\fslabel(\\astop), and all backward event labels into a backward stop label \\rslabel(\\astop). Similar to distance labels, each stop label~$\\stoplabel(\\astop)$ is a list of pairs~$(\\ahub, \\atime_\\astop(\\ahub))$, each containing a hub and a time, sorted by hub. For a forward label, $\\atime_\\astop(\\ahub)$ encodes the latest departure time from~\\astop to reach hub~\\ahub. More precisely, let $h$ be a hub in an event label~$\\flabel(\\event_i(\\astop))$: we add the pair~$(\\ahub, \\eventTime(\\event_i(\\astop)))$ to the stop label \\fslabel(\\astop) only if $\\ahub \\notin \\flabel(\\event_j(\\astop)), j>i$, \\ie, only if \\ahub does not appear in the label of another event with a later departure time at the stop. Analogously, for backward stop labels, $\\atime_\\astop(\\ahub)$ encodes the earliest arrival time at $p$ from~$h$.\n\nBy restricting ourselves to these entries, we effectively discard dominated~(non-tight) journeys to these hubs. It is easy to see that these stop labels obey a \\emph{tight journey cover property}: for each pair of stops \\sourceStop and \\targetStop, $\\fslabel(\\sourceStop) \\cap \\rslabel(\\targetStop)$ contains at least one hub on each tight journey between them (or any equivalent journey that departs and arrives at the same time; recall from Section~\\ref{sec:prelim} that we allow arbitrary tie-breaking). This property does \\emph{not}, however, imply that the label intersection \\emph{only} contains tight journeys: for example, $\\fslabel(\\sourceStop)$ and $\\rslabel(\\targetStop)$ could share a hub that is important for long distance travel, but not to get from~\\sourceStop~to~\\targetStop. The remainder of this section discusses how we handle this fact during queries.\n\n\\paragraph{Stop Label Profile Queries.} To run an (\\sourceStop,\\targetStop)-profile query on stop labels, we perform a coordinated sweep over both labels~\\fslabel(\\sourceStop) and \\rslabel(\\targetStop). For every matching hub~$\\ahub$, \\ie, $(\\ahub, \\atime_\\sourceStop(\\ahub)) \\in \\fslabel(\\sourceStop)$ and $(\\ahub, \\atime_\\targetStop(\\ahub)) \\in \\rslabel(\\targetStop)$, we consider the journey induced by~$(\\atime_\\sourceStop(\\ahub), \\atime_\\targetStop(\\ahub))$ for output. However, since we are only interested in reporting tight journeys, we maintain~(during the algorithm) a tentative set of tight journeys, removing dominated journeys from it on-the-fly. (We found this to be faster than adding all journeys during the sweep and only discarding dominated journeys at the end.) We can further improve the efficiency of this approach in practice by (globally) reassigning hub IDs by the time of day. Note that every hub~$\\ahub$ of a stop label is still also an event and carries an event time~$\\eventTime(\\ahub)$. (Not to be confused with $\\atime_\\sourceStop(\\ahub)$ and $\\atime_\\targetStop(\\ahub)$.) We assign sequential IDs to all hubs~$\\ahub$ in order of increasing $\\eventTime(\\ahub)$, thus ensuring that hubs in the label intersection are enumerated chronologically. Note that this does not imply that journeys are enumerated in order of departure or arrival time, since each hub $\\ahub$ may appear anywhere along its associated journey. However, preliminary experiments have shown that this approach leads to fewer insertions into the tentative set of tight journeys, reducing query time. Moreover, as in shortest path labels~\\cite{dgpw-rdqmn-14}, we improve cache efficiency by storing the values for hubs and times separately in a stop label, accessing times only for matching hubs.\n\nOverall, stop and event labels have different trade-offs: maintaining the profile requires less effort with event labels~(any discovered journey is already tight), but fewer hubs are scanned with stop labels~(there are no duplicate hubs).\n\n\\paragraph{Stop Label Earliest Arrival Queries.} Reassigned hub IDs also enable fast $(\\sourceStop,\\targetStop,\\depTime)$-EA queries. We use binary search in $\\fslabel(\\sourceStop)$ and $\\rslabel(\\targetStop)$ to find the earliest relevant hub~\\ahub, \\ie, with $\\eventTime(\\ahub) \\geq \\depTime$. From there, we perform a linear coordinated sweep as in the profile query, finding $(\\ahub, \\atime_\\sourceStop(\\ahub)) \\in \\fslabel(\\sourceStop)$ and $(\\ahub, \\atime_\\targetStop(\\ahub)) \\in \\rslabel(\\targetStop)$. However, instead of maintaining tentative profile entries~$(\\atime_\\sourceStop(\\ahub), \\atime_\\targetStop(\\ahub))$, we ignore solutions that depart too early~(\\ie, $\\atime_\\sourceStop(\\ahub) < \\depTime$), while picking the hub~$\\ahub^*$ that minimizes the tentative best arrival time~$\\atime_\\targetStop(\\ahub^*)$. (Note that $\\eventTime(\\ahub) \\geq \\depTime$ does not imply $\\atime_\\sourceStop(\\ahub) \\geq \\depTime$.) Once we scan a hub~$h$ with~\\mbox{$\\eventTime(\\ahub) \\geq \\atime_\\targetStop(\\ahub^*)$}, the tentative best arrival time cannot be improved anymore, and we stop the query. For practical performance, \\emph{pruning} the scan, so that we only sweep hubs~\\ahub between $\\depTime \\leq \\atime(\\ahub) \\leq \\atime_\\targetStop(\\ahub^*)$, is very important.\n\n\\section{Practical Extensions} \\label{sec:practical}\n\nSo far, we presented stop-to-stop queries, which report the departure and arrival times of the quickest journey(s). In this section, we address multicriteria queries, general location-to-location requests, and obtaining detailed journey descriptions.\n\n\\paragraph{Multicriteria Optimization and Minimum Transfer Time.} Besides optimizing arrival time, many users also prefer journeys with fewer transfers. To solve the underlying multicriteria optimization problem, we adapt our labeling approach by (1)~encoding transfers as arc costs in the graph, (2)~computing shortest path labels based on these costs (instead of reachability labels on an unweighted graph), and (3)~adjusting the query algorithm to find the Pareto set of solutions.\n\nReconsider the earliest arrival graph from Section~\\ref{sec:basic}. As before, we add a vertex for each unique event, linking consecutive events at the same stop with waiting arcs of cost~0. However, each connection arc~$(u,w)$ in the graph is subdivided by an intermediate \\emph{connection vertex}~$v$, setting the cost of arc~$(u,v)$ to 0 and the cost of arc~$(v,w)$ to 1. By interpreting costs of 1 as leaving a vehicle, we can count the number of trips taken along any path. To model staying in the vehicle, consecutive connection vertices of the same trip are linked by zero-cost arcs.\n\nA shortest path labeling on this graph now encodes the number of transfers as the shortest path distance between two events, while the duration of the journey can still be deduced from the time difference of the events. Consider a fixed source event~$\\event(\\sourceStop)$ and the arrival events of a target stop~$\\event_0(\\targetStop), \\event_1(\\targetStop), \\ldots$ in order of increasing time. The minimum number of transfers required to reach the target stop~$\\targetStop$ never increases with arrival times. (Hence, the whole Pareto set~$P$ of multicriteria solutions can be computed with a single Dijkstra run~\\cite{pswz-emtip-08}.)\n\nWe exploit this property to compute~$(\\sourceStop,\\targetStop,\\depTime)$-EA multicriteria~(MC) queries from the labels as follows. We initialize~$P$ as the empty set. We then perform an~$(\\sourceStop,\\targetStop,\\depTime)$-EA query~(with all optimizations described in Section~\\ref{sec:basic}) to compute the \\emph{fastest} journey in the solution, \\ie, the one with most transfers. We add this journey to~$P$. We then check~(by performing distance label queries) for each subsequent event at~$\\targetStop$ whether there is a journey with fewer transfers~(than the most recently added entry of~$P$), in which case we add the journey to~$P$ and repeat. The MC~query ends once the last event at the target stop has been processed. We can stop earlier with the following optimization: we first run a distance label query on the \\emph{last} event at~\\targetStop\\ to obtain the \\emph{smallest} possible number of transfers to travel from~$\\sourceStop$ to~$\\targetStop$. We may then already stop the MC~query once we add a journey to~$P$ with this many transfers. Note that, since we do not need to check for domination in~$P$ explicitly, our algorithm maintains~$P$ in constant time per added journey.\n\n\\paragraph{Minimum Transfer Times.} Transit agencies often model an entire station with multiple platforms as a single stop and account for the time required to change trips inside the station by associating a \\emph{minimum transfer time}~$\\mtt(\\astop)$ with each stop~$\\astop$. To incorporate them into the EA graph, we first locally replace each affected stop~$\\astop$ by a \\emph{set} of new stops~$\\superstop$, distributing \\emph{conflicting} trips~(between which transferring is impossible due to~$\\mtt(\\astop)$) to different stops of~$\\superstop$. We then add footpaths between all pairs of stops in~$\\superstop$ with length~$\\mtt(\\astop)$. A small set~$\\superstop$ can be computed by solving an appropriate coloring problem~\\cite{dkp-pcbcp-12}. For the MC graph, we need not change the input. Instead, it is sufficient to \\emph{shift} each arrival event~$\\event \\in \\stopEventList{\\astop}$ by adding~$\\mtt(\\astop)$ to~$\\eventTime(\\event)$ before creating the vertices.\n\n\\paragraph{Location-to-Location Queries.}\n\nA query between arbitrary locations~$\\sourceLocation$ and~$\\targetLocation$, which may employ walking or driving as the first and last legs of the journey, can be handled by a two-stage approach. It first computes sets~$\\sourceSet$ and~$\\targetSet$ of relevant stops near the origin~$\\sourceLocation$ and destination~$\\targetLocation$ that can be reached by car or on foot. With that information, a \\emph{forward superlabel}~\\cite{adfgw-hldbl-12} is built from all forward stop labels associated with \\sourceSet. For each entry $(\\ahub, \\atime_\\astop(\\ahub)) \\in \\fslabel(\\astop)$ in the label of stop~$\\astop \\in \\sourceSet$, we adjust the departure time~$\\atime_\\sourceLocation(\\ahub) = \\atime_\\astop(\\ahub) - \\dist(\\sourceLocation, \\astop)$ so that the journey starts at \\sourceLocation and add $(\\ahub, \\atime_\\sourceLocation(\\ahub))$ to the superlabel. For duplicate hubs that occur in multiple stop labels, we keep only the latest departure time from \\sourceLocation. This can be achieved with a coordinated sweep, always adding the next hub of minimum ID. A \\emph{backward superlabel} (for $\\targetSet$) is built analogously. For location-to-location queries, we then simply run our stop-label-based EA and profile query algorithms using the superlabels. In practice, we need not build superlabels explicitly but can simulate the building sweep during the query~(which in itself is a coordinated sweep over two labels). A similar approach is possible for event labels. Moreover, point-of-interest queries~(such as finding the closest restaurants to a given location) can be computed by applying known techniques~\\cite{adfgw-hldbl-12} to these superlabels.\n\n\\paragraph{Journey Descriptions.} While for many applications it suffices to report departure and arrival times (and possibly the number of transfers) per journey, sometimes a more detailed description is needed. We could apply known path unpacking techniques~\\cite{adfgw-hldbl-12} to retrieve the full sequence of connections~(and transfers), but in public transit it is usually enough to report the list of trips with associated transfer stops. We can accomplish that by storing with each hub the sequences of trips~(and transfer stops) for travel between the hub and its label vertex.\n\n\\section{Experiments} \\label{sec:exp}\n\n\\paragraph{Setup.}\n\nWe implemented all algorithms in C++ using Visual Studio 2013 with full optimization. All experiments were conducted on a machine with two 8-core Intel Xeon E5-2690 CPUs and 384\\,GiB of DDR3-1066 RAM, running Windows 2008R2 Server. All runs are \\emph{sequential}. We use at most 32 bits for distances.\n\n\\begin{table} \\centering \\caption{\\label{tab:sizes}Size of timetables and the earliest arrival~(EA) and multicriteria~(MC) graphs.} \\setlength{\\tabcolsep}{0.9ex} \\begin{tabular}{@{}lrrrrrrrrr@{}} \\toprule &&&&&& \\multicolumn{2}{c}{EA Graph} & \\multicolumn{2}{c}{MC Graph} \\\\ \\cmidrule(lr){7-8}\\cmidrule(l){9-10} Instance & Stops & Conns & Trips & Footp. & Dy. & $|V|$ & $|A|$ & $|V|$ & $|A|$ \\\\ \\midrule London&20.8\\,k&5,133\\,k&133\\,k&45.7\\,k&1&4,719\\,k&51,043\\,k&9,852\\,k&72,162\\,k\\\\ Madrid&4.7\\,k&4,527\\,k&165\\,k&1.3\\,k&1&3,003\\,k&13,730\\,k&7,530\\,k&34,505\\,k\\\\ Sweden&51.1\\,k&12,657\\,k&548\\,k&1.1\\,k&2&8,151\\,k&34,806\\,k&20,808\\,k&93,194\\,k\\\\ Switzerland&27.1\\,k&23,706\\,k&2,198\\,k&29.8\\,k&2&7,979\\,k&49,656\\,k&31,685\\,k&170,503\\,k\\\\ \\bottomrule \\end{tabular} \\end{table} We consider four realistic inputs: the metropolitan networks of London (\\url{data.london.gov.uk}) and Madrid (\\url{emtmadrid.es}), and the national networks of Sweden (\\url{trafiklab.se}) and Switzerland (\\url{gtfs.geops.ch}). London includes all modes of transport, Madrid contains only buses, and the national networks contain both long-distance and local transit. We consider 24-hour timetables for the metropolitan networks, and two days for national ones (to enable overnight journeys). Footpaths were generated using a known heuristic~\\cite{dkp-pcbcp-12} for Madrid; they are part of the input for the other networks. See~\\tablename~\\ref{tab:sizes} for size figures of the timetables and resulting graphs. The average number of unique events per stop ranges from 160 for Sweden to 644 for Madrid. (Recall from Section~\\ref{sec:basic} that we merge all coincident events at a stop.) Note that no two instances dominate each other~(\\wrt number of stops, connections, trips, events per stop, and footpaths).\n\n\\paragraph{Preprocessing.}\n\n\\begin{table}[b] \\setlength{\\tabcolsep}{0.9ex} \\centering \\caption{Preprocessing figures. Label sizes are averages of forward and backward labels.} \\label{tab:prepro} \\begin{tabular}{@{}lrrrrrrrrrr@{}} \\toprule & \\multicolumn{6}{c}{\\textbf{Earliest Arrival}} & \\multicolumn{4}{c}{\\textbf{Multicriteria}}\\\\ \\cmidrule(lr){2-7}\\cmidrule(l){8-11} & & \\multicolumn{3}{c}{Event Labels} & \\multicolumn{2}{c}{Stop Labels} & & \\multicolumn{3}{c}{Event Labels} \\\\ \\cmidrule(lr){3-5}\\cmidrule(lr){6-7}\\cmidrule(l){9-11} & RXL & Hubs & Hubs & Space & Hubs & Space & RXL & Hubs & Hubs & Space \\\\ Instance & [h:m] & p.\\,lbl & p.\\,stop & [MiB] & p.\\,stop & [MiB] & [h:m] & p.\\,lbl & p.\\,stop & [MiB]\\\\ \\midrule London&0:54&70&15,480&1,334&7,075&1,257&49:19&734&162,565&26,871\\\\ Madrid&0:25&77&49,247&963&9,830&403&10:55&404&258,008&10,155\\\\ Sweden&0:32&37&5,630&1,226&1,536&700&36:14&190&29,046&12,637\\\\ Switzerland&0:42&42&11,189&1,282&2,970&708&61:36&216&58,022&12,983\\\\ \\bottomrule \\end{tabular} \\end{table}\n\n\\tablename~\\ref{tab:prepro} reports preprocessing figures for the unweighted earliest arrival graph~(which also enables profile queries) and the multicriteria graph. For earliest arrival~(EA), preprocessing takes well below an hour and generates about one gigabyte, which is quite practical. Although there are only 37--70 hubs per label, the total number of hubs per stop (\\ie, the combined size of all labels) is quite large~(5,630--49,247). By eliminating redundancy (\\cf~Section~\\ref{sec:leverage}), stop labels have only a fifth as many hubs (for Madrid). Even though they need to store an additional distance value per hub, total space usage is still smaller. In general, \\emph{average} labels sizes (though not total space) are higher for metropolitan instances. This correlates with the higher number of daily journeys in these networks.\n\nPreprocessing the multicriteria~(MC) graph is much more expensive: times increase by a factor of~26.2--54.8 for the metropolitan and~67.9--88 for the national networks. On Madrid, Sweden, and Switzerland labels are five times larger compared to EA, and on London the factor is even more than ten. This is immediately reflected in the space consumption, which is up to 26\\,GiB~(London).\n\n\\paragraph{Queries.}\n\nWe now evaluate query performance. For each algorithm, we ran~100,000 queries between random source and target stops, at random departure times between~0:00 and~23:59~(of the first day). \\tablename~\\ref{tab:ea-queries} reports detailed figures, organized in three blocks: event label EA queries, stop label EA queries, and profile queries (with both event and stop labels). We discuss MC queries later.\n\nWe observe that event labels result in extremely fast EA queries (6.9--14.7\\,\\musec), even without optimizations. As expected, pruning and hashing reduce the number of accesses to labels and hubs~(see columns ``Lbls.'' and ``Hubs''). Although binary search cannot stop as soon as a matching hub is found~(see the ``='' column), it accesses fewer labels and hubs, achieving query times below~3\\,\\musec\\ on all instances.\n\nUsing stop labels~(\\cf~Section~\\ref{sec:leverage}) in their basic form is significantly slower than using event labels. With pruning enabled, however, query times (3.6--6.2\\,\\musec) are within a factor of two of the event labels, while saving a factor of 1.1--2.4 in space. For profile queries, stop labels are clearly the best approach. It scans up to a factor of 5.1 fewer hubs and is up to 3.3 times faster, computing the profile of the full timetable period in under 80\\,\\musec~on all instances. The difference in factors is due to the overhead of maintaining the Pareto set during the stop label query.\n\n\\begin{table} \\setlength{\\tabcolsep}{0.7ex} \\centering \\caption{Evaluating earliest arrival queries. 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Work done mostly while all authors were at Microsoft Research Silicon Valley.}}\\author[1]{Daniel Delling}\\author[2]{Julian Dibbelt}\\author[3]{Thomas Pajor}\\author[4]{Renato~F.~Werneck}\\affil[1]{Sunnyvale, USA, \\email{daniel.delling@gmail.com}}\\affil[2]{Karlsruhe Institute of Technology, Germany, \\email{dibbelt@kit.edu}}\\affil[3]{Microsoft Research, USA, \\email{tpajor@microsoft.com}}\\affil[4]{San Francisco, USA, \\email{rwerneck@acm.org}}\\date{May 5, 2015}\\begin{document&0.5&12.7&0.0121\\\\ PTL&Switzerland&27.1&23.7&2&\\ensuremath{\\bullet}}\\newcommand{\\disabled}{\\ensuremath{\\circ}}\\newcommand{\\feature}[1]{\\begin{rotate}{60}\\hspace{-.33ex}#1\\end{rotate}}\\newcommand{\\Xcomment}[1]{}\\newcommand{\\tabhead}{\\sc}\\newcommand{\\musec}{\\textmu{}s\\xspace}\\newcommand{\\wrt}{w.\\,r.\\,t.\\xspace}\\newcommand{\\eg}{e.\\,g.\\xspace}\\newcommand{\\ie}{i.\\,e.\\xspace}\\newcommand{\\cf}{cf.\\xspace}\\newcommand{\\depTime}{\\ensuremath{\\tau}\\xspace}\\newcommand{\\astop}{\\ensuremath{p}\\xspace}\\newcommand{\\superstop}{\\ensuremath{p^*}\\xspace}\\newcommand{\\sourceStop}{\\ensuremath{s}\\xspace}\\newcommand{\\targetStop}{\\ensuremath{t}\\xspace}\\newcommand{\\sourceSet}{\\ensuremath{\\cal S}\\xspace}\\newcommand{\\targetSet}{\\ensuremath{\\cal T}\\xspace}\\newcommand{\\sourceLocation}{\\ensuremath{s^*}\\xspace}\\newcommand{\\targetLocation}{\\ensuremath{t^*}\\xspace}\\newcommand{\\event}{\\ensuremath{e}}\\newcommand{\\lab}{\\ensuremath{L}}\\newcommand{\\flabel}{\\ensuremath{\\lab_f}}\\newcommand{\\rlabel}{\\ensuremath{\\lab_b}}\\newcommand{\\stoplabel}{\\ensuremath{SL}}\\newcommand{\\fslabel}{\\ensuremath{\\stoplabel_f}}\\newcommand{\\rslabel}{\\ensuremath{\\stoplabel_b}}\\newcommand{\\reachLabelTrue}{\\ensuremath{\\texttt{reach}_\\lab}}\\newcommand{\\reachLabelFalse}{\\ensuremath{\\texttt{!reach}_\\lab}}\\newcommand{\\eventTime}{\\ensuremath{\\texttt{time}}}\\newcommand{\\eventStop}{\\ensuremath{\\texttt{stop}}}\\newcommand{\\stopSet}{\\ensuremath{S}}\\newcommand{\\routeSet}{\\ensuremath{R}}\\newcommand{\\tripSet}{\\ensuremath{T}}\\newcommand{\\footSet}{\\ensuremath{F}}\\newcommand{\\eventSet}{\\ensuremath{E}}\\newcommand{\\conn}{\\ensuremath{c}}\\newcommand{\\journey}{\\ensuremath{j}}\\newcommand{\\ajourney}{\\ensuremath{\\journey_1}\\xspace}\\newcommand{\\bjourney}{\\ensuremath{\\journey_2}\\xspace}\\newcommand{\\stopEventList}[1]{\\ensuremath{E(#1)}}\\newcommand{\\avertex}{\\ensuremath{u}\\xspace}\\newcommand{\\bvertex}{\\ensuremath{v}\\xspace}\\newcommand{\\cvertex}{\\ensuremath{w}\\xspace}\\newcommand{\\ahub}{\\ensuremath{h}\\xspace}\\newcommand{\\atime}{\\eventTime\\xspace}\\DeclareMathOperator{\\dist}{dist}\\DeclareMathOperator{\\rank}{rank}\\DeclareMathOperator{\\mtt}{mtt}\\def\\comment#1{}\\def\\comment}\\def\\withcomments{\\newcounter{mycommentcounter}\\def\\comment##1{\\refstepcounter{mycommentcounter}\\ifhmode\\unskip{\\dimen1=\\baselineskip \\divide\\dimen1 by 2 \\raise\\dimen1\\llap{\\tiny\\bfseries \\textcolor{red}{-\\themycommentcounter-}}}\\fi\\marginpar[{\\renewcommand{\\baselinestretch}{0.8}\\hspace*{-1em}\\begin{minipage}{10em}\\footnotesize [\\themycommentcounter] \\raggedright ##1\\end{minipage}}]{\\renewcommand{\\baselinestretch}{0.8}\\begin{minipage}{10em}\\footnotesize [\\themycommentcounter]: \\raggedright ##1\\end{minipage}}}}\\renewcommand\\Affilfont{\\small}\\renewcommand\\Authands{ and }\\newcommand{\\email}[1]{\\texttt{#1}}\\title{Public Transit Labeling\\thanks{An extended abstract of this paper has been accepted at the 14th International Symposium on Experimental Algorithms~(SEA'15). Work done mostly while all authors were at Microsoft Research Silicon Valley.}}\\author[1]{Daniel Delling}\\author[2]{Julian Dibbelt}\\author[3]{Thomas Pajor}\\author[4]{Renato~F.~Werneck}\\affil[1]{Sunnyvale, USA, \\email{daniel.delling@gmail.com}}\\affil[2]{Karlsruhe Institute of Technology, Germany, \\email{dibbelt@kit.edu}}\\affil[3]{Microsoft Research, USA, \\email{tpajor@microsoft.com}}\\affil[4]{San Francisco, USA, \\email{rwerneck@acm.org}}\\date{May 5, 2015}\\begin{document{\\comment}\\def\\withcomments{\\newcounter{mycommentcounter}\\def\\comment##1{\\refstepcounter{mycommentcounter}\\ifhmode\\unskip{\\dimen1=\\baselineskip \\divide\\dimen1 by 2 \\raise\\dimen1\\llap{\\tiny\\bfseries \\textcolor{red}{-\\themycommentcounter-}}}\\fi\\marginpar[{\\renewcommand{\\baselinestretch}{0.8}\\hspace*{-1em}\\begin{minipage}{10em}\\footnotesize [\\themycommentcounter] \\raggedright ##1\\end{minipage}}]{\\renewcommand{\\baselinestretch}{0.8}\\begin{minipage}{10em}\\footnotesize [\\themycommentcounter]: \\raggedright ##1\\end{minipage}}}}\\renewcommand\\Affilfont{\\small}\\renewcommand\\Authands{ and }\\newcommand{\\email}[1]{\\texttt{#1}}\\title{Public Transit Labeling\\thanks{An extended abstract of this paper has been accepted at the 14th International Symposium on Experimental Algorithms~(SEA'15). Work done mostly while all authors were at Microsoft Research Silicon Valley.}}\\author[1]{Daniel Delling}\\author[2]{Julian Dibbelt}\\author[3]{Thomas Pajor}\\author[4]{Renato~F.~Werneck}\\affil[1]{Sunnyvale, USA, \\email{daniel.delling@gmail.com}}\\affil[2]{Karlsruhe Institute of Technology, Germany, \\email{dibbelt@kit.edu}}\\affil[3]{Microsoft Research, USA, \\email{tpajor@microsoft.com}}\\affil[4]{San Francisco, USA, \\email{rwerneck@acm.org}}\\date{May 5, 2015}\\begin{document&\\disabled&\\ensuremath{\\bullet}}\\newcommand{\\disabled}{\\ensuremath{\\circ}}\\newcommand{\\feature}[1]{\\begin{rotate}{60}\\hspace{-.33ex}#1\\end{rotate}}\\newcommand{\\Xcomment}[1]{}\\newcommand{\\tabhead}{\\sc}\\newcommand{\\musec}{\\textmu{}s\\xspace}\\newcommand{\\wrt}{w.\\,r.\\,t.\\xspace}\\newcommand{\\eg}{e.\\,g.\\xspace}\\newcommand{\\ie}{i.\\,e.\\xspace}\\newcommand{\\cf}{cf.\\xspace}\\newcommand{\\depTime}{\\ensuremath{\\tau}\\xspace}\\newcommand{\\astop}{\\ensuremath{p}\\xspace}\\newcommand{\\superstop}{\\ensuremath{p^*}\\xspace}\\newcommand{\\sourceStop}{\\ensuremath{s}\\xspace}\\newcommand{\\targetStop}{\\ensuremath{t}\\xspace}\\newcommand{\\sourceSet}{\\ensuremath{\\cal S}\\xspace}\\newcommand{\\targetSet}{\\ensuremath{\\cal T}\\xspace}\\newcommand{\\sourceLocation}{\\ensuremath{s^*}\\xspace}\\newcommand{\\targetLocation}{\\ensuremath{t^*}\\xspace}\\newcommand{\\event}{\\ensuremath{e}}\\newcommand{\\lab}{\\ensuremath{L}}\\newcommand{\\flabel}{\\ensuremath{\\lab_f}}\\newcommand{\\rlabel}{\\ensuremath{\\lab_b}}\\newcommand{\\stoplabel}{\\ensuremath{SL}}\\newcommand{\\fslabel}{\\ensuremath{\\stoplabel_f}}\\newcommand{\\rslabel}{\\ensuremath{\\stoplabel_b}}\\newcommand{\\reachLabelTrue}{\\ensuremath{\\texttt{reach}_\\lab}}\\newcommand{\\reachLabelFalse}{\\ensuremath{\\texttt{!reach}_\\lab}}\\newcommand{\\eventTime}{\\ensuremath{\\texttt{time}}}\\newcommand{\\eventStop}{\\ensuremath{\\texttt{stop}}}\\newcommand{\\stopSet}{\\ensuremath{S}}\\newcommand{\\routeSet}{\\ensuremath{R}}\\newcommand{\\tripSet}{\\ensuremath{T}}\\newcommand{\\footSet}{\\ensuremath{F}}\\newcommand{\\eventSet}{\\ensuremath{E}}\\newcommand{\\conn}{\\ensuremath{c}}\\newcommand{\\journey}{\\ensuremath{j}}\\newcommand{\\ajourney}{\\ensuremath{\\journey_1}\\xspace}\\newcommand{\\bjourney}{\\ensuremath{\\journey_2}\\xspace}\\newcommand{\\stopEventList}[1]{\\ensuremath{E(#1)}}\\newcommand{\\avertex}{\\ensuremath{u}\\xspace}\\newcommand{\\bvertex}{\\ensuremath{v}\\xspace}\\newcommand{\\cvertex}{\\ensuremath{w}\\xspace}\\newcommand{\\ahub}{\\ensuremath{h}\\xspace}\\newcommand{\\atime}{\\eventTime\\xspace}\\DeclareMathOperator{\\dist}{dist}\\DeclareMathOperator{\\rank}{rank}\\DeclareMathOperator{\\mtt}{mtt}\\def\\comment#1{}\\def\\comment}\\def\\withcomments{\\newcounter{mycommentcounter}\\def\\comment##1{\\refstepcounter{mycommentcounter}\\ifhmode\\unskip{\\dimen1=\\baselineskip \\divide\\dimen1 by 2 \\raise\\dimen1\\llap{\\tiny\\bfseries \\textcolor{red}{-\\themycommentcounter-}}}\\fi\\marginpar[{\\renewcommand{\\baselinestretch}{0.8}\\hspace*{-1em}\\begin{minipage}{10em}\\footnotesize [\\themycommentcounter] \\raggedright ##1\\end{minipage}}]{\\renewcommand{\\baselinestretch}{0.8}\\begin{minipage}{10em}\\footnotesize [\\themycommentcounter]: \\raggedright ##1\\end{minipage}}}}\\renewcommand\\Affilfont{\\small}\\renewcommand\\Authands{ and }\\newcommand{\\email}[1]{\\texttt{#1}}\\title{Public Transit Labeling\\thanks{An extended abstract of this paper has been accepted at the 14th International Symposium on Experimental Algorithms~(SEA'15). Work done mostly while all authors were at Microsoft Research Silicon Valley.}}\\author[1]{Daniel Delling}\\author[2]{Julian Dibbelt}\\author[3]{Thomas Pajor}\\author[4]{Renato~F.~Werneck}\\affil[1]{Sunnyvale, USA, \\email{daniel.delling@gmail.com}}\\affil[2]{Karlsruhe Institute of Technology, Germany, \\email{dibbelt@kit.edu}}\\affil[3]{Microsoft Research, USA, \\email{tpajor@microsoft.com}}\\affil[4]{San Francisco, USA, \\email{rwerneck@acm.org}}\\date{May 5, 2015}\\begin{document{\\comment}\\def\\withcomments{\\newcounter{mycommentcounter}\\def\\comment##1{\\refstepcounter{mycommentcounter}\\ifhmode\\unskip{\\dimen1=\\baselineskip \\divide\\dimen1 by 2 \\raise\\dimen1\\llap{\\tiny\\bfseries \\textcolor{red}{-\\themycommentcounter-}}}\\fi\\marginpar[{\\renewcommand{\\baselinestretch}{0.8}\\hspace*{-1em}\\begin{minipage}{10em}\\footnotesize [\\themycommentcounter] \\raggedright ##1\\end{minipage}}]{\\renewcommand{\\baselinestretch}{0.8}\\begin{minipage}{10em}\\footnotesize [\\themycommentcounter]: \\raggedright ##1\\end{minipage}}}}\\renewcommand\\Affilfont{\\small}\\renewcommand\\Authands{ and }\\newcommand{\\email}[1]{\\texttt{#1}}\\title{Public Transit Labeling\\thanks{An extended abstract of this paper has been accepted at the 14th International Symposium on Experimental Algorithms~(SEA'15). Work done mostly while all authors were at Microsoft Research Silicon Valley.}}\\author[1]{Daniel Delling}\\author[2]{Julian Dibbelt}\\author[3]{Thomas Pajor}\\author[4]{Renato~F.~Werneck}\\affil[1]{Sunnyvale, USA, \\email{daniel.delling@gmail.com}}\\affil[2]{Karlsruhe Institute of Technology, Germany, \\email{dibbelt@kit.edu}}\\affil[3]{Microsoft Research, USA, \\email{tpajor@microsoft.com}}\\affil[4]{San Francisco, USA, \\email{rwerneck@acm.org}}\\date{May 5, 2015}\\begin{document&0.7&31.5&0.0245\\\\ \\bottomrule \\end{tabular} \\end{table}\n\n\\tablename~\\ref{tab:comparison} compares our new algorithm (indicated as \\emph{PTL}, for Public Transit Labeling) to the state of the art and also evaluates multicriteria queries. In this experiment, PTL uses event labels with pruning, hashing and binary search for earliest arrival~(and multicriteria) queries, and stop labels for profile queries. We compare PTL to~CSA~\\cite{dpsw-isftr-13} and RAPTOR~\\cite{dpw-rbptr-14}~(currently the fastest algorithms without preprocessing), as well as Accelerated CSA~(ACSA)~\\cite{sw-csa-13}, Timetable Contraction Hierarchies~(CH)~\\cite{g-ctnrt-10}, and Transfer Patterns~(TP)~\\cite{bceghrv-frvlp-10,bs-fbspt-14}~(which make use of preprocessing). Since RAPTOR always optimizes transfers~(by design), we only include it for the MC problem. Note that the following evaluation should be taken with a grain of salt, as no standardized benchmark instances exist, and many data sets used in the literature are proprietary. Although precise numbers are not available for several competing methods, it is safe to say they use less space than PTL, particularly for the MC problem.\n\n\\tablename~\\ref{tab:comparison} shows that PTL queries are very efficient. Remarkably, they are faster on the national networks than on the metropolitan ones: the latter are smaller in most aspects, but have more frequent journeys (that must be covered). Compared to other methods, PTL is~\\mbox{2--3}~orders of magnitude faster on London than CSA and RAPTOR for EA~(factor~643), profile~(factor~2,167), and MC~(factor~203) queries. We note, however, that PTL is a point-to-point algorithm~(as are ACSA, TP, and CH); for one-to-all queries, CSA and RAPTOR would be faster.\n\nPTL has 1--2~orders of magnitude faster preprocessing and queries than TP for the EA and profile problems. On Madrid, EA queries are 233~times faster while preprocessing is faster by a factor of~48. Note that Sweden~(PTL) and Germany~(TP) have a similar number of connections, but PTL queries are~95~times faster. (Germany does have more stops, but recall that PTL query performance depends more on the frequency of trips.) For the MC problem, the difference is smaller, but both preprocessing and queries of PTL are still an order of magnitude faster than TP~(up to 48~times for MC queries on Madrid).\n\nCompared to ACSA and CH~(for which figures are only available for the EA and profile problems), PTL has slower preprocessing but significantly faster queries~(even when accounting for different network sizes).\n\n\\section{Conclusion} \\label{sec:conclusion}\n\nWe introduced PTL, a new preprocessing-based algorithm for journey planning in public transit networks, by revisiting the time-expanded model and adapting the Hub Labeling approach to it. By further exploiting structural properties specific to timetables, we obtained simple and efficient algorithms that outperform the current state of the art on large metropolitan and country-sized networks by orders of magnitude for various realistic query types. Future work includes developing tailored algorithms for hub computation~(instead of using RXL as a black box), compressing the labels~(\\eg, using techniques from~\\cite{bs-fbspt-14} and~\\cite{dgpw-rdqmn-14}), exploring other hub representations~(\\eg, using trips instead of events, as in 3-hop labeling~\\cite{yaiy-fsrqg-13}), using multicore- and instruction-based parallelism for preprocessing and queries, and handling dynamic scenarios~(\\eg, temporary station closures and train delays or cancellations~\\cite{bdgmpsww-rptn-14}).\n\n\\bibliographystyle{plain} \\begin{small} ","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nMany conjectures on the moduli of Higgs bundles require a deeper understanding of the ends of the moduli space, among which the most well-known are the GMN (Gaiotto-Moore-Neitzke) conjecture \\cite{gaiotto2010four} on the asymptotic metric and Hausel's conjecture \\cite{hausel2001geometry} on the vanishing of the $L^2$-cohomology. A consequence of the GMN conjecture is that $g_{L^2}-g_{\\mathrm{sf}}=O(\\mathrm{e}^{-c t})$ along a generic ray parametrized by $t\\in \\mathbb{R}^+$, where $g_{L^2}$ is the natural complete hyperk\u00e4hler metric on the moduli space, and $g_{\\mathrm{sf}}$ is the semiflat metric \\cite{freed1999special} arising from the algebraic integrability. With analytic techniques, many progresses have been made towards proving this result. The pioneering work \\cite{mazzeo2019asymptotic} used fiducial solutions and limiting configurations in \\cite{mazzeo_swoboda_weiss_witt_2016} to establish the polynomial decay of the above difference. In \\cite{dumas2019asymptotics}, Dumas and Neitzke used the local biholomorphic flow to show that the decay is exponential on the Hitchin section. These techniques were combined in \\cite{Fredrickson:2018fun, fredrickson_2019} to prove the exponential decay for $\\mathrm{SL}(n,\\mathbb{C})$-Higgs bundles. Recently in \\cite{fredrickson2022asymptotic}, the authors extended this result to the moduli space of parabolic $\\mathrm{SL}(2,\\mathbb{C})$-Higgs bundles, where the Higgs fields admit simple poles, and they showed that the moduli space of $\\mathrm{SL}(2,\\mathbb{C})$-Higgs bundles with four strongly parabolic points over $\\mathbb{C}P^1$ is an ALG gravitational instanton.\n\nThe goal of this paper is to extend the result of \\cite{fredrickson2022asymptotic} to the moduli space of irregular (wild) Higgs bundles, which means the Higgs fields can have higher order poles. Irregular Higgs bundles have received an increasing interest in recent years, because of their appearances in isomonodromic deformations \\cite{boalch2001symplectic}, wild character varieties \\cite{boalch2014geometry}, and the geometric Langlands program \\cite{witten2008gauge}. In \\cite{biquard_boalch_2004}, Biquard and Boalch established the wild non-abelian Hodge correspondence, which exhibits the moduli space of irregular Higgs bundles as a complete hyperk\u00e4hler manifold. When the moduli space is four-dimensional, it is expected to be a gravitational instanton of ALG type \\cite{boalch2012hyperkahler}. In this paper we consider the moduli space $\\mathcal{M}$ of rank two irregular Higgs bundles over $\\mathbb{C}P^1$. Using the methods of \\cite{fredrickson2022asymptotic} and the analytic results of \\cite{biquard_boalch_2004}, we obtain the following main result.\n\n\\begin{theorem}\\label{Main_thm}\n  Fix a generic curve $[(\\bar{\\partial}_E,\\Phi_t)]$ in $\\mathcal{M}$, and an infinitesimal deformation $[(\\dot{\\eta},\\dot{\\Phi})]\\in T_{[(\\bar{\\partial}_E,\\Phi_t)]}\\mathcal{M}$. As $t\\to \\infty$, there exist positive constants $c,\\sigma$ such that\n  \\[\\lVert [(\\dot{\\eta},\\dot{\\Phi})]\\rVert_{g_{L^2}}^2-\\lVert [(\\dot{\\eta},\\dot{\\Phi})]\\rVert_{g_{\\mathrm{sf}}}^2=O(\\mathrm{e}^{-ct^{\\sigma}}).\\]\n\\end{theorem}\nThe precise definition of the curve $[(\\bar{\\partial}_E,\\Phi_t)]$ will be given in Section \\ref{approxsol_sec}. Later in Section 6.3, we will specialize to the case when the moduli space is (real) four dimensional, with this result, we are able to show that $g_{L^2}$ is polynomially close to an ALG\/ALG$^\\ast$ model metric $g_{\\mathrm{model}}$. Moreover, the constants in the difference $g_{L^2}-g_{\\mathrm{model}}$ can be shown to be independent of the choice of the curve $[(\\bar{\\partial}_E,\\Phi_t)]$. The correspondences between the singularity configurations and the types of the model metrics are listed below (see also \\cite[p.~448]{boalch2018wild}). The tilde $(\\tilde{~})$ on the pole indicates a twisted irregular type (defined in Section 2).\n\\renewcommand{\\arraystretch}{1.3}\n\\begin{table}[h]\n  \\centering\n  \\begin{tabular}{|c|c|c|c|}\n    \\hline\n Kodaira type&$\\uppercase\\expandafter{\\romannumeral2}^\\ast$ & $\\uppercase\\expandafter{\\romannumeral3}^\\ast$ & $\\uppercase\\expandafter{\\romannumeral4}^\\ast$\\\\\nDynkin diagram &$A_0$&$A_1$&$A_2$\\\\\n $\\beta$&$\\frac{5}{6}$&$\\frac{3}{4}$&$\\frac{2}{3}$\\\\\n$\\tau$&$\\mathrm{e}^{2\\pi \\mathrm{i}\/3}$&$\\mathrm{i}$&$\\mathrm{e}^{2\\pi \\mathrm{i}\/3}$\\\\\n $D$&$4\\cdot\\{\\tilde{0}\\}$&$4\\cdot\\{0\\}$ or $3\\cdot\\{\\tilde{0}\\}+\\{\\infty\\}$&$3\\cdot\\{0\\}+\\{\\infty\\}$\\\\\n \\hline\n  \\end{tabular}\n  \\vspace{0.5em}\n  \\caption{ALG}\n\\end{table}\n\\vspace{-1em}\n\nIn the previous table, $(\\beta, \\tau)$ means that $\\mathcal{M}$ is asymptotic to the standard ALG model $(X,G_{\\beta,\\tau})$ of type $(\\beta,\\tau)$. Here $X$ is obtained by identifying two boundary components of \\[\\{u\\in \\mathbb{C}\\,|\\,\\mathrm{Arg}(u)\\in [0,2\\pi\\beta]\\text{ and }|u|\\geq R\\}\\times \\mathbb{C}_v\/(\\mathbb{Z}\\oplus \\mathbb{Z}\\tau)\\] via the gluing map $(u,v)\\sim (\\mathrm{e}^{2\\pi \\mathrm{i}\\beta}u,\\mathrm{e}^{-2\\pi \\mathrm{i}\\beta}v)$. $G_{\\beta,\\tau}$ is a flat hyperk\u00e4hler metric on $X$ such that $\\omega^1=\\mathrm{i}\/2 (\\mathrm{d}u\\wedge \\mathrm{d}\\bar{u}+\\mathrm{d}v\\wedge \\mathrm{d}\\bar{v})$ and $\\omega^2+\\mathrm{i}\\omega^3=\\mathrm{d}u\\wedge \\mathrm{d}v$. Kodaira type means that $\\mathcal{M}$ is biholomorphic to a rational elliptic surface minus a fiber with the given Kodaira type \\cite{chen_chen_2020}. Dynkin diagram means that $H^2(\\mathcal{M})$ is generated by the given extended Dynkin diagram. This makes sense because ALG gravitational instantons with the same $\\beta$ are diffeomorphic to each other \\cite{chen2021gravitational}.\n\n\\begin{table}[h]\n  \\centering\n  \\begin{tabular}{|c|c|c|c|c|}\n    \\hline\n Kodaira type&$\\uppercase\\expandafter{\\romannumeral1}_4^\\ast$&$\\uppercase\\expandafter{\\romannumeral1}_3^\\ast$&$\\uppercase\\expandafter{\\romannumeral1}_2^\\ast$&$\\uppercase\\expandafter{\\romannumeral1}_1^\\ast$\\\\\nDynkin diagram &$D_0$&$D_1$&$D_2$&$D_3=A_3$\\\\\n $D$&$2\\cdot\\{\\tilde{0}\\}+2\\cdot\\{\\tilde{\\infty}\\}$&$2\\cdot\\{0\\}+2\\cdot\\{\\tilde{\\infty}\\}$&\\makecell{$2\\cdot\\{0\\}+2\\cdot\\{\\infty\\}$ \\\\or $\\{0\\}+\\{1\\}+2\\cdot\\{\\tilde{\\infty}\\}$}&$\\makecell{\\{0\\}+\\{1\\}\\\\+2\\cdot\\{\\infty\\}}$\\\\\n \\hline\n  \\end{tabular}\n  \\vspace{0.5em}\n  \\caption{ALG$^\\ast$}\n\\end{table}\n\\vspace{-1em}\n\nIn the previous table,\nKodaira type means that $\\mathcal{M}$ is biholomorphic to a rational elliptic surface minus a fiber with the given Kodaira type \\cite{chen2021gravitational}. Dynkin diagram means that $H^2(\\mathcal{M})$ is generated by the given extended Dynkin diagram. This makes sense because ALG$^*$ gravitational instantons with the same Kodaira type at infinity are diffeomorphic to each other \\cite{chen2021gravitational}.\n\nTo prove Theorem \\ref{Main_thm}, we wish to use a similar strategy as that in \\cite{fredrickson2022asymptotic}:\n\\begin{itemize}\n\\item\tConstruct the harmonic metric $h_t$ for $(\\bar{\\partial}_E,t\\Phi)$ by a gluing method, with the following building blocks.\n\\begin{itemize}\n  \\item Limiting metric $h_\\infty$, which is singular at the points where the spectral cover is ramified.\n  \\item Fiducial solutions $h_{t}^{\\mathrm{model}}$ in some fixed disks around the ramification points.\n  \\item Desingularize $h_\\infty$ by $h_{t}^{\\mathrm{model}}$ to obtain $h_t^{\\mathrm{app}}$, and perturb $h_t^{\\mathrm{app}}$ to $h_t$.\n\\end{itemize}\n\\item Interpolate $g_{\\mathrm{app}}$ between $g_{L^2}$ and $g_{\\mathrm{sf}}$, compare tangent vectors in Coulomb gauges using $L^2$ descriptions of these metrics.\n\\end{itemize}\nHowever, there are two major new challenges in our case. The first is the lack of a natural $\\mathbb{C}^\\ast$ action on $\\mathcal{M}$ since the irregular types are fixed. Fortunately, irregular Higgs bundles over $\\mathbb{C}P^1$ can be explicitly described, and we can find some curve $[(\\bar{\\partial}_E,\\Phi_t)]$ in $\\mathcal{M}$ tending to infinity in place of the ray $[(\\bar{\\partial}_E,t\\Phi)]$ considered in \\cite{fredrickson2022asymptotic}. The second is that the analysis of the linearized operator requires modifications because of the presence of irregular singularities. We use the analytic tools from \\cite{biquard_boalch_2004} to handle these singularities.\n\nThis paper is organized as follows. In Section 2, we give some definitions relating to irregular Higgs bundles, and describe the Hitchin base and generic Hitchin fibers explicitly. In Section 3, we introduce the building blocks, the fiducial solutions and the limiting metric, and construct an approximate solution by gluing these together. Note that in our case the zeros of $\\mathrm{det}\\,\\Phi_t$ are moving with $t$, so the gluing regions also depend on $t$. The decaying rates of the error terms in different regions depend on some local parameters, which are called \\emph{local masses} by analogy with those appearing in \\cite{foscolo2017gluing}. We analyze the linearized Hitchin equation in Section 4, and nonlinear terms in Section 5, then a genuine solution to the Hitchin equation is obtained. Finally in Section 6, we prove Theorem \\ref{Main_thm}, and analyze the semiflat metrics in detail for the four-dimensional moduli spaces.\n\n\n\\section{Preliminaries}\\label{Prelim_sec}\nLet $E$ be a complex rank two vector bundle over $C=\\mathbb{C}P^1$, $S$ be a set of points on $C$. Cover $C$ by the usual coordinate charts $U=\\mathbb{C}_z$, $V=\\mathbb{C}_w$ with $w=1\/z$, and such that $\\{w=0\\}\\notin S$. For $x\\in S$, we also denote its $z$-coordinate by $x\\in \\mathbb{C}$. Furthermore, we fix the \\emph{parabolic structure} at each $x\\in S$, which consists of a one-dimensional subspace $L_x\\subset E_x$ (a full flag) and the associated parabolic weights $1\/2<\\alpha_{x,2}<1$, $\\alpha_{x,1}=1-\\alpha_{x,2}$.\n\nRecall that an $\\mathrm{SL}(2,\\mathbb{C})$-irregular Higgs bundle \\cite{biquard_boalch_2004} is a pair $(\\bar{\\partial}_E,\\Phi)$, where $\\bar{\\partial}_E$ is a holomorphic structure on $E$, and the Higgs field $\\Phi\\in H^0(C,\\mathfrak{sl}(E)\\otimes K(D))$, for the divisor $D=\\sum_{x\\in S} m_x\\cdot \\{x\\}$ with $m_x\\in \\mathbb{Z}^+$. Write $S=I\\cup T$, where $m_x>1$ for $x\\in I$ and $m_x=1$ for $x\\in T$. Then $\\Phi$ is a meromorphic section of $\\Omega^{1,0}(\\mathfrak{sl}(E))$ with singularities at points in $S$, which are called \\emph{irregular singularities} when they belong to $I$, otherwise called \\emph{tame singularities}. We require that $\\Phi$ has a fixed \\emph{irregular type} at each $x\\in I$, meaning that there exists a holomorphic trivialization of $(E,\\bar{\\partial}_E)$ over a neighborhood $U_x$ of $x$ such that\n\\[\\Phi=\\left(\\frac{\\phi_{x,m_x}}{z_x^{m_x}}+\\cdots+\\frac{\\phi_{x,1}}{z_x}+\\text{holomorphic terms}\\right)\\,\\mathrm{d}z_x,\\]\nwhere $z_x=z-x$, and $\\phi_{x,m_x},\\ldots,\\phi_{x,1}\\in \\mathfrak{sl}(2,\\mathbb{C})$ are given. Moreover, $\\Phi$ is adapted to the parabolic structure, i.e., $\\phi_{x,m_x}(L_x)\\subset L_x$. We assume that $\\Phi$ is \\emph{strongly parabolic} at each $x\\in T$ (see Remark \\ref{WeakParabolic_rmk}), meaning that $\\phi_{x,1}$, the residue of $\\Phi$ at $x$, is nilpotent and not fixed.\n\\begin{definition}\nAn irregular Higgs bundle $(\\bar{\\partial}_E,\\Phi)$ is \\emph{stable} if for every rank one $\\Phi$-invariant holomorphic subbundle $F$ of $(E,\\bar{\\partial}_E)$,\n\\[\\mathrm{pdeg}\\,F<\\frac{1}{2}\\mathrm{pdeg}\\,E, \\]\nwhere $\\mathrm{pdeg}\\,E=\\mathrm{deg}\\,E+|S|$, and $\\mathrm{pdeg}\\,F=\\mathrm{deg}\\,F+\\sum_{x\\in S}\\alpha_{x,F}$, $\\alpha_{x,F}=\\alpha_{x,2}$ if $F_x=L_x$, otherwise $\\alpha_{x,F}=\\alpha_{x,1}$.\n\\end{definition}\n\n\\begin{definition}\n  The \\emph{moduli space} of irregular Higgs bundles is\n\\begin{align*}\n  \\mathcal{M}=\\{(\\bar{\\partial}_E,\\Phi) \\text{ stable and compatible with the parabolic }&\\\\\\text{structure and the irregular type at each }x\\in S\\}&\\,\/\\,\\mathscr{G}_{\\mathbb{C}},\n\\end{align*}\nwhere $\\mathscr{G}_{\\mathbb{C}}=\\Gamma(\\mathrm{ParEnd}(E)\\cap SL(E))$, which consists of sections of $\\mathrm{SL}(E)$ preserving $L_x$ for each $x\\in S$. Alternatively,\n $\\mathcal{M}=\\{(\\bar{\\partial}_E,\\Phi,L)\\}\\,\/\\,\\Gamma(SL(E))$, where $L=\\{L_x\\}_{x\\in S}$, $L_x\\in \\mathbb{CP}^1_x$ which parametrizes one-dimensional subspaces of $E_x$.\n  The gauge group acts as\n\\[g\\cdot (\\bar{\\partial}_E,\\Phi,L)=(g^{-1}\\comp \\bar{\\partial}_E\\comp g,g^{-1}\\Phi g,g^{-1}\\cdot L).\\]\nBy \\cite[Th.~3.1]{inaba_michi_2016}, $\\mathrm{dim}_{\\mathbb{C}}\\,\\mathcal{M}=2(N-3)$, where $N=\\mathrm{deg}\\,D$. We assume that $N\\geqslant 4$.\n\\end{definition}\n\nWe further decompose $I=I_u\\cup I_t$. For $x\\in I_u$, the leading term $\\phi_{x,m_x}$ is diagonalizable with opposite nonzero eigenvalues $\\pm \\rho_{x,m_x}$ ($x$ is called an \\emph{untwisted singularity}), while for $x\\in I_t$, $\\phi_{x,m_x}$ is nilpotent ($x$ is called a \\emph{twisted singularity}). In the triple $(\\bar{\\partial}_E,\\Phi,L)$, as $\\Phi$ is compatible with $L$, we can determine $L_x$ form $\\Phi$ as the kernel of $\\phi_{x,m_x}$ for $x\\in I_t\\cup T$, and as one of the eigenspace of $\\phi_{x,m_x}$ for $x\\in I_u$. There are $2^{|I_u|}$ choices, depending on whether $L_x$ corresponds to $\\rho_{x,m_x}$ or $-\\rho_{x,m_x}$, where $0\\leqslant \\mathrm{Arg}(\\rho_{x,m_x})<\\pi$. These $2^{|I_u|}$ components of $\\mathcal{M}$ are equivalent, so we only need to consider the one where $L_x$ corresponds to $\\rho_{x,m_x}$, and still denote this component by $\\mathcal{M}$. We also suppress $[(\\bar{\\partial}_E,\\Phi,L_{\\Phi})]\\in\\mathcal{M}$ as $[(\\bar{\\partial}_E,\\Phi)]$.\n\n\\begin{definition}\n  The \\emph{Hitchin fibration (map)} is defined as \\[H:\\mathcal{M}\\to\\mathcal{B},~[(\\bar{\\partial}_E,\\Phi)]\\mapsto \\det\\Phi.\\]\n  Here $\\mathcal{B}\\subset H^0(C,K(D)^2)$ consists of quadratic differentials $\\nu$ which has the expansion\n  \\[\\nu=\\left(\\frac{\\mu_{x,2m_x}}{z_x^{2m_x}}+\\cdots+\\frac{\\mu_{x,m_x+1}}{z_x^{m_x+1}}+\\text{higher order terms}\\right)\\,\\mathrm{d}z_x^2,\\]\n  near each $x\\in I$. $\\mu_{x,j}$'s are complex numbers fixed by the irregular type at $x$, and clearly $\\mu_{x,2m_{x}}\\neq 0$ for $x\\in I_u$, while $\\mu_{x,2m_{x}}= 0$ for $x\\in I_t$. Moreover, we assume that $\\mu_{x,2m_{x}-1}\\neq 0$ for $x\\in I_t$ (when it is zero, by choosing a different extension of the holomorphic structure over $x$, this case can be reduced to the nonzero case or the case where the leading coefficient is diagonalizable \\cite[p.~53]{witten2008gauge}).\n\\end{definition}\n\\begin{lemma}\n  The \\emph{Hitchin base} $\\mathcal{B}$ is a dimension $N-3$ affine subspace of $H^0(C,K(D)^2)$, which can be explicitly written as\n  \\begin{align}\n    \\mathcal{B}=\\left\\{\\,\\left(\\sum_{x\\in I_{\\geqslant 3}}\\sum_{a=m_x+1}^{2m_x}\\frac{\\mu_{x,a}}{(z-x)^a}+\\sum_{x\\in I_{=2}}\\left( \\frac{\\mu_{x,4}}{(z-x)^4}+\\frac{\\mu_{x,3}(x-y_x)}{(z-x)^3(z-y_x)}\\right)\\hspace{1.15cm}\\right.\\right.\\notag\\\\+\\left.\\left.\\left.\\frac{\\sum_{b=0}^{N-4}\\nu_{b}z^b}{\\prod_{x\\in S}(z-x)^{m_x}}\\right)\\,\\mathrm{d}z^2\\,\\right|\\, \\nu_0,\\ldots,\\nu_{N-4}\\in \\mathbb{C} \\right\\},\\label{HitBase_eq}\n  \\end{align}\n    where $I_{\\geqslant 3}$ consists of poles with order $\\geqslant 3$, and $I_{=2}$ consists of order two poles. If $I_{=2}\\neq \\varnothing$, then we can choose $y_x\\in S\\backslash\\{x\\}\\neq \\varnothing$ since $N\\geqslant 4$. Choosing a different $y_x$ for $x\\in I_{=2}$ amounts to shifting $(\\nu_0,\\ldots,\\nu_{N-4})$ by an element of $\\mathbb{C}^{N-3}$.\n\\end{lemma}\n\\begin{proof}\nBy considering the difference of two elements in $\\mathcal{B}$, one can see that $\\mathcal{B}$ is an affine space modeled on $H^0(C,K^2\\otimes \\mathcal{O}(D))$, which has dimension $N-3$. Let the right hand side of \\eqref{HitBase_eq} be $\\mathcal{B}_1$, then $\\mathcal{B}_1\\subset \\mathcal{B}$. Since $\\prod_{x\\in S}(z-x)^{-m_x}z^i$ ($i=0,\\ldots,N-4$) are linearly independent, we have $\\mathrm{dim}\\,\\mathcal{B}_1=N-3$ and $\\mathcal{B}_1=\\mathcal{B}$. For $y_{x,1},y_{x,2}\\in S\\backslash \\{x\\}$, \\[\\frac{\\mu_{x,3}(x-y_{x,1})}{(z-x)^3(z-y_{x,1})}-\\frac{\\mu_{x,3}(x-y_{x,2})}{(z-x)^3(z-y_{x,2})}=\\frac{\\mu_{x,3}(y_{x,2}-y_{x,1})}{(z-x)^2(z-y_{x,1})(z-y_{x,2})},\\]\nthe last statement follows.\n\\end{proof}\n\\begin{remark}\n  Let $\\widetilde{\\mathcal{M}}$ be a moduli space of $\\mathrm{GL}(2,\\mathbb{C})$-irregular Higgs bundles, defined in the same way as for $\\mathcal{M}$ except that $\\Phi$ belongs to $H^0(C,\\mathfrak{gl}(E)\\otimes K(D))$. Then for two different Higgs fields $\\Phi_1,\\Phi_2$ in $\\widetilde{\\mathcal{M}}$, since the irregular types are fixed, we have $\\mathrm{tr}(\\Phi_1)-\\mathrm{tr}(\\Phi_2)\\in H^0(C,K)$ which is trivial. This means that Higgs fields in $\\widetilde{\\mathcal{M}}$ have constant traces, and there is no loss in considering $\\mathrm{SL}(2,\\mathbb{C})$-irregular Higgs bundles over $C$.\n\\end{remark}\nLet $\\pi:K(D)\\to C$ be the projection, and $\\lambda$ be the tautological section of $\\pi^\\ast K(D)$. Each point $\\nu$ in $\\mathcal{B}$ defines a section of $(\\pi^\\ast K(D))^2$ of the form $\\lambda^2+\\pi^\\ast \\nu$ whose zero locus $S_\\nu$ is called the \\emph{spectral curve} determined by $\\nu$. From now on we assume that $S_\\nu$ is smooth, which is true for generic $\\nu$ by Bertini's theorem. $\\pi: S_\\nu\\to C$ is a two-fold ramified cover with ramification divisor $R_\\nu$. By the Riemann-Hurwitz formula, the genus of $S_\\nu$ is $N-3$. For $\\Phi$ with $\\mathrm{det}\\,\\Phi=\\nu$, one can associate the \\emph{spectral line bundle} $\\mathcal{L}_\\Phi$ over $S_\\nu$ such that $\\mathcal{L}_\\Phi(-R_\\nu)\\subset \\pi^{\\ast} E$ and $\\pi_\\ast \\mathcal{L}_\\Phi=E$. Away from the support of $R_\\nu$, the fiber $\\mathcal{L}_{\\Phi,q}$ is the eigenspace in $E_{\\pi(q)}$ of $\\Phi$ with the eigenvalue $q$. By the Grothendieck\u2013Riemann\u2013Roch theorem \\cite[p.~96]{logares_martens_2010},\n\\begin{equation}\\label{SpecLinedeg_eq}\n  \\mathrm{deg}\\,\\mathcal{L}_\\Phi=\\mathrm{deg}\\,E+N-2.\n\\end{equation}\n\nAssume for simplicity that $\\mathrm{pdeg}\\,E=0$ (if $\\mathrm{pdeg}\\,E=2d$ for some $d\\in \\mathbb{Z}$, one can replace $E$ by $E\\otimes \\mathcal{O}(-d)$ \\cite[p.~624]{fredrickson2021moduli}). Then $\\mathrm{deg}\\,E=-|S|$ and $\\mathcal{E}=(E,\\bar{\\partial}_E)\\cong \\mathcal{O}(m)\\oplus \\mathcal{O}(-|S|-m)$, $m\\geqslant -|S|\/2$, by the Birkhoff\u2013Grothendieck theorem. Now \\[\\mathrm{End}\\,\\mathcal{E}\\cong \\begin{pmatrix}\n  \\mathcal{O}&\\mathcal{O}(2m+|S|)\\\\ \\mathcal{O}(-|S|-2m)&\\mathcal{O}\n\\end{pmatrix}.\\]\nIn a trivialization of $\\mathrm{End}\\,\\mathcal{E}$ over $U$, we have\n\\begin{equation}\\label{GlobalHiggs_eq}\n  \\Phi=\\frac{\\mathrm{d}z}{\\prod_{x\\in S}(z-x)^{m_x}}\\begin{pmatrix}\n    a(z)&b(z)\\\\ c(z)&-a(z)\n  \\end{pmatrix},\\text{ where }a(z),b(z),c(z)\\in \\mathbb{C}[z].\n\\end{equation}\n\n\\begin{lemma}\nEvery $\\Phi$ in \\eqref{GlobalHiggs_eq} with $\\det\\Phi\\in\\mathcal{B}$ of \\eqref{HitBase_eq} is compatible with some fixed irregular data at points in $I$, i.e.,\n\\begin{enumerate}[label=(\\roman*)]\n  \\item there exists a local holomorphic frame around $x\\in I_u$ in which\n    \\begin{equation}\\label{DiagLocHiggs_eq}\n      \\Phi=\\sum_{j=1}^{m_{x}}\\rho_{x,j}z_{x}^{-j}\\sigma_3\\,\\mathrm{d}z_{x}+\\text{holomorphic terms},\n    \\end{equation}\n  where $\\sigma_3$ is the Pauli matrix $\\mathrm{diag}(1,-1)$ and $\\rho_{x,j}$'s are determined by $\\mu_{x,a}$'s in \\eqref{HitBase_eq};\n  \\item there exists a local holomorphic frame around $x\\in I_t$ in which\n    \\begin{equation}\\label{NilLocHiggs_eq}\n      \\Phi=\\begin{pmatrix}\n        0&-\\sum_{j=1}^{m_{x}-1}\\mu_{x,m_{x}+j}z_{x}^{-j}\\\\ z_{x}^{-m_{x}}&0\n      \\end{pmatrix}\\,\\mathrm{d}z_{x}+\\text{holomorphic terms}.\n    \\end{equation}\n\\end{enumerate}\n\\end{lemma}\n\\begin{proof}\n  (\\romannumeral1) This is essentially \\cite[Lem.~1.1]{biquard_boalch_2004}. By \\eqref{HitBase_eq} and \\eqref{GlobalHiggs_eq}, in a neighborhood of $x$, we have the expansion\n  \\begin{align}\n    \\det\\Phi(z_{x})&=\\bigg(\\sum_{j=m_{x}+1}^{2m_{x}}\\mu_{x,j}z_{x}^{-j}+\\text{higher order terms}\\bigg)\\,\\mathrm{d}z_{x}^2,\\label{DiagLocDetExp_eq}\\\\\n    \\Phi(z_{x})&=\\bigg(\\sum_{j=1}^{m_{x}}\\phi_{x,j}z^{-j}+\\text{holomorphic terms}\\bigg)\\,\\mathrm{d}z_{x},\\label{DiagLocHiggsExp_eq}\n  \\end{align}\n $\\mu_{x,2m_{x}}=\\det\\phi_{x,m_{x}}\\neq 0$, so we can find a frame where $\\phi_{x,m_{x}}=\\rho_{x,m_{x}}\\sigma_3$ with $\\rho_{x,m_{x}}=\\mu_{x,2m_{x}}^{1\/2}$. Then we use gauge transformations of the form $g=1-g_jz_{x}^j$ ($j=1,\\ldots,m_{x}-1$) to successively cancel the off-diagonal terms of $\\phi_{x,m_{x}-j}$. In each stage, if necessary, we shrink the neighborhood of $0$ so that $g$ is invertible. For example, for $g=1-g_1z$, \\[g^{-1}\\Phi g=\\left(\\phi_{x,m_{x}}z^{-m_{x}}+(\\phi_{x,m_{x}-1}-[\\phi_{x,m_{x}},g_1])z^{-m_{x}+1}+O\\left(z^{-m_{x}+2}\\right)\\right)\\,\\mathrm{d}z_{x},\\]\n  and $\\phi_{x,m_{x}-1}-[\\phi_{x,m_{x}},g_1]$ becomes diagonal if we choose\n  \\[g_1=\\begin{pmatrix}\n  0&\\phi_{x,m_{x}-1,12}\/2\\\\-\\phi_{x,m_{x}-1,21}\/2&0\n\\end{pmatrix},\\text{ where }\\phi_{x,m_{x}-1,ij}\\text{ is the }(i,j)-\\text{entry of }\\phi_{x,m_{x}-1}.\\]\nTherefore, in some local holomorphic frame around $x$, $\\Phi$ has the form in \\eqref{DiagLocHiggs_eq}. Now $\\rho_{x,m_{x}}=\\mu_{x,2m_{x}}^{1\/2}\\neq 0$, and $\\rho_{x,j}$ ($j=m_{x}-1,\\ldots,1$) can be recursively determined as\n\\[\\rho_{x,j}=(-2\\rho_{x,m_{x}})^{-1}\\bigg(\\mu_{x,j+m_{x}}+\\sum_{k=1}^{m_{x}-j-1}\\rho_{x,j+k}\\rho_{x,m_{x}-k}\\bigg).\\]\n(\\romannumeral2) The proof is similar to (\\romannumeral1). Around $x$, we can expand $\\det\\Phi$ and $\\Phi$ as in \\eqref{DiagLocDetExp_eq}, \\eqref{DiagLocHiggsExp_eq}. Since $\\mu_{x,2m_{x}}=0$ and $\\mu_{x,2m_{x}-1}\\neq 0$, then $\\phi_{x,m_{x}}\\neq 0$ and is nilpotent, so we can find a local frame in which $\\phi_{x,m_{x}}=\\left(\\begin{smallmatrix}\n    0&0\\\\1&0\n  \\end{smallmatrix}\\right)$. Then we use gauge transformations of the form $g=1-g_jz_{x}^j$ ($j=1,\\ldots,m_{x}-1$) to successively cancel the lower-triangular part of $\\phi_{x,m_{x}-j}$. For example if we choose \\[g_1=\\begin{pmatrix}\n    \\phi_{x,m_{x}-1,21}&\\phi_{x,m_{x}-1,22}\\\\0&0\n  \\end{pmatrix}\\]\n  in $g=1-g_1z$, then $\\phi_{x,m_{x}-1}-[\\phi_{x,m_{x}},g_1]$ becomes strictly upper-triangular. Therefore, in some local holomorphic frame around $x$, $\\Phi$ has the form in \\eqref{NilLocHiggs_eq}.\n\\end{proof}\nIn $V$, we can write $\\Phi$ as\n\\[\\Phi=\\frac{-w^{N-2}\\,\\mathrm{d}w}{\\prod_{x\\in S}(1-xw)^{m_x}}\\begin{pmatrix}\n  a(w^{-1})&w^{2m+|S|}b(w^{-1})\\\\ w^{-2m-|S|}c(w^{-1})&-a(w^{-1})\n\\end{pmatrix}.\\]\n$\\Phi$ is regular at $\\infty$, so $\\mathrm{deg}\\,a\\leqslant N-2$, $\\mathrm{deg}\\,b\\leqslant 2m+|S|+N-2$, $\\mathrm{deg}\\,c\\leqslant -2m-|S|+N-2$. Since $ \\mathrm{deg}\\,c\\geqslant 0$, and recall that $m\\geqslant -|S|\/2$, we have\n\\[-|S|\\leqslant 2m\\leqslant -|S|+N-2,\\]\nwhich yields a stratification of $\\mathcal{M}$:\n\\begin{equation}\\label{Strata_eq}\n  \\mathcal{M}=\\bigsqcup_{m=\\lceil -|S|\/2\\rceil}^{\\lfloor (-|S|+N-2)\/2 \\rfloor}\\mathcal{M}_m,\n\\end{equation}\nwhere $[(\\bar{\\partial}_E,\\Phi)]\\in \\mathcal{M}_m$ if $(E,\\bar{\\partial}_E)\\cong \\mathcal{O}(m)\\oplus \\mathcal{O}(-|S|-m)$. Let $H_m$ be the restriction of $H$ to $\\mathcal{M}_m$. Note that\n\\begin{equation}\\label{detHiggs_eq}\n  -a(z)^2-b(z)c(z)=\\prod_{x\\in S}(z-x)^{2m_x}\\nu(z):=\\tilde{\\nu}(z) \\text{ where } \\det\\Phi=\\nu=\\nu(z)\\,\\mathrm{d}z^2,\n\\end{equation}\n then $b(z)=(-a(z)^2-\\tilde{\\nu}(z))\/c(z)$. The spectral cover $S_\\nu\\to C$ ramifies exactly at the $2N-4$ zeros (counted with multiplicity) of $\\tilde{\\nu}(z)$. We say that $\\nu\\in\\mathcal{B}'$, the \\emph{regular locus}, if $\\tilde{\\nu}(z)$ only has simple zeros. When $\\nu=\\det\\Phi\\in \\mathcal{B}'$, the Higgs bundle is stable \\cite[p.~7]{mazzeo_swoboda_weiss_witt_2016} and $S_\\nu$ is smooth \\cite[p.~10]{fredrickson2022asymptotic}. Fix $\\nu\\in\\mathcal{B}'$, and next we study the fiber $H_m^{-1}(\\nu)$. Parabolic Higgs bundles have been studied in \\cite{fredrickson2022asymptotic}, so from now on we suppose that $I\\neq\\varnothing$. If $I_u\\neq \\varnothing$, by a coordinate change when necessary, we may assume that $0\\in I_u$, or else we assume that $0\\in I_t$.\n\nIf $m=-|S|\/2$, then $|S|$ is even, the holomorphic gauge transformations are of the form\n\\[g=\\begin{pmatrix}\n  g_{11}&g_{12}\\\\g_{21}&g_{22}\n\\end{pmatrix},\\quad g_{11},g_{12},g_{21},g_{22}\\in \\mathbb{C},\\det g=1.\\]\nLet $a_i,b_i,c_i,\\tilde{\\nu}_i$ be the coefficients of $z^i$ in $a(z),b(z),c(z),\\tilde{\\nu}(z)$, and make the generic assumption that $\\tilde{\\nu}_{2N-4},\\tilde{\\nu}_{2N-5}\\neq 0$. By some gauge transformation $g$ above, we can make $b_0=c_0=0$ when $0\\in I_u$, or $a_0=c_0=0$ when $0\\in I_t$. Now $c(z)\\neq 0$, otherwise $\\tilde{v}(z)=-a(z)^2$, contradicting that $\\tilde{\\nu}(z)$ only has simple zeros. If $\\mathrm{deg}\\,c=N-2$, using the diagonal gauge transformation $g=\\mathrm{diag}(c_{N-2}^{-1\/2},c_{N-2}^{1\/2})$ we can make $c(z)$ monic. Then we apply \\[g=\\begin{pmatrix}\n  1& a_{N-2}\\\\ 0&1\n\\end{pmatrix}\\]\nto make $a_{N-2}=0$. There is no gauge freedom left. The remaining $N-2$ (if $0\\in I_u$, or $N-3$ if $0\\in I_t$) coefficients of $a(z)$ can be determined (up to finitely many choices) by $c_1,c_2,\\ldots,c_{N-3}$ and the relation \\eqref{detHiggs_eq}. For example, if $c(z)$ has distinct roots $x_1=0,\\ldots,x_{N-2}$, then $a_0,a_1,\\ldots,a_{N-3}$ satisfy $N-2$ linear equations \\[a(x_i)=y_i,~ i=1,\\ldots, N-2,\\text{ where }y_i=\\pm (-\\tilde{\\nu}(x_i))^{1\/2}.\\] If $\\mathrm{deg}\\,c<N-2$, then $a_{N-2}^2= -\\tilde{\\nu}_{2N-4}$ and $\\mathrm{deg}\\,c=N-3$. In fact, if $a_{N-2}^2\\neq -\\tilde{\\nu}_{2N-4}$, then $\\mathrm{deg}(-a(z)^2-\\tilde{\\nu}(z))=2N-4>2N-5\\geqslant\\mathrm{deg}(b(z)c(z))$, a contradiction. Then we have $ \\mathrm{deg}(-a(z)^2-\\tilde{\\nu}(z))=2N-5$, and $\\mathrm{deg}\\,c=2N-5-\\mathrm{deg}\\,b\\geqslant N-3$, and $\\mathrm{deg}\\,c=N-3$. Applying gauge transformations as above we can make $c(z)$ monic and $a_{N-3}=0$. As before, the remaining coefficients of $a(z)$ are determined by $c_1,\\ldots, c_{N-4}$. Therefore \\[\\mathrm{dim}(\\mathcal{M}_{-|S|\/2})=N-3.\\]\n\nIf $m>-|S|\/2$, the holomorphic gauge transformations are of the form\n\\[g=\\begin{pmatrix}\n  g_{11}&g_{12}(z)\\\\\n  0&g_{11}^{-1}\n\\end{pmatrix},\\quad g_{11}\\in \\mathbb{C},g_{12}(z)\\in \\mathbb{C}[z], \\mathrm{deg}\\,g_{12}(z)\\leqslant 2m+|S|.\\]\nIf $\\mathrm{deg}\\,c=-2m-|S|+N-2$, we may assume that $c(z)$ is monic after applying a diagonal gauge transformation as above. Then we successively apply\n\\[g=\\begin{pmatrix}\n  1& a_{N-2-i}z^{2m+|S|-i}\\\\0&1\n\\end{pmatrix},\\quad i=0,1,\\ldots, 2m+|S|,\\]\nto make $a_{N-2}=a_{N-3}=\\cdots=a_{N-2-2m-|S|}=0$. There is no gauge freedom left. The remaining $N-2-2m-|S|$ coefficients of $a(z)$ are determined by $c(z)$ and \\eqref{detHiggs_eq}. If $\\mathrm{deg}\\,c<-2m-|S|+N-2$, then $a_{N-2}^2=-\\tilde{\\nu}_{2N-4}$ and $\\mathrm{deg}\\,c=-2m-|S|+N-3$. Again we can find a representative in each gauge orbit with $c(z)$ monic and $a_{N-3}=\\cdots=a_{N-3-2m-|S|}=0$. The remaining coefficients of $a(z)$ are determined by $c(z)$. Therefore \\[\\mathrm{dim}(\\mathcal{M}_{m})=-2m-|S|+N-2.\\]\n\n\\section{Approximate Solutions}\\label{approxsol_sec}\nWe consider the case $I_u\\neq \\varnothing$, then $0\\in I_u$ by assumption. Fix $\\nu\\in\\mathcal{B}'$ satisfying the generic conditions $\\tilde{\\nu}_{2N-4}, \\tilde{\\nu}_{2N-5}\\neq 0$, and $\\nu_{N-4}\\neq 0$. By \\eqref{HitBase_eq}, we can represent $\\nu$ by $(\\nu_0,\\ldots,\\nu_{N-4})\\in \\mathbb{C}^{N-3}$. Consider a curve $\\nu_t=(f_0(t)\\nu_0,\\ldots,f_{N-4}(t)\\nu_{N-4})$ in $\\mathcal{B}$ with $\\nu_{t=1}=\\nu$ and its norm tending to $\\infty$, where $f_i(t)$ is positive and nondecreasing in $t$. For simplicity, let $f_{N-4}(t)=t$ and $f_i(t)=o(t^{(m_0+i)\/(N-4+m_0)})$ for $i=0,\\ldots,N-5$, so that the following holds.\n\\begin{lemma}\\label{HiggsDetRoot_lem}\n  For $t$ sufficiently large, $\\tilde{\\nu}_t(z)$ has $2N-4$ distinct roots, with leading terms given by\n\\begin{align*}\n  z_{0,j}(t)&\\sim \\Bigg(-\\frac{\\mu_{0,2m_0}}{\\nu_{N-4}}\\prod_{y\\in S\\backslash\\{0\\}}(-y)^{m_y}\\Bigg)^{\\frac{1}{m_0+N-4}}\\mathrm{e}^{\\frac{2j\\pi \\mathrm{i}}{m_0+N-4}}t^{-\\frac{1}{m_0+N-4}},~ j=0,\\ldots, m_0+N-5;\\\\\n  z_{x,j}(t)&\\sim x+ \\Bigg(-\\frac{\\mu_{x,2m_x}}{\\nu_{N-4}}x^{4-N}\\prod_{y\\in S\\backslash \\{x\\}}(x-y)^{m_y}\\Bigg)^{\\frac{1}{m_x}}\\mathrm{e}^{\\frac{2j\\pi \\mathrm{i}}{m_x}}t^{-\\frac{1}{m_x}},~j=0,\\ldots,m_x-1,\\\\\\text{ for }x&\\in I_u\\backslash\\{0\\};\\\\\n  z_{x,j}(t)&\\sim x+ \\Bigg(-\\frac{\\mu_{x,2m_x-1}}{\\nu_{N-4}}x^{4-N}\\prod_{y\\in S\\backslash \\{x\\}}(x-y)^{m_y}\\Bigg)^{\\frac{1}{m_x-1}}\\mathrm{e}^{\\frac{2j\\pi \\mathrm{i}}{m_x-1}}t^{-\\frac{1}{m_x-1}},~j=0,\\ldots,m_x-2,\\\\\\text{ for }x&\\in I_t;\\text{ and }\n  z_{x,m_x-1}(t)=x,\\text{ for }x\\in I_t\\cup T.\n\\end{align*}\nHere the notation $A\\sim B$ means $A=B+o(B)$. Therefore $\\nu_t\\in \\mathcal{B}'$ for $t$ large enough.\n\\end{lemma}\n\\begin{proof}\n  By \\eqref{HitBase_eq}, we can write\n  \\[\\tilde{\\nu}_t(z)=\\bigg(\\sum_{b=0}^{N-4}f_b(t)\\nu_b z^b\\bigg)z^{m_0}\\prod_{y\\in S\\backslash \\{0\\}}(z-y)^{m_y}+\\mu_{0,2m_0}\\prod_{y\\in S\\backslash\\{0\\}}(z-y)^{2m_y}+zg_0(z),\\]\n  where $g_0(z)$ is some polynomial independent of $t$. By Rouch\u00e9's theorem, there is a constant $\\kappa$ such that in the disk $B_{\\kappa t^{-1\/(m_0+N-4)}}(0)$, $\\tilde{\\nu}_t(z)$ has the same number of zeros as that for\n  \\[\\tilde{\\nu}_t^{(0)}(z):=\\nu_{N-4}tz^{m_0+N-4}\\prod_{y\\in S\\backslash\\{0\\}}(-y)^{m_y}+\\mu_{0,2m_0}\\prod_{y\\in S\\backslash\\{0\\}}(-y)^{2m_y},\\]\n  for $t$ large enough. Choose $\\kappa$, so that $\\tilde{\\nu}_t^{(0)}$ has $m_0+N-4$ distinct roots in the disk. Using the scaling $z\\mapsto t^{1\/(m_0+N-4)}z$, we see that the roots $z_{0,j}(t)$ ($j=0,\\ldots,m_0+N-5$) of $\\tilde{\\nu}_t(z)$ in the disk are asymptotic to those of $\\tilde{\\nu}_t^{(0)}(z)$, which are given in the statement of the lemma. The analysis near $x\\in I\\backslash\\{0\\}$ is similar, using the coordinate $z_x$ in pace of $z$.\n\\end{proof}\n\nThe hyperk\u00e4hler metric on $\\mathcal{M}$ is defined through the nonabelian Hodge correspondence, which realizes $\\mathcal{M}$ as a hyperk\u00e4hler quotient, i.e., unitary gauge equivalence classes of solutions to Hitchin's equations. Thus, we need to find the solution of the following \\emph{Hitchin's equation} for $[(\\bar{\\partial}_E,\\Phi)]\\in\\mathcal{M}$:\n\\[F_h+[\\Phi,\\Phi^{\\ast_h}]=0,\\]\nwhere $F_h$ is the curvature of the Chern connection $D(\\bar{\\partial}_E,h)$. $h$ is called a \\emph{harmonic metric} if it satisfies the above equation and adapted to the parabolic structure, which means for each $x$ the filtration of the fiber $E_x$ given by $E_{x,\\beta}:=\\{s(0):s\\text{ holomorphic},|s(z)|_{h}=O(|z|^\\beta)\\}$ coincides with that of the parabolic structure. When $I_t=\\varnothing$, such a metric $h$ always exists on a stable Higgs bundle by \\cite{biquard_boalch_2004}. However, if $I_t\\neq \\varnothing$, the existence of $h$ will impose further constraints on the parabolic weights $\\alpha_{x,1}$, $\\alpha_{x,2}$ for $x\\in I_t$.\n\\begin{lemma}\\label{TIrregWeight_lem}\n  Let $(E,\\Phi)$ be an $\\mathrm{SL}(2,\\mathbb{C})$-Higgs bundle over a Riemann surface $C$, where $\\Phi$ has an order $n\\geqslant 2$ pole at $x$, with nilpotent leading coefficient $\\phi_{-n}$. The kernel $L$ of $\\phi_{-n}$ determines a filtration $0\\subset L\\subset E_x$ and let the associated parabolic weights be $1>\\alpha_2>\\alpha_1>0$, $\\alpha_1+\\alpha_2=1$. Suppose $\\mathrm{pdeg}\\,E=0$. If $h$ is a harmonic metric adapted to the parabolic structure, then $\\alpha_1=1\/4,\\alpha_2=3\/4$.\n\\end{lemma}\n\\begin{proof}\nAs in \\cite[Lem.~3.6]{mazzeo_swoboda_weiss_witt_2016}, we first find a standard form of $\\Phi$ near $x$. Since $\\phi_{-n}$ is nilpotent, $\\det\\Phi$ has an order $2n-1$ pole at $p_0$. By \\cite[p.~28]{strebel_1984}, one can find a local coordinate $z$ centered at $p_0$, such that $\\det\\Phi=-z^{1-2n}\\,\\mathrm{d}z^2$. Again by the nilpotency of the leading term, in some local holomorphic frame\n  \\[z^n\\phi(z)=\\begin{pmatrix}\na(z)&b(z)\\\\c(z)&-a(z)\n\\end{pmatrix},\\quad z^n\\phi(z)|_{z=0}=\\begin{pmatrix}\n    0&0\\\\1&0\n  \\end{pmatrix}, \\text{ where }\\Phi(z)=\\phi(z)\\,\\mathrm{d}z.\\]\n  Now $\\sqrt{c(z)}$ is well-defined near $0$ since $c(0)=1$, and \\[g^{-1}\\Phi g=\\begin{pmatrix}\n    0&\\frac{1}{z^{n-1}}\\\\\\frac{1}{z^n}&0\n  \\end{pmatrix}\\,\\mathrm{d}z,\\quad\\text{where }g=\\frac{1}{\\sqrt{c(z)}}\\begin{pmatrix}\n    1&a(z)\\\\0&c(z)\n  \\end{pmatrix}.\\]\n  In the local holomorphic frame determined by $g$, we have \\[h=\\begin{pmatrix}\n    h_{11}(z)&h_{12}(z)\\\\\\widebar{{h}_{12}(z)}&h_{22}(z)\n  \\end{pmatrix},\\quad h_{11}(z)\\sim c_1r^{2\\alpha_1},h_{22}(z)\\sim c_2r^{2\\alpha_2},\\det h=Q(z)r^{2},h_{12}(z)=O(r),\\]\n  where $c_1,c_2$ are positive constants, $r=|z|$, $Q(z)$ is determined by $h_{\\det E}$, the metric on $\\det E$ induced by $h$, $Q(0)>0$. The $(1,1)$-entry of $F_h$ is $O(r^{-2})$ while that of $[\\Phi,\\Phi^{\\ast_h}]$ is $O(r^{4\\alpha_1-2n})+O(r^{4\\alpha_2-2-2n})$ (the leading terms in these $O(\\cdot)$'s are nonzero), which grows faster than $r^{-2}$ near $0$ if the sum of the leading terms is nonzero. Therefore $F_h+[\\Phi,\\Phi^{\\ast_h}]=0$ implies that $4\\alpha_1-2n=4\\alpha_2-2-2n$, and $\\alpha_1=1\/4,\\alpha_2=3\/4$.\n\\end{proof}\n\nHenceforth, we assume that $\\alpha_{x,1}=1\/4, \\alpha_{x,2}=3\/4$, for $x\\in I_t$. To get a more explicit description of the hyperk\u00e4hler metric near the ends of $\\mathcal{M}$, we need more information on $h$, which can be obtained via gluing constructions as in \\cite{fredrickson2022asymptotic}, and we also follow this strategy. Fix $[(\\bar{\\partial}_E,\\Phi)]\\in H^{-1}(\\nu)$, and choose $[(\\bar{\\partial}_E,\\Phi_t)]\\in H^{-1}(\\nu_t)$, such that $\\Phi_t(z)$ and $\\Phi(z)$ have the same $c(z)$ when written as in \\eqref{GlobalHiggs_eq} (by the previous section, there are only finitely many gauge inequivalent choices of $\\Phi_t$). Note that if $[(\\bar{\\partial}_E,\\Phi)]$ belongs to $\\mathcal{M}_m$ of \\eqref{Strata_eq}, then the whole curve $[(\\bar{\\partial}_E,\\Phi_t)]$ remains in $\\mathcal{M}_m$ as $t$ varies.\n\nNow we aim to find a solution $h_t$ of $F_{h_t}+[\\Phi_t,\\Phi_t^{\\ast_{h_t}}]=0$ for large $t$. First we construct an approximate solution $h_t^{\\mathrm{app}}$, and later we will deform it into a genuine solution. The construction of $h_t^{\\mathrm{app}}$ is similar to that in \\cite[Sec.~3]{fredrickson2022asymptotic}. Since we are considering $\\mathrm{SL}(2,\\mathbb{C})$ Higgs bundles, the data on $\\mathrm{det}\\,E\\simeq \\mathcal{O}(-|S|)$ is fixed. In other words, we fix the holomorphic structure induced from $\\bar{\\partial}_E$ and the harmonic metric $h_{\\mathrm{det}\\,E}=\\prod_{x\\in S}|z-x|^2 $ adapted to the induced parabolic weight $\\alpha_{x,1}+\\alpha_{x,2}=1$ at $x$ (in $V$, $h_{\\mathrm{det}\\,E}=\\prod_{x\\in S}|1-xw|^2$). Let $(S_t,\\mathcal{L}_t)$ be the spectral data associated with $(\\bar{\\partial}_E,\\Phi_t)$. The spectral cover $\\pi:S_t\\to C$ ramifies at $Z_t$, the zero locus of $\\tilde{\\nu}_t(z)$. By \\eqref{SpecLinedeg_eq}, $\\mathrm{deg}\\,\\mathcal{L}_t=-|S|+N-2$. Now $\\mathcal{L}_t$ has rank one, a parabolic structure at a point is equivalent to a parabolic weight since there is only a trivial filtration. For each of the two points in $\\pi^{-1}(x)$, $x\\in I_u$, let the weight be $\\alpha_{x,1}$ if it corresponds to the eigenvalue $\\rho_{x,m_{x}}$ of $\\phi_{x,m_{x}}$ and otherwise be $\\alpha_{x,2}$. For each point in $\\pi^{-1}(Z_t\\backslash(I_t\\cup T))$, let the weight be $-1\/2$. For each point $\\pi^{-1}(x)$, $x\\in I_t\\cup T$, let the weight be $\\alpha_{x,1}+\\alpha_{x,2}-1\/2=1\/2$. Endowed with this parabolic structure we have\n\\[\\mathrm{pdeg}\\,\\mathcal{L}_t=-|S|+N-2+|I_u|+|I_t|\/2+|T|\/2-(2N-4-|I_t|-|T|)\/2 =0=\\mathrm{pdeg}\\, E.\\] By \\cite{simpson_1990}, there is a unique (up to scaling) Hermitian-Einstein metric $h_{\\mathcal{L}_t}$ on $\\mathcal{L}_t$ adapted to the parabolic structure, which is flat in our setting. Finally, let $h_{t,\\infty}$ be the metric on $E|_{C\\backslash (Z_t\\cup S)}$ defined as the orthogonal pushforward of $h_{\\mathcal{L}_t}$. Then $h_{t,\\infty}$ solves the decoupled Hitchin's equations:\n\\[F_{h_{t,\\infty}}=0,\\quad [\\Phi_t,\\Phi_t^{\\ast_{h_{t,\\infty}}}]=0.\\]\n The metric on $\\mathrm{det}\\,E|_{C\\backslash (Z_t\\cup S)}$ induced by $h_{t,\\infty}$ can be extended to $C\\backslash\\{S\\}$ and is a harmonic metric. By uniqueness, we may assume that this metric is $h_{\\mathrm{det}\\,E}$ defined above, by rescaling $h_{\\mathcal{L}_t}$ if necessary. Near $x\\in Z_t\\cup S$, $\\Phi_t$ and $h_{t,\\infty}$ can be explicitly described using the following normal forms.\n\\begin{proposition}\\label{NormalForm_prop}\n  There exist constants $1>\\kappa,\\kappa_1,\\kappa_2>0$, such that for $t$ large enough, the followings hold.\n  \\begin{enumerate}[label=(\\roman*)]\n    \\item There is a holomorphic coordinate $\\zeta_{x,j,t}$ centered at $z_{x,j}(t)$ for $x\\in I$, $j=0,\\ldots,j(x)-1$ where\n    \\[j(x)=\\begin{cases}\n      m_0+N-4,&\\text{ if }x=0,\\\\\n      m_x,&\\text{ if }x\\in I_u\\backslash\\{0\\},\\\\\n      m_x-1,&\\text{ if }x\\in I_t,\n    \\end{cases}\\]\n    and a local holomorphic frame of $(E,\\bar{\\partial}_E)$ over $\\widetilde{B}_{x,j,t}:=\\{\\,|\\zeta_{x,j,t}|<\\kappa t^{-1\/j(x)} \\,\\}$ in which\n    \\begin{equation}\\label{NormalFormZ_eq}\n      \\bar{\\partial}_E=\\bar{\\partial},\\quad \\Phi_t=\\lambda_{x,j}(t)\\begin{pmatrix}\n        0&1\\\\\\zeta_{x,j,t}&0\n      \\end{pmatrix}\\,\\mathrm{d}\\zeta_{x,j,t},\\quad h_{t,\\infty}=\\begin{pmatrix}\n        |\\zeta_{x,j,t}|^{1\/2}&0\\\\0&|\\zeta_{x,j,t}|^{-1\/2}\n      \\end{pmatrix}.\n    \\end{equation}\n    Here the \\emph{local mass} $\\lambda_{x,j}(t)=|\\nu_{x,j,t}(z_{x,j}(t))|^{1\/2}$, where $\\nu_{x,j,t}(z)=(z-z_{x,j}(t))^{-1}\\nu_t(z)$.\n    \\item There is a holomorphic coordinate $\\zeta_{x,t}$ centered at $x$ for $x\\in T$, and a local holomorphic frame of $(E,\\bar{\\partial}_E)$ over $\\widetilde{B}_{x,t}:=\\{\\,|\\zeta_{x,t}|<\\kappa\\,\\}$ in which\n  \\begin{equation}\\label{NormalFormT_eq}\n    \\bar{\\partial}_E=\\bar{\\partial},\\quad \\Phi_t=\\lambda_{x}(t)\\begin{pmatrix}\n        0&1\\\\ \\frac{1}{\\zeta_{x,t}}&0\n      \\end{pmatrix}\\,\\mathrm{d}\\zeta_{x,t},\\quad h_{t,\\infty}=\\begin{pmatrix}\n        |\\zeta_{x,t}|^{1\/2}&0\\\\0&|\\zeta_{x,t}|^{3\/2}\n      \\end{pmatrix}.\n  \\end{equation}\n    Here $\\lambda_{x}(t)=|\\nu_{x,t}(x)|^{1\/2}$, and $\\nu_{x,t}(z)=(z-x)\\nu_t(z)$.\n    \\item There is a holomorphic coordinate $\\zeta_{x,t}$ centered at $x$ for $x\\in I_t$, and a local holomorphic frame of $(E,\\bar{\\partial}_E)$ over $\\widetilde{B}_{x,t}:=\\{\\,|\\zeta_{x,t}|<\\kappa t^{-1\/(m_x-1)}\\,\\}$ in which\n\\begin{equation}\\label{NormalFormIt_eq}\n  \\bar{\\partial}_E=\\bar{\\partial},\\quad \\Phi_t=\\lambda_{x}(t)\\begin{pmatrix}\n    0&\\frac{1}{\\zeta_{x,t}^{m_x-1}}\\\\ \\frac{1}{\\zeta_{x,t}^{m_x}}&0\n  \\end{pmatrix}\\,\\mathrm{d}\\zeta_{x,t},\\quad h_{t,\\infty}=\\begin{pmatrix}\n    |\\zeta_{x,t}|^{1\/2}&0\\\\0&|\\zeta_{x,t}|^{3\/2}\n  \\end{pmatrix}.\n\\end{equation}\n    Here $\\lambda_{x}(t)=|\\nu_{x,t}(x)|^{1\/2}$, and $\\nu_{x,t}(z)=(z-x)^{2m_x-1}\\nu_t(z)$.\n    \\item For $x\\in I_u$, there is a local holomorphic frame of $(E,\\bar{\\partial}_E)$ over $\\widetilde{B}_{x,t}:=\\{\\,|z-x|<\\kappa t^{-1\/j(x)}\\,\\}$ in which\n    \\begin{equation}\\label{NormalFormIu_eq}\n      \\bar{\\partial}_E=\\bar{\\partial},\\quad \\Phi_t=\\frac{1}{z_x^{m_x}}(z_x^{2m_x}\\nu_t(z))^{1\/2}\\sigma_3\\,\\mathrm{d}z,\\quad h_{t,\\infty}=\\begin{pmatrix}\n        |z_x|^{2\\alpha_{x,1}}&0\\\\0&|z_x|^{2\\alpha_{x,2}}\n      \\end{pmatrix}\n    \\end{equation}\n  \\end{enumerate}\nThe disks $\\widetilde{B}_{x(,j),t}$ are disjoint, and $B_{x(,j),t,\\kappa_1} \\subset \\widetilde{B}_{x(,j),t}\\subset B_{x(,j),t,\\kappa_2}$, where $B_\\bullet$ are disks defined using the coordinate $z$, for example, $B_{x,j,t,\\kappa_1}=\\{\\,|z-z_{x,j}(t)|<\\kappa_1 t^{-1\/j(x)}\\,\\}$, for $x\\in I, j=0,\\ldots,j(x)-1$.\n\\end{proposition}\n\\begin{proof}\n  By \\cite[Th.~6.1]{strebel_1984}, any quadratic differential having a critical point of odd order can be written in a standard form using some local holomorphic coordinate around that point. This can be applied to (\\romannumeral1)-(\\romannumeral3), where the quadratic differential $\\det\\Phi_t(z)$ has a critical point of order $1$, $-1$, $-(2m_x-1)$ respectively.\n\n  (\\romannumeral1) By a straightforward adaptation of the proof in \\cite{strebel_1984}, one can find a biholomorphic map $\\sigma_{x,j,t}(z-z_{x,j}(t))=\\zeta_{x,j,t}$ from some disk $B_{x,j,t,\\kappa_2}$ to its image, such that $\\det \\Phi_t=-\\lambda_{x,j}(t)^2 \\zeta_{x,j,t}\\,\\mathrm{d}\\zeta_{x,j,t}^2$. $\\sigma_{x,j,t}(0)=0$, and moreover for $t$ large enough, $|\\sigma_{x,j,t}'|$ is bounded above and below by some positive constants. Therefore one can find constants $\\kappa,\\kappa_1$ so that $\\widetilde{B}_{x,j,t}$ lies inside the image $\\sigma_{x,j,t}(B_{x,j,t,\\kappa_2})$ and its preimage contains a smaller disk $B_{x,j,t,\\kappa_1}$, for all sufficiently large $t$. As in \\cite[Lem.~3.6]{mazzeo_swoboda_weiss_witt_2016}, we will write $\\Phi_t$ in a standard form using local holomorphic gauge transformations, well defined over $\\widetilde{B}_{x,j,t}$. Recall that $c(z)\\neq 0$ in $\\Phi_t$ is fixed, for $t$ large, $c(z)$ is nonzero on $\\widetilde{B}_{x,j,t}$. So we can write $\\Phi_t$ in the coordinate $\\zeta_{x,j,t}$ as $\\Phi_t=\\phi_{x,j,t}(\\zeta_{x,j,t})\\,\\mathrm{d}\\zeta_{x,j,t}$,   with $\\phi_{x,j,t,21}(\\zeta_{x,j,t})$ nonvanishing on $\\widetilde{B}_{x,j,t}$, $\\phi_{x,j,t,ij}(\\zeta_{x,j,t})$ being the $(i,j)$-entry of $\\phi_{x,j,t}(\\zeta_{x,j,t})$. By a constant gauge transformation\n  \\[\\begin{pmatrix}\n    \\phi_{x,j,t,11}(0)&1\\\\\\phi_{x,j,t,21}(0)&0\n  \\end{pmatrix},\\]\n  we can make $\\Phi_t=\\phi^{(1)}_{x,j,t}(\\zeta_{x,j,t})\\,\\mathrm{d}\\zeta_{x,j,t}$, with\n\\[\\phi^{(1)}_{x,j,t}(0)=\\begin{pmatrix}\n  0&1\\\\0&0\n\\end{pmatrix},\\text{ and }\\phi^{(1)}_{x,j,t,12}=\\phi_{x,j,t,21}(\\zeta_{x,j,t})\/\\phi_{x,j,t,21}(0),\\]\nNow for $\\kappa$ suitably small, $|\\phi^{(1)}_{x,j,t,12}-1|<1\/2$ and $\\sqrt{\\phi^{(1)}_{x,j,t,12}}$ is well defined on $\\widetilde{B}_{x,j,t}$. Take the gauge transformation\n\\[\\frac{1}{\\sqrt{\\phi^{(1)}_{x,j,t,12}}}\\begin{pmatrix}\n  \\phi^{(1)}_{x,j,t,12}&0\\\\-\\phi^{(1)}_{x,j,t,11}&1\n\\end{pmatrix}, \\text{ such that }\\Phi_t=\\begin{pmatrix}\n  0&1\\\\ \\lambda_{x,j}(t)^2 \\zeta_{x,j,t}&0\n\\end{pmatrix}\\,\\mathrm{d}\\zeta_{x,j,t}.\\]\nUse the constant gauge transformation\n\\[\\begin{pmatrix}\n  1&0\\\\0&\\lambda_{x,j}(t)\n\\end{pmatrix}, \\text{ and then }\\Phi_t=\\lambda_{x,j}(t)\\begin{pmatrix}\n  0&1\\\\\\zeta_{x,j,t}&0\n\\end{pmatrix}\\,\\mathrm{d}\\zeta_{x,j,t}.\\]\nFinally, as in \\cite[Prop.~3.5]{Fredrickson:2018fun}, $h_{t,\\infty}$ has the form given above, after applying a holomorphic gauge transformation on $\\widetilde{B}_{x,j,t}$ preserving $\\Phi_t$.\n\n(\\romannumeral2) is similar to \\cite[Prop.~3.5]{fredrickson2022asymptotic}. (\\romannumeral3) is similar to Lemma \\ref{TIrregWeight_lem} and (\\romannumeral2). (\\romannumeral4) is similar to \\cite[Prop.~3.7]{fredrickson2022asymptotic}, noting that $(z_x^{2m_x}\\nu_t(z))^{1\/2}$ is well defined on $\\widetilde{B}_{x,t}$ for $\\kappa$ suitably small. We can also choose $\\kappa$ so that all the disks above are disjoint.\n\\end{proof}\n\nNear the points in (\\romannumeral3) and (\\romannumeral4) above, $h_{t,\\infty}$ is already adapted to the parabolic structure. Near the points in (\\romannumeral1), $h_{t,\\infty}$ is singular, while near the points in (\\romannumeral2), $h_{t,\\infty}$ is not adapted to the parabolic weights. So we need to desingularize $h_{t,\\infty}$ using the following \\emph{fiducial solutions} $h_{t}^{\\mathrm{model}}$ \\cite[Prop.s~3.8~\\&~3.9]{fredrickson2022asymptotic}, and obtain the approximate metric $h_{t}^{\\mathrm{app}}$.\n\\begin{definition}\\label{ApproxMet_def}\n  \\begin{enumerate}[label=(\\roman*)]\n    \\item For $x\\in I$, in the above holomorphic frame over $\\widetilde{B}_{x,j,t}$, define\n    \\[h_{t}^{\\mathrm{model}}=\\begin{pmatrix}\n      |\\zeta_{x,j,t}|^{1\/2}\\mathrm{e}^{l_{x,j,t}(|\\zeta_{x,j,t}|)}&0\\\\0&|\\zeta_{x,j,t}|^{-1\/2}\\mathrm{e}^{-l_{x,j,t}(|\\zeta_{x,j,t}|)}\n    \\end{pmatrix},\\]\n    where $l_{x,j,t}(r)=\\psi_1(8\\lambda_{x,j}(t)r^{3\/2}\/3)$, and $\\psi_1$ satisfies the ODE\n    \\begin{equation}\\label{PainleveODE_eq}\n      \\left(\\frac{\\mathrm{d}^2}{\\mathrm{d}\\rho^2}+\\frac{1}{\\rho}\\frac{\\mathrm{d}}{\\mathrm{d}\\rho}\\right)\\psi_1(\\rho)=\\frac{1}{2}\\sinh(2\\psi_1(\\rho)),\n    \\end{equation}\n    with asymptotics\n    \\[\\psi_1(\\rho)\\sim \\frac{1}{\\pi}K_0(\\rho)\\text{ as }\\rho\\to\\infty,\\quad \\psi_1(\\rho)\\sim -\\frac{1}{3}\\log\\rho\\text{ as }\\rho\\to 0.\\]\n  Here  $K_0$ is the modified Bessel function of the second kind, $K_0(\\rho)\\sim \\rho^{-1\/2}\\mathrm{e}^{-\\rho}$ as $\\rho\\to\\infty$. Then we define\n\\begin{equation}\\label{approxMetricZ_eq}\n  h_t^{\\mathrm{app}}=\\begin{pmatrix}\n      |\\zeta_{x,j,t}|^{1\/2}\\mathrm{e}^{\\chi(|\\zeta_{x,j,t}|\\kappa^{-1}t^{1\/j(x)})l_{x,j,t}(|\\zeta_{x,j,t}|)}&0\\\\0&|\\zeta_{x,j,t}|^{-1\/2}\\mathrm{e}^{-\\chi(|\\zeta_{x,j,t}|\\kappa^{-1}t^{1\/j(x)})l_{x,j,t}(|\\zeta_{x,j,t}|)}\n    \\end{pmatrix},\n\\end{equation}\n  where $\\chi$ is a nonincreasing smooth cutoff function with $\\chi(r)=1$ for $r\\leqslant 1\/2$ and $\\chi(r)=0$ for $r\\geqslant 1$.\n  \\item For $x\\in T$, in the above holomorphic frame over $\\widetilde{B}_{x,t}$, define\n  \\[h_{t}^{\\mathrm{model}}=\\begin{pmatrix}\n    |\\zeta_{x,t}|^{1\/2}\\mathrm{e}^{m_{x,t}(|\\zeta_{x,t}|)}&0\\\\0&|\\zeta_{x,t}|^{3\/2}\\mathrm{e}^{-m_{x,t}(|\\zeta_{x,t}|)}\n  \\end{pmatrix},\\]\n  where $m_{x,t}(r)=\\psi_{2,x}(8\\lambda_x(t)r^{1\/2})$, and $\\psi_{2,x}$ satisfies \\eqref{PainleveODE_eq} with asymptotics\n  \\[\\psi_{2,x}(\\rho)\\sim \\frac{1}{\\pi}K_0(\\rho)\\text{ as }\\rho\\to\\infty,\\quad \\psi_{2,x}(\\rho) \\sim (1+2\\alpha_{x,1}-2\\alpha_{x,2})\\log\\rho\\text{ as }\\rho\\to 0.\\]\n  Then we define\n\\begin{equation}\\label{approxMetricT_eq}\n  h_t^{\\mathrm{app}}=\\begin{pmatrix}\n      |\\zeta_{x,t}|^{1\/2}\\mathrm{e}^{\\chi(|\\zeta_{x,t}|\\kappa^{-1})m_{x,t}(|\\zeta_{x,t}|)}&0\\\\0&|\\zeta_{x,t}|^{3\/2}\\mathrm{e}^{-\\chi(|\\zeta_{x,t}|\\kappa^{-1})m_{x,t}(|\\zeta_{x,t}|)}\n    \\end{pmatrix}.\n\\end{equation}\n    \\end{enumerate}\n  It is proved in \\cite{mccoy1977painleve} that there exist unique functions $\\psi_1$, $\\psi_{2,x}$ solving \\eqref{PainleveODE_eq} with the prescribed asymptotics. Finally, set $h_{t}^{\\mathrm{app}}=h_{t,\\infty}$ on the complement of the above disks.\n\\end{definition}\nIt is straightforward to check that $h_{t}^{\\mathrm{model}}$ satisfies Hitchin's equation over $\\widetilde{B}_{x(,j),t}$, smooth near $z_{x,j}(t)\\in Z_t\\backslash(I_t\\cup T)$ and adapted to the parabolic structure at $x\\in T$. $h_t^{\\mathrm{app}}$ approximately solves Hitchin's equation in the following sense.\n\\begin{proposition}\n  There exists positive constants $t_0$, $c,c'$, such that for any $t\\geqslant t_0$,\n  \\begin{equation}\\label{ErrorEst_eq}\n  \\left\\lVert F_{h_t^\\mathrm{app}}+\\left[\\Phi_t,\\Phi_t^{\\ast_{h_t^\\mathrm{app}}}\\right] \\right\\rVert_{L^2}\\leqslant c\\mathrm{e}^{-c' t^{\\sigma}},\n\\end{equation}\nwhere the $L^2$-norm is taken with respect to $h_t^{\\mathrm{app}}$ and a fixed Riemannian metric on $C$, and\n\\[\\sigma=\\min\\left(\\frac{m_0-1}{m_0+N-4},\\min_{x\\in I_u\\backslash\\{0\\}}\\frac{m_x-1}{m_x},\\min_{x\\in I_t}\\frac{2m_x-3}{2(m_x-1)},\\frac{1}{2}\\delta_{T\\neq \\varnothing}\\right),\\]\n$\\delta_{T\\neq \\varnothing}=1$ if $T\\neq \\varnothing$, otherwise it equals to $2$.\n\\end{proposition}\n\\begin{proof}\n  Since $h_t^{\\mathrm{app}}$ coincides with $h_{t}^{\\mathrm{model}}$ and $h_{t,\\infty}$ away from some annuli, solving Hitchin's equation, we only need to estimate the error in these regions. First, we consider $\\mathrm{Ann}_{x,j}=\\{\\,\\kappa t^{-1\/j(x)}\/2<|\\zeta_{x,j,t}|<\\kappa t^{-1\/j(x)}\\,\\}$. Denote $|\\zeta_{x,j,t}|$ by $r$ for simplicity.\n  \\begin{align*}\n  F_{h_t^{\\mathrm{app}}}&=-\\frac{1}{4}\\left(\\Delta l_{x,j,t}(r)\\chi\\left(r\\kappa^{-1}t^{1\/j(x)}\\right)\\right)\\sigma_3\\, \\mathrm{d}\\zeta_{x,j,t}\\mathrm{d}\\bar{\\zeta}_{x,j,t}\\\\\n  &=-\\frac{1}{4}\\left(\\vphantom{\\frac{1}{r}}l_{x,j,t}''(r)\\chi\\left(r\\kappa^{-1}t^{1\/j(x)}\\right)+2\\kappa^{-1}t^{1\/j(x)}l_{x,j,t}'(r)\\chi'\\left(r\\kappa^{-1}t^{1\/j(x)}\\right)\\right.\\\\&\\quad+\\kappa^{-2}t^{2\/j(x)}l_{x,j,t}(r)\\chi''\\left(r\\kappa^{-1}t^{1\/j(x)}\\right)\\\\&\\quad+\\left.\\frac{1}{r}l_{x,j,t}'(r)\\chi\\left(r\\kappa^{-1}t^{1\/j(x)}\\right)+\\frac{1}{r}\\kappa^{-1}t^{1\/j(x)}l_{x,j,t}(r)\\chi'\\left(r\\kappa^{-1}t^{1\/j(x)}\\right)\\right)\\sigma_3\\,\\mathrm{d}\\zeta_{x,j,t}\\mathrm{d}\\bar\\zeta_{x,j,t}.\n\\end{align*}\nOn the other hand,\n\\[  \\left[\\Phi,\\Phi^{\\ast_{h_t^{\\mathrm{app}}}}\\right]=2\\lambda_{x,j}(t)^2r\\sinh\\left(2l_{x,j,t}(r)\\chi\\left(r\\kappa^{-1}t^{1\/j(x)}\\right)\\right)\\sigma_3\\,\\mathrm{d}\\zeta_{x,j,t}\\mathrm{d}\\bar\\zeta_{x,j,t}.\\]\nBy \\eqref{PainleveODE_eq} and the definition of $l_{x,j,t}$, we have\n\\[l''_{x,j,t}(r)+r^{-1}l_{x,j,t}'(r)=8\\lambda_{x,j}(t)^2 r\\sinh(2l_{x,j,t}),\\quad l_{x,j,t}=O(\\mathrm{e}^{-c'\\lambda_{x,j}(t)t^{-3\/2j(x)}}).\\]\nWe can write $F_{h_t^\\mathrm{app}}+\\left[\\Phi_t,\\Phi_t^{\\ast_{h_t^\\mathrm{app}}}\\right]=E_{x,j,t}\\sigma_3\\,\\mathrm{d}\\zeta_{x,j,t}\\mathrm{d}\\bar\\zeta_{x,j,t}$, the $L^2$ norm of which is bounded by $ct^{-1\/j(x)}\\sup|E_{x,j,t}|$. Here and below, $c,c'>0$ are generic constants which may vary, but does not depend on $t$. By Lemma \\ref{HiggsDetRoot_lem}, as $t\\to\\infty$,\n\\[\\lambda_{x,j}(t)\\sim t^{1\/2}\\prod_{y\\in S}|z_{x,j}(t)-y|^{-m_x}\\prod_{(y,k)\\neq(x,j)}|z_{x,j}(t)-z_{y,k}(t)|^{1\/2}\\sim \\begin{cases}\n  c' t^{\\frac{2m_x+1}{2j(x)}}&\\text{ if }x\\in I_u,\\\\\n  c' t^{\\frac{m_x}{j(x)}}&\\text{ if }x\\in I_t.\n\\end{cases}\\]\nThen for $x\\in I_u$,\n\\begin{align*}\n  |E_{x,j,t}|&\\leqslant c\\left(\\lambda_{x,j}(t)^2r(\\left|\\sinh(2l_{x,j,t})-\\sinh(2l_{x,j,t}\\chi)\\right|+|\\chi-1||\\sinh(2l_{x,j,t})|)\\right.\\\\&\\left.\\quad+t^{1\/j(x)}|l'_{x,j,t}|+t^{2\/j(x)}|l_{x,j,t}|\\right)\\\\\n  &\\leqslant c\\left(t^{2m_x\/j(x)}\\cosh(2l_{x,j,t})|l_{x,j,t}|+t^{1\/j(x)}|l'_{x,j,t}|+t^{2\/j(x)}|l_{x,j,t}|\\right)\\\\&\\leqslant c\\left(t^{2m_x\/j(x)}|l_{x,j,t}|+t^{2\/j(x)}|l'_{x,j,t}|\\right)\\leqslant c \\mathrm{e}^{-c' t^{(m_x-1)\/j(x)}}.\n\\end{align*}\nFor $x\\in I_t$, $|E_{x,j,t}|\\leqslant c\\mathrm{e}^{-c' t^{(2m_x-3)\/2j(x)}}$. Let $\\mathrm{Ann}_x=\\{\\,\\kappa \/2<|\\zeta_{x,t}|<\\kappa \\,\\}$ for $x\\in T$, and denote $r=|\\zeta_{x,t}|$. Similar as before,\n\\begin{align*}\n  F_{h_t^{\\mathrm{app}}}  &=-\\frac{1}{4}\\left(\\vphantom{\\frac{1}{r}}m_{x,t}''(r)\\chi\\left(r\\kappa^{-1}\\right)+2\\kappa^{-1}m_{x,t}'(r)\\chi'\\left(r\\kappa^{-1}\\right)+\\kappa^{-2}m_{x,t}(r)\\chi''\\left(r\\kappa^{-1}\\right)\\right.\\\\&\\quad+\\left.\\frac{1}{r}m_{x,t}'(r)\\chi\\left(r\\kappa^{-1}\\right)+\\frac{1}{r}\\kappa^{-1}m_{x,t}(r)\\chi'\\left(r\\kappa^{-1}\\right)\\right)\\sigma_3\\,\\mathrm{d}\\zeta_{x,t}\\mathrm{d}\\bar{\\zeta}_{x,t},\\\\\n  \\left[\\Phi,\\Phi^{\\ast_{h_t^{\\mathrm{app}}}}\\right]\n &=2\\lambda_{x}(t)^2r^{-1}\\sinh\\left(2m_{x,t}(r)\\chi\\left(r\\kappa^{-1}\\right)\\right)\\sigma_3\\,\\mathrm{d}\\zeta_{x,t}\\mathrm{d}\\bar{\\zeta}_{x,t}.\n\\end{align*}\nNow we have\n\\[m_{x,t}''(r)+r^{-1}m_{x,t}'(r)=8\\lambda_{x}(t)^2 r^{-1}\\sinh(2m_{x,t}),\\quad m_{x,j,t}=O(\\mathrm{e}^{-c'\\lambda_{x}(t)}),\\]\nwhere $\\lambda_{x}(t)\\sim c' t^{1\/2}$. For $E_{x,t}$ defined similarly as above, we have the estimate \\[|E_{x,t}|\\leqslant c\\mathrm{e}^{-c' t^{1\/2}}.\\]\nCombining all the estimates above, the result follows.\n\\end{proof}\n\\begin{remark}\\label{WeakParabolic_rmk}\n  If we allow tame untwisted singularities (also called weakly parabolic), then $\\lambda_{x,j}(t)\\sim c' t^{3\/2}$, and $l_{x,j,t}=O(\\mathrm{e}^{-c' t^{3\/2}t^{-3\/2}})=O(1)$ which does not decay to $0$ as $t\\to\\infty$. Thus, the error cannot be arbitrarily small when tame untwisted poles exist.\n\\end{remark}\n\n\\section{Linear Analysis}\nWe wish to obtain the solution $h_t$ to Hitchin's equation by perturbing $h_t^{\\mathrm{app}}$. This is equivalent to finding an $\\mathrm{SL}(E)$-valued $h_t^{\\mathrm{app}}$-Hermitian section $H_t$ such that $h_t(v,w)=h_t^{\\mathrm{app}}(H_tv,w)$. Let $\\mathrm{e}^{\\gamma_t}=H_t^{-1\/2}$, then $\\mathrm{e}^{\\gamma_t}\\cdot h_t(v,w)=h_t(H_t^{-1\/2}v,H_t^{-1\/2}w)=h_t^{\\mathrm{app}}(v,w)$. By the gauge invariance of Hitchin's equations, $(\\bar{\\partial}_E,\\Phi_t,h_t)$ is a solution is equivalent to that \\[\\mathrm{e}^{\\gamma_t}\\cdot (\\bar{\\partial}_E,\\Phi_t,h_t)=(\\mathrm{e}^{-\\gamma_t}\\bar{\\partial}_E\\mathrm{e}^{\\gamma_t},\\mathrm{e}^{-\\gamma_t}\\Phi_t \\mathrm{e}^{\\gamma_t},h_t^{\\mathrm{app}})\\]\nis a solution. Therefore, we need to find $\\gamma_t\\in \\Omega^0(i\\mathfrak{su}(E))$ (the bundle is defined over $C\\backslash S$ using $E$ and $h_t^{\\mathrm{app}}$), such that $F_t(\\gamma_t)=0$, where\n\\[F_t(\\gamma):=F_{A_t^{\\exp(\\gamma)}}+\\big[\\mathrm{e}^{-\\gamma}\\Phi_t\\mathrm{e}^{\\gamma},\\mathrm{e}^{\\gamma}\\Phi_t^{\\ast_{h_t^{\\mathrm{app}}}}\\mathrm{e}^{-\\gamma}\\big],\\]\n$A_t$ is the Chern connection $D(\\bar{\\partial}_E,h_t^{\\mathrm{app}})$ and $A_t^{\\exp(\\gamma)}=D(\\mathrm{e}^{-\\gamma}\\bar{\\partial}_E\\mathrm{e}^{\\gamma},h_t^{\\mathrm{app}})$. Let $\\mathrm{d}_{A_t}=\\partial_{A_t}+\\bar{\\partial}_E$ be the exterior derivative associated with $A_t$, $\\nabla_{A_t}$ be the covariant derivative. We have\n\\[ F_{A_t^{\\mathrm{exp}(\\gamma)}}=\\mathrm{e}^{-\\gamma}\\left(F_{A_t}+\\bar{\\partial}_{A_t}\\left(\\mathrm{e}^{2\\gamma} \\partial_{A_t}\\left(\\mathrm{e}^{-2\\gamma}\\right)\\right)\\right)\\mathrm{e}^\\gamma.\\]\nLinearize $F_t$ at $\\gamma=0$ we get\n\\begin{align*}\n  DF_t(\\gamma)&=[F_{A_t},\\gamma]-2\\bar{\\partial}_{A_t}\\partial_{A_t}\\gamma+[[\\Phi_t,\\gamma],\\Phi_t^{\\ast_{h_t^{\\mathrm{app}}}}]+[\\Phi_t,[\\gamma,\\Phi_t^{\\ast_{h_t^{\\mathrm{app}}}}]]\\\\&=\\mathrm{i}\\star\\Delta_{A_t}\\gamma+M_{\\Phi_t}\\gamma,\n\\end{align*}\nwhere $\\Delta_{A_t}=\\mathrm{d}_{A_t}^\\ast \\mathrm{d}_{A_t}$ on $\\Omega^0(i\\mathfrak{su}(E))$, $M_{\\Phi_t}\\gamma=[\\Phi_t^{\\ast_{h_t^{\\mathrm{app}}}},[\\Phi_t,\\gamma]]-[\\Phi_t,[\\Phi_t^{\\ast_{h_t^{\\mathrm{app}}}},\\gamma]]$, and we used the identity $2\\bar{\\partial}_A\\partial_A=F_A-\\mathrm{i}\\star\\Delta_A$. By composing $-\\mathrm{i}\\star: \\Omega^2(\\mathfrak{su}(E))\\to\\Omega^0(i\\mathfrak{su}(E))$ we get the operator\n\\begin{equation}\\label{LinOp_eq}\n  L_t(\\gamma)=\\Delta_{A_t}\\gamma-\\mathrm{i}\\star M_{\\Phi_t}\\gamma.\n\\end{equation}\nConsider the irregular connection defined by $D_t^\\gamma=\\mathrm{d}_{A_t^{\\mathrm{exp}(\\gamma)}}+\\mathrm{e}^{-\\gamma}\\Phi_t\\mathrm{e}^{\\gamma}+\\mathrm{e}^{\\gamma}\\Phi_t^{\\ast}\\mathrm{e}^{-\\gamma}$, here and below we drop the subscript of $\\ast_{h_t^\\mathrm{app}}$. Then its curvature $\\left(D_t^\\gamma\\right)^2=F_t(\\gamma)$, since $\\bar{\\partial}_E \\Phi_t=\\partial_{A_t}\\Phi_t^\\ast=0$. So we can also regard the problem as finding $\\gamma$ so that this connection is flat. When $\\gamma=0$, write the operator as $D_t$, which is the sum of the unitary part $\\mathrm{d}_{A_t}$ and the self-adjoint part $\\Psi_t:=\\Phi_t+\\Phi_t^\\ast$.\n\\begin{lemma}(Weitzenb\u00f6ck formulas)\\label{weitz_lemma}\n  Suppose $\\gamma\\in\\Omega^0(i\\mathfrak{su}(E))$, $w\\in \\Omega^p(\\mathrm{End}\\,E)$, then \\begin{enumerate}[label=(\\roman*)]\n    \\item $(D_t^\\ast D_t+D_tD_t^\\ast)w=\\nabla_{A_t}^\\ast\\nabla_{A_t} w+(\\Psi_t\\otimes )^\\ast\\Psi_t\\otimes w+\\mathscr{F}(w)+\\mathscr{R}(w)$, where $\\mathscr{F},\\mathscr{R}$ are curvature operators, $\\Psi_t\\otimes\\cdot$ is a combination of tensor product on the manifold part and Lie bracket on the bundle part. More explicitly, let $e_1,e_2$ be a local orthonormal frame of $TC$ and $e^1,e^2$ be the dual frame, write the curvature $D_t^2$ as $F_{t,12}e^1\\wedge e^2$, then $\\mathscr{F}(w)=[F_{t,12},(e^1\\wedge i_{e_2}-e^2\\wedge  i_{e_1})w]$, and\n  \\begin{align*}\n    \\mathscr{R}(w)(X_1,\\ldots,X_p)&=\\sum_{j=1}^pw(X_1,\\ldots,\\mathrm{Ric}(X_j),\\ldots,X_p)\\\\&+\\sum_{a=1}^2\\sum_{j<k}(-1)^{j+k+1}w(e_a,R(X_j,X_k)e_a,X_1,\\ldots,\\hat{X}_j,\\ldots,\\hat{X}_k,\\ldots,X_p).\\hspace{-1.5cm}\n  \\end{align*}\n    \\item $D_t^\\ast D_t\\gamma=\\mathrm{d}_{A_t}^\\ast \\mathrm{d}_{A_t}\\gamma+([\\Psi_t,\\cdot])^\\ast[\\Psi_t,\\gamma]=L_t\\gamma$.\n    \\item $\\mathrm{Re}\\left(\\langle \\mathrm{d}_{A_t}\\gamma,[\\Psi_t,\\gamma]\\rangle\\right)=0$, so the following pointwise equality holds\n   \\[\\langle D_t\\gamma,D_t\\gamma\\rangle= |D_t\\gamma|^2=|\\mathrm{d}_{A_t}\\gamma|^2+|[\\Psi_t,\\gamma]|^2.\\]\n  \\end{enumerate}\n\\end{lemma}\n\\begin{proof}\n  (\\romannumeral1) This follows from \\cite[Theorem 5.4]{biquard_1997}, since $\\mathrm{d}_{A_t}^\\ast \\Psi_t=0$.\\\\\n  (\\romannumeral2) The first equality is a special case of (\\romannumeral1). The adjoint of $[\\Phi_t,\\cdot]$ is $-\\bar{\\ast}[\\Phi_t,\\bar{\\ast}\\cdot]$, where $\\bar{\\ast}$ is the conjugate linear operator combining $\\bar{\\star}$ and the Hermitian adjoint. Actually, for a fixed holomorphic coordinate $z$ where $\\Phi=\\varphi\\,\\mathrm{d}z$ (subscript $t$ omitted),\n  \\begin{align*}\n    \\langle [\\Phi,a],b\\,\\mathrm{d}z\\rangle&=\\langle [\\varphi,a]\\,\\mathrm{d}z,b\\,\\mathrm{d}z\\rangle=2\\langle [\\varphi,a],b\\rangle,\\\\\n    \\langle a,-\\bar{\\ast}[\\Phi,\\bar{\\ast} b\\,\\mathrm{d}z]\\rangle&=\\langle a,-\\bar{\\ast} [\\Phi,ib^\\ast \\mathrm{d}\\bar{z}]\\rangle=\\langle a,-[b,\\varphi^\\ast]\\bar{\\star} (\\mathrm{i}\\,\\mathrm{d}z \\mathrm{d}\\bar{z})\\rangle=2\\langle a,[\\varphi^\\ast,b]\\rangle=2\\langle [\\varphi,a],b\\rangle.\n  \\end{align*}\nSimilarly, the adjoint of $[\\Phi_t^\\ast,\\cdot]$ is $-\\bar{\\ast}[\\Phi_t^\\ast,\\bar{\\ast}\\cdot]$. Then (recall that $\\bar{\\ast}\\mathrm{d}z=\\mathrm{i}\\,\\mathrm{d}\\bar{z}\n$, $\\bar{\\ast}\\mathrm{d}\\bar{z}=-\\mathrm{i}\\,\\mathrm{d}z$)\n\\begin{align*}\n  ([\\Psi_t,\\cdot])^\\ast[\\Psi_t,\\gamma]&=-\\bar{\\ast}[\\Phi_t,\\bar{\\ast}[\\Phi_t,\\gamma]]-\\bar{\\ast}[\\Phi_t^\\ast,\\bar{\\ast}[\\Phi_t^\\ast,\\gamma]]\\\\\n  &=-\\bar{\\ast}[\\Phi_t,\\mathrm{i}[\\Phi_t^\\ast,\\gamma]]+\\bar{\\ast}[\\Phi_t^\\ast,\\mathrm{i}[\\Phi_t,\\gamma]]\\\\\n  &=-\\mathrm{i}\\star [\\Phi_t^\\ast,[\\Phi_t,\\gamma]]+\\mathrm{i}\\star [\\Phi_t,[\\Phi_t^\\ast,\\gamma]]=-\\mathrm{i}\\star M_{\\Phi_t}\\gamma.\n\\end{align*}\n(\\romannumeral3) Write $\\mathrm{d}_{A_t}\\gamma=\\gamma_z\\,\\mathrm{d}z+\\gamma_{\\bar{z}}\\,\\mathrm{d}\\bar{z}$. $\\gamma_{\\bar{z}}=\\gamma_z^\\ast$, since $\\gamma=\\gamma^\\ast$ and $\\mathrm{d}_{A_t}$ is a Hermitian connection. As above, write $\\Phi=\\varphi\\, \\mathrm{d}z$, then \\begin{align*}\n  \\mathrm{Re}\\left(\\langle \\mathrm{d}_{A_t}\\gamma,[\\Phi+\\Phi^\\ast,\\gamma]\\rangle\\right)&=2\\mathrm{Re}\\left(\\mathrm{tr}(\\gamma_z[\\varphi,\\gamma]^\\ast)+\\mathrm{tr}(\\gamma_{\\bar{z}}[\\varphi^\\ast,\\gamma]^\\ast)\\right)\\\\\n  &=2\\mathrm{Re}\\left(\\mathrm{tr}(\\gamma_z[\\varphi,\\gamma]^\\ast)+\\mathrm{tr}(\\gamma_z^\\ast[\\gamma,\\varphi])\\right)\\\\\n  &=2\\mathrm{Re}\\left(\\mathrm{tr}(\\gamma_z[\\varphi,\\gamma]^\\ast)-\\overline{\\mathrm{tr}(\\gamma_z[\\varphi,\\gamma]^\\ast)}\\right)=0.\n  \\end{align*}\n\\end{proof}\nChoose a local holomorphic trivialization as in Proposition \\ref{NormalForm_prop} (\\romannumeral3). Consider the map $\\sigma_{x,t}:D_{x,t}\\to \\widetilde{B}_{x,t},~\\xi_{x,t}\\to \\xi_{x,t}^2=\\zeta_{x,t}$, where $D_{x,t}=\\{\\,|\\xi_{x,t}|<\\kappa^{1\/2}t^{-1\/2(m_x-1)}\\,\\}$. Then for the pullbacks $\\widetilde{E}=\\sigma_{x,t}^\\ast E$, $\\widetilde{\\Phi}_t=\\sigma_{x,t}^\\ast \\Phi_t$, $\\tilde{h}_t^{\\mathrm{app}}=\\sigma_{x,t}^\\ast h_t^{\\mathrm{app}}$, we have\n\\[ \\tilde{\\bar{\\partial}}_E=\\bar{\\partial},\\quad \\widetilde{\\Phi}_t=2\\lambda_x(t)\\begin{pmatrix}\n    0&\\frac{1}{\\xi_{x,t}^{2m_x-3}}\\\\\\frac{1}{\\xi_{x,t}^{2m_x-1}}&0\n\\end{pmatrix}\\,\\mathrm{d}\\xi_{x,t},\\quad \\tilde{h}_t^{\\mathrm{app}}=\\begin{pmatrix}\n  |\\xi_{x,t}|&0\\\\0&|\\xi_{x,t}|^3\n\\end{pmatrix}. \\]\nAfter applying the gauge transformation $\\left(\\begin{smallmatrix}\n  \\xi_{x,t}&-\\xi_{x,t}\\\\1&1\n\\end{smallmatrix}\\right)$ (noninvertible at $0$),\n\\begin{equation}\\label{DiagHiggs2Cover_eq}\n  \\widetilde{\\Phi}_t=2\\lambda_x(t)\\begin{pmatrix}\n      \\frac{1}{\\xi_{x,t}^{2(m_x-1)}}&0\\\\0&\\frac{1}{\\xi_{x,t}^{2(m_x-1)}}\n  \\end{pmatrix}\\,\\mathrm{d}\\xi_{x,t},\\quad \\tilde{h}_t^{\\mathrm{app}}=2\\begin{pmatrix}\n    |\\xi_{x,t}|^3&0\\\\0&|\\xi_{x,t}|^3\n  \\end{pmatrix}.\n\\end{equation}\nFor $x\\in T$, denote $\\hat{B}_{x,t}:=\\{\\,|\\zeta_{x,t}|<\\lambda_x(t)^{-2}\\,\\}$. Choose $t_0\\geqslant 1$ so that when $t\\geqslant t_0$, we have $\\lambda_x(t)^{-2}<\\kappa\/2$, and then $h_t^{\\mathrm{app}}=h_t^{\\mathrm{model}}$ over $\\hat{B}_{x,t}$. From now on, we assume that $t\\geqslant t_0$.\n\\begin{definition}\n  Let $r_t$ be a smooth weight function on $C$ such that\n\\begin{enumerate}[label=(\\roman*)]\n  \\item  $r_t=1$ in $C\\backslash (\\bigcup_{x\\in I}\\widetilde{B}_{x,t}\\cup\\bigcup_{x\\in T}\\hat{B}_{x,t})$.\n  \\item  In $\\widetilde{B}_{x,t}$, $x\\in S$, $r_t$ is nondecreasing in $|\\zeta_{x,t}|$ (or $|z_x|$ for $x\\in I_u$).\n  \\item \\[r_t=\\begin{cases}\n      |t^{1\/j(x)}z_x|\\quad &\\text{if }x\\in I_u,~|z_x|\\leqslant \\kappa t^{-1\/j(x)}\/2,  \\\\\n      |t^{1\/(m_x-1)}\\zeta_{x,t}|^{1\/2}\\quad &\\text{if }x\\in I_t,~|\\zeta_{x,t}|\\leqslant \\kappa t^{-1\/(m_x-1)}\/2,\\\\\n      |\\lambda_x(t)^2\\zeta_{x,t}|\\quad &\\text{if }x\\in T,~|\\zeta_{x,t}|\\leqslant \\lambda_x(t)^{-2}\/2.\n    \\end{cases}\\]\n\\end{enumerate}\n\\end{definition}\n  Denote $B_x=\\{\\,|z_x|<\\kappa_2\\,\\}$, where $\\kappa_2$ is from Proposition \\ref{NormalForm_prop}, and we can assume that $\\kappa_2$ is small enough so that the disks $B_{x}$ are disjoint.\n\\begin{definition}\\label{Metricgt_def}\n  Let $g_t=f_t g_C$ be the metric on $C$ conformal to the round metric $g_C$ (Hitchin's equations are conformally invariant, we consider $L_t$ above with $\\star$ and $\\Delta_{A_t}$ defined using $g_t$), which satisfies\n\\begin{enumerate}[label=(\\roman*)]\n  \\item $f_t=1$ in $C\\backslash(\\bigcup_{x\\in S}B_x)$, independent of $t$ in $C\\backslash(\\bigcup_{x\\in I_t\\cup T}B_x)$.\n  \\item $g_t$ induces the volume form $(i\/8|\\zeta_{x,t}|)\\,\\mathrm{d}\\zeta_{x,t} \\mathrm{d}\\bar{\\zeta}_{x,t}$ when restricted to $\\widetilde{B}_{x,t}$ for $x\\in I_t$, its pullback under $\\sigma_{x,t}$ is $(i\/2)\\,\\mathrm{d}\\xi_{x,t} \\mathrm{d}\\bar{\\xi}_{x,t}$, the Euclidean volume form on $D_{x,t}$.\n  \\item $g_t$ is Euclidean in $B_x$ for $x\\in I_u$.\n  \\item $g_t$ is Euclidean in $\\widetilde{B}_{x,t}$ for $x\\in T$.\n  \\item $|\\nabla_{g_C}^j \\log\\,f_t|_{g_C}\\leqslant c$ in $B_x$ for $x\\in I_u\\cup T$, $j=0,1,2$, and $c^{-1}|z_x|^{-1}\\leqslant f_t\\leqslant c|z_x|^{-1}$, $|\\nabla_{g_C}^j \\log\\,f_t|_{g_C}\\leqslant ct^{j\/(m_x-1)}$ in $B_x\\backslash \\widetilde{B}_{x,t}$ for $x\\in I_t$, $j=1,2$.\n\\end{enumerate}\n\\end{definition}\n\n\\begin{definition}\n  For smooth sections $w\\in\\Omega^p(\\mathrm{End} E)$, define\n\\begin{equation}\\label{SobolevSpace_eq}\n\\lVert w\\rVert_{L_\\delta^p}=\\lVert r_t^{-\\delta-2\/p} w\\rVert_{L^p},~  \\lVert w\\rVert_{L_\\delta^{k,p}}^p=\\lVert w\\rVert_{L_\\delta^{k-1,p}}^p+\\lVert \\nabla_{A_t}w\\rVert_{L_\\delta^{k-1,p}}^p+\\lVert \\Psi_t\\otimes w\\rVert_{L_\\delta^{k-1,p}}^p\n\\end{equation}\ninductively for $k=1,2\\ldots$ Here the $L^p$ norms are taken with respect to $g_t$ and $h_t^{\\mathrm{app}}$. Let the spaces $L_\\delta^p$ and $L_\\delta^{k,p}$ be the completions of smooth sections over $C\\backslash S$ with respect to these norms. We also introduce spaces with additional weighted conditions,\n\\[\\hat{L}_{\\delta}^{k,p}=\\left\\{\\,f\\in L_{\\delta}^{k,p},r_t^{j-k}\\nabla_{A_t}^j f\\in L_\\delta^p\\text{ for }0\\leqslant j\\leqslant k\\right\\}.\\]\nBy using cutoff functions of the form $1-\\chi(nr_t)$, one can see that $C_0^\\infty(C\\backslash S)$ is dense in $\\hat{L}_\\delta^{k,p}$.\n\\end{definition}\n\nIn $\\widetilde{B}_{x,t}$ for $x\\in I_u$, we can decompose $\\mathrm{End}\\, E$ as $\\mathrm{End}_D\\, E\\oplus \\mathrm{End}_T\\, E$ in the holomorphic frame above, where $\\mathrm{End}_D$ is the diagonal part. Then $\\mathrm{End}_D\\, E=\\ker \\mathrm{ad}(\\Phi_t), \\mathrm{End}_T\\, E=\\left(\\ker \\mathrm{ad}(\\Phi_t)\\right)^\\perp$. $\\Phi_t$ acts on $u\\in\\mathrm{End}_T\\, E$ as\n\\[[\\Phi_t,u]= 2 \\frac{\\sqrt{-z_x^{2m_x}\\nu_t(z)}}{z_x^{m_x}}\\begin{pmatrix}\n0&u_{12}\\\\-u_{21}&0\n\\end{pmatrix}\\mathrm{d}z\\Rightarrow |[\\Phi_t,u]|=2\\frac{|-z_x^{2m_x}\\nu_t(z)|^{1\/2}}{|z_x|^{m_x}}|u|,\\]\nand then\n\\begin{equation}\\label{HiggsActBd_eq}\nc^{-1}t^{m_x\/j(x)} |u|r_t^{-m_x}  \\leqslant |[\\Phi_t,u]|\\leqslant ct^{m_x\/j(x)} |u|r_t^{-m_x}\\text{ in }\\widetilde{B}_{x,t}.\n\\end{equation}\n Therefore, over $\\widetilde{B}_{x,t}$, the spaces defined above are equivalent (may not be uniformly in $t$) to the spaces defined in \\cite{biquard_boalch_2004}, using the irregular Hermitian connection $A_t+\\Phi_t-\\Phi_t^\\ast$ and the decomposition of $\\mathrm{End}\\, E$ into components $\\mathrm{End}(E)_0=\\mathrm{End}_D\\, E$, $\\mathrm{End}(E)_{m_x}=\\mathrm{End}_T\\, E$, from which we collect the following facts over $\\widetilde{B}_{x,t}$.\n\\begin{lemma}\\label{SobEmbed_lem}\n\\begin{enumerate}[label=(\\roman*)]\n\\item For functions, we have $\\hat{L}_{\\delta-1}^{1,p}\\hookrightarrow L_\\delta^q$ if $1\/2\\geqslant 1\/p-1\/q$ (with strict inequality for $q=\\infty$). In particular, for $p>2$ we have $\\hat{L}_{\\delta-1}^{1,p}\\hookrightarrow C_\\delta^0$.\n\\item If $\\delta<0$ and the function $f$ vanishes on $\\partial \\widetilde{B}_{x,t}$, or if $\\delta>0$ and $f$ vanishes near $x$, then\n\\[\\lVert \\partial f\/\\partial r\\rVert_{L_{\\delta-1}^p}\\geqslant c^{-1}t^{1\/j(x)}\\lVert f\\rVert_{L_{\\delta}^p},\\]\nwhere $r=|z_x|$. This implies that $\\hat{L}_{\\delta-1}^{1,p}=L_{\\delta-1}^{1,p}$ for $\\delta<0$ and $p\\in[1,\\infty)$.\n\\item If $u\\in L_{-2+\\delta}^{2,2}$ is a section of $\\mathrm{End}_D\\, E$, then $u$ is continuous and $u-u(x)\\in\\hat{L}_{-2+\\delta}^{2,2}\\subset C_\\delta^0\\cap L_\\delta^p$.\n\\item For $\\delta'<\\delta$, $p>2$, there are compact Sobolev embeddings for sections of $\\mathrm{End}_T\\, E$: \\[\nL_{\\delta}^{1,2}\\hookrightarrow L_{\\delta'+1+(2m_x-2)\/p}^p,\\quad L_{\\delta}^{1,p}\\hookrightarrow C_{\\delta'+m_x-2(m_x-1)\/p}^0,\\quad\nL_{\\delta}^{2,2}\\hookrightarrow C_{\\delta'+m_x+1}^0.\n\\]\n\\end{enumerate}\n\\end{lemma}\n\nNow we consider the Dirichlet problem on $\\widetilde{B}_{x,t}$ and derive some a priori estimates. The proofs of the following two results are adapted from \\cite[Cor.~4.2,~Lem.~4.4]{biquard_boalch_2004}, with the $t$-dependence of the operator $L_t$ taken into account.\n\n\\begin{lemma}\\label{aprioriEst_lem}\n  Suppose $w\\in L_{-2+\\delta}^{1,2}(\\Omega^1(\\mathrm{End}_T\\, E))$, which vanishes on $\\partial \\widetilde{B}_{x,t}$, then\n  \\[\\lVert D_t w\\rVert_{L_{-2+\\delta}^2}+\\lVert D_t^\\ast w\\rVert_{L_{-2+\\delta}^2}\\geqslant c^{-1}\\big(\\lVert\\nabla_{A_t} w\\rVert_{L_{-2+\\delta}^2}+\\lVert \\Psi_t\\otimes w\\rVert_{L_{-2+\\delta}^2}\\big),\\]\n  for all weights $\\delta$, when $t$ is sufficiently large.\n\\end{lemma}\n\\begin{proof}\n   By Lemma \\ref{weitz_lemma} (\\romannumeral1), noting that the curvature terms vanish,\n   \\[\\lVert (D_t+D_t^\\ast)(r_t^{1-\\delta}w)\\rVert_{L^2}^2=\\lVert \\nabla_{A_t}(r_t^{1-\\delta}w)\\rVert_{L^2}^2+\\lVert r_t^{1-\\delta}\\Psi_t\\otimes w\\rVert_{L^2}^2.\\]\n   The commutator can be controlled as\n \\[|[D_t+D_t^\\ast,r_t^{1-\\delta}]w|+|[\\nabla_{A_t},r_t^{1-\\delta}]w|\\leqslant ct^{1\/j(x)}|r_t^{-\\delta}w|.\\]\n Then we have \\[\\lVert r_t^{1-\\delta}(D_t+D_t^\\ast)w\\rVert_{L^2}\\geqslant c^{-1}\\left(\\lVert r_t^{1-\\delta}\\nabla_{A_t} w\\rVert_{L^2}+\\lVert r_t^{1-\\delta}\\Psi_t\\otimes w\\rVert_{L^2}\\right)-ct^{1\/j(x)}\\lVert r_t^{-\\delta}w\\rVert_{L^2}.\\]\nOn the other hand, \\[\\lVert r_t^{-\\delta}w\\rVert_{L^2}\\leqslant \\lVert r_t^{1-\\delta}r_t^{-m_x}w\\rVert_{L^2}\\leqslant ct^{-m_x\/j(x)} \\lVert r_t^{1-\\delta}\\Psi_t\\otimes w\\rVert_{L^2}.\\]\n Thus \\begin{align*}\n   \\lVert r_t^{1-\\delta}(D_t+D_t^\\ast)w\\rVert_{L^2}&\\geqslant (c^{-1}-ct^{-(m_x-1)\/j(x)})\\left(\\lVert r_t^{1-\\delta}\\nabla_{A_t} w\\rVert_{L^2}+\\lVert r_t^{1-\\delta}\\Psi_t\\otimes w\\rVert_{L^2}\\right)\\\\\n   &\\geqslant (c^{-1}\/2)\\left(\\lVert r_t^{1-\\delta}\\nabla_{A_t} w\\rVert_{L^2}+\\lVert r_t^{1-\\delta}\\Psi_t\\otimes w\\rVert_{L^2}\\right),\n \\end{align*}\n when $t$ is large enough.\n\\end{proof}\n\\begin{proposition}\\label{analysisIu_prop}\n  On $\\widetilde{B}_{x,t}$ for $x\\in I_u$, $L_t:L_{-2+\\delta}^{2,2}(i\\mathfrak{su}(E))\\to L_{-2+\\delta}^{2}(i\\mathfrak{su}(E))$ is an isomorphism with zero Dirichlet boundary condition for small weights $\\delta>0$. When restricted to the off-diagonal part, the same holds for any $\\delta$. Moreover, the norm of $L_t^{-1}$ is uniformly bounded in $t$.\n\\end{proposition}\n\\begin{proof}\nObserve that the decomposition $\\mathrm{End}\\, E=\\mathrm{End}_D\\, E\\oplus \\mathrm{End}_T\\, E$ is preserved by $L_t$, we can consider the two components separately. For the off-diagonal part, we find a solution of $L_tu=v$ by minimizing the functional (recall Lemma \\ref{weitz_lemma} (\\romannumeral3))\n\\[S(u)=\\frac{1}{2}\\int_{\\widetilde{B}_{x,t}} |D_t u|^2\\,\\mathrm{dvol}-\\langle u,v\\rangle_{L^2}=\\frac{1}{2}\\lVert \\mathrm{d}_{A_t}u\\rVert_{L^2}^2+\\lVert[\\Phi_t,u]\\rVert_{L^2}^2-\\langle u,v\\rangle_{L^2}\\]\namong $u\\in L_{-1}^{1,2}(i\\mathfrak{su}_T(E))$ vanishing on the boundary, where $v\\in L_{-m_x-1}^2(i\\mathfrak{su}_T(E))$. $S(u)$ is continuous: \\[S(u)\\leqslant \\lVert \\nabla_{A_t} u\\rVert_{L^2}^2+\\lVert \\Psi_t\\otimes u\\rVert_{L^2}^2+\\lVert r_t^{-m_x}u \\rVert_{L^2}\\lVert r_t^{m_x} v\\rVert_{L^2}\\leqslant \\lVert u\\rVert_{L_{-1}^{1,2}}^2+c\\lVert u\\rVert_{L_{-1}^{1,2}}\\lVert v\\rVert_{L_{-m_x-1}^2},\\]\nwhere we used \\eqref{HiggsActBd_eq} in the last inequality. On the other hand,\n\\begin{align}\n  S(u)&=\\frac{1}{2}\\left(\\lVert \\mathrm{d}_{A_t}u\\rVert_{L^2}^2+2\\lVert [\\Phi_t,u]\\rVert_{L^2}^2\\right)-\\langle u,v\\rangle_{L^2}\\notag\\\\&\\geqslant c^{-1}\\left(\\lVert \\nabla_{A_t} u\\rVert_{L^2}^2+\\lVert \\Psi_t\\otimes u\\rVert_{L^2}^2+t^{2m_x\/j(x)} \\left\\lVert r_t^{-m_x}u\\right\\rVert_{L^2}^2\\right)- \\lVert r_t^{-m_x}u\\rVert_{L^2}\\lVert v\\rVert_{L_{-m_x-1}^2}\\notag\\\\\n  &\\geqslant c^{-1}\\lVert u\\rVert_{L_{-1}^{1,2}}^2-c\\lVert u\\rVert_{L_{-1}^{1,2}}\\lVert v\\rVert_{L_{-m_x-1}^2}.\\label{lowerbdS_eq}\n\\end{align}\nTherefore $S(u)$ is coercive. If $\\mathrm{d}_{A_t}u=0$, then $|u|^2$ is constant and $u=0$ since it vanishes on the boundary. This implies that $S$ is strictly convex, and a unique minimum of $S$ can be found, being a weak solution of the equation of the form \\[\\Delta_{A_t}\\left( r_t^{m_x} u\\right)=r_t^{m_x}v+P_0\\left(r_t^{-m_x}u\\right)+P_1\\left(\\nabla_{A_t} u\\right),\\]\nwhere $P_0,P_1:L^2\\to L^2$ are bounded. By elliptic regularity, $r_t^{m_x} \\nabla_{A_t}^2 u\\in L^2$, and then $u\\in L_{-m_x-1}^{2,2}$. With the Dirichlet boundary condition, we have obtained an isomorphism $L_t: L_{-m_x-1}^{2,2}\\to L_{-m_x-1}^2$. The solution satisfies $\\langle D_tu,D_tf\\rangle_{L^2}=\\langle v,f\\rangle_{L^2}$ for any $f\\in L_{-1}^{1,2}$ vanishing on the boundary. Choose $f=u$ (the first inequality follows by setting $v=0$ in \\eqref{lowerbdS_eq}),\n\\begin{align}\n  \\lVert u\\rVert_{L_{-1}^{1,2}}^2\\leqslant c\\lVert D_tu\\rVert_{L^2}^2= c\\langle v,u\\rangle_{L^2}&\\leqslant ct^{-m_x\/j(x)}\\lVert \\Psi_t\\otimes u\\rVert_{L^2}\\lVert r_t^{m_x}v\\rVert_{L^2}\\notag\\\\&\\leqslant ct^{-m_x\/j(x)}\\lVert u\\rVert_{L_{-1}^{1,2}}\\lVert v\\rVert_{L_{-m_x-1}^2}\\notag\\\\\\Rightarrow~\\lVert u\\rVert_{L_{-1}^{1,2}}&\\leqslant ct^{-m_x\/j(x)}\\lVert v\\rVert_{L_{-m_x-1}^2}.\\label{estimate1_eq}\n\\end{align}\nFor a one form $w$ with coefficient in $i\\mathfrak{su}(E)$, by Lemma \\ref{aprioriEst_lem} we have \\[\\lVert\\nabla_{A_t} w\\rVert_{L_{-m_x-1}^2}\\leqslant c\\left(\\lVert D_t w\\rVert_{L_{-m_x-1}^2}+\\lVert D_t^\\ast w\\rVert_{L_{-m_x-1}^2}\\right).\\]\nThen for $w=\\nabla_{A_t}u$,\n\\begin{align*}\n  D_tw&=(\\mathrm{d}_{A_t}+[\\Psi_t,\\cdot])(\\mathrm{d}_{A_t}u)=[\\Psi_t,\\mathrm{d}_{A_t}u],\\\\\n  D_t^\\ast w&=D_t^\\ast D_tu-D_t^\\ast[\\Psi_t,u]=D_t^\\ast D_tu+\\bar{\\ast}[\\bar{\\ast}\\Psi_t,\\mathrm{d}_{A_t}u]-[\\Psi_t,\\cdot]^\\ast[\\Psi_t,u],\\\\\n  \\lVert r_t^{m_x} (\\Psi_t\\otimes\\cdot)^2 u\\rVert_{L^2}&\\leqslant t^{m_x\/j(x)}\\lVert \\Psi_t\\otimes u\\rVert_{L^2}\\leqslant t^{m_x\/j(x)}\\lVert u\\rVert_{L_{-1}^{1,2}}, \\\\\n  \\lVert r_t^{m_x}\\Psi_t\\otimes \\nabla_{A_t}u\\rVert_{L^2}&\\leqslant t^{m_x\/j(x)}\\lVert \\nabla_{A_t}u\\rVert_{L^2}\\leqslant t^{m_x\/j(x)}\\lVert u\\rVert_{L_{-1}^{1,2}},\\\\\n  \\lVert r_t^{m_x}\\nabla_{A_t}^2 u\\rVert_{L^2}&\\leqslant c\\left(t^{m_x\/j(x)}\\lVert u\\rVert_{L_{-1}^{1,2}}+\\lVert D_t^\\ast D_tu\\rVert_{L_{-m_x-1}^2}\\right)\\leqslant c\\lVert v\\rVert_{L_{-m_x-1}^2},\n\\end{align*}\nby \\eqref{estimate1_eq}. Similarly, with $w=[\\Psi_t,u]$ one deduces that $\\lVert r_t^{m_x} \\nabla_{A_t}[\\Psi_t,u]\\rVert_{L^2}\\leqslant c\\lVert v\\rVert_{L_{-m_x-1}^2}$. Hence\n\\begin{align*}\n  \\lVert u\\rVert_{L_{-m_x-1}^{2,2}}&\\leqslant c\\left(\\lVert u\\rVert_{L_{-1}^{1,2}}+\\lVert r_t^{m_x}\\nabla_{A_t}^2u\\rVert_{L^2}+\\lVert r_t^{m_x}\\nabla_{A_t}[\\Psi_t,u]\\rVert_{L^2}\\right.\\\\&+\\left.\\lVert r_t^{m_x}(\\Psi_t\\otimes\\cdot)^2u\\rVert_{L^2}+\\lVert r_t^{m_x}\\Psi_t\\otimes \\nabla_{A_t}u\\rVert_{L^2}\\right)\\leqslant c\\lVert v\\rVert_{L_{-m_x-1}^2}.\n\\end{align*}\n\nNext we prove that the isomorphism extends to all weights, by showing that the inverse is continuous in other weighted spaces. For any weight $\\delta$, it suffices to prove\n\\[\\lVert r_t^\\delta D_t^\\ast D_t u\\rVert_{L_{-m_x-1}^2}=\\lVert D_t^\\ast D_t u\\rVert_{L_{-m_x-1-\\delta}^2}\\geqslant c^{-1}\\lVert u\\rVert_{L_{-m_x-1-\\delta}^{2,2}},\\]\nusing the established a priori estimates\n\\begin{equation}\n  \\lVert r_t^\\delta u\\rVert_{L_{-m_x-1}^{2,2}}\\leqslant c\\lVert D_t^\\ast D_t r_t^\\delta u\\rVert_{L_{-m_x-1}^2}.\\label{wtApriori_eq}\n\\end{equation}\nWe have\n \\begin{align*}\n  \\lVert [D_t^\\ast D_t,r_t^\\delta]u\\rVert_{L_{-m_x-1}^2}&\\leqslant ct^{2\/j(x)}\\lVert r_t^{\\delta-2} u\\rVert_{L_{-m_x-1}^2}\\\\&\\quad+ct^{1\/j(x)}\\left(\\lVert r_t^{\\delta-1} \\nabla_{A_t} u\\rVert_{L_{-m_x-1}^2}+\\lVert r_t^{\\delta-1} [\\Phi_t,u]\\rVert_{L_{-m_x-1}^2}\\right)\\\\\n  &\\leqslant ct^{2\/j(x)} \\lVert r_t^\\delta r_t^{-2m_x} u\\rVert_{L_{-m_x-1}^2}\\\\&\\quad+ct^{1\/j(x)}\\left(\\lVert r_t^\\delta r_t^{-m_x}\\nabla_{A_t} u\\rVert_{L_{-m_x-1}^2}+\\lVert r_t^\\delta r_t^{-m_x}[\\Phi_t, u] \\rVert_{L_{-m_x-1}^2}\\right)\\\\\n  &\\leqslant c t^{(1-m_x)\/j(x)} \\left(\\lVert r_t^\\delta \\Psi_t\\otimes \\nabla_{A_t}u\\rVert_{L_{-m_x-1}^2}+\\lVert r_t^\\delta (\\Psi_t\\otimes \\cdot)^2u\\rVert_{L_{-m_x-1}^2}\\right)\\\\\n  &\\leqslant ct^{(1-m_x)\/j(x)} \\lVert u\\rVert_{L_{-m_x-1-\\delta}^{2,2}}.\n\\end{align*}\n Similarly, noting that $|[\\nabla_{A_t},r_t^\\delta]w|\\leqslant c t^{(1-m_x)\/j(x)}|r_t^\\delta \\Psi_t\\otimes w|$,\n\\begin{align*}\n  \\lVert u\\rVert_{L_{-m_x-1-\\delta}^{2,2}}&\\leqslant \\lVert r_t^\\delta u\\rVert_{L_{-m_x-1}^{2,2}}+\\lVert  [\\nabla_{A_t}^2,r_t^\\delta]u\\rVert_{L_{-m_x-1}^2}+\\lVert [\\nabla_{A_t},r_t^\\delta]u\\rVert_{L_{-m_x-1}^2}\\\\&\\hspace{2.75cm}+\\lVert \\Psi_t\\otimes [\\nabla_{A_t},r_t^\\delta]u\\rVert_{L_{-m_x-1}^2}+\\lVert [\\nabla_{A_t},r_t^\\delta]\\Psi_t\\otimes u\\rVert_{L_{-m_x-1}^2} \\\\\n  &\\leqslant \\lVert r_t^\\delta u\\rVert_{L_{-m_x-1}^{2,2}}+ct^{(1-m_x)\/j(x)} \\lVert u\\rVert_{L_{-m_x-1-\\delta}^{2,2}}.\n\\end{align*}\nFinally by \\eqref{wtApriori_eq},\n\\begin{align*}\n  \\lVert r_t^\\delta D_t^\\ast D_t u\\rVert_{L_{-m_x-1}^2}&\\geqslant \\lVert D_t^\\ast D_tr_t^\\delta u \\rVert_{L_{-m_x-1}^2}-\\lVert [D_t^\\ast D_t,r_t^\\delta]u\\rVert_{L_{-m_x-1}^2}\\\\&\\geqslant c^{-1}\\lVert r_t^\\delta u\\rVert_{L_{-m_x-1}^{2,2}}-\\lVert [D_t^\\ast D_t,r_t^\\delta]u\\rVert_{L_{-m_x-1}^2}\\\\\n  &\\geqslant (c^{-1}-ct^{(1-m_x)\/j(x)})\\lVert u\\rVert_{L_{-m_x-1-\\delta}^{2,2}}\\geqslant (c^{-1}\/2)\\lVert u\\rVert_{L_{-m_x-1-\\delta}^{2,2}},\n\\end{align*}\nfor $t$ large. We have proved that for the off-diagonal part, $L_t: L_{-2+\\delta}^{2,2}\\to L_{-2+\\delta}^2$ is an isomorphism for all weights $\\delta$ with the norm of the inverse uniformly bounded in $t$.\n\nFor the diagonal part, the equation becomes the Laplace equation $\\Delta u=v$ with the Dirichlet boundary condition. Let $\\tau: \\widetilde{B}_{x,t}\\to B_{0}(\\kappa):=\\{\\,|\\tilde{z}|<\\kappa\\,\\},~ z_x\\mapsto t^{1\/j(x)}z_x:=\\tilde{z}$ be the rescaling map, and $\\tilde{u}={(\\tau^{-1})}^\\ast u$, $\\tilde{v}= {(\\tau^{-1})}^\\ast v$. Then $\\Delta u=v$ is equivalent to $t^{2\/j(x)}\\Delta \\tilde{u}=\\tilde{v}$ on $B_{0}(\\kappa)$, where $\\Delta: L_{-2+\\delta}^{2,2}\\to L_{-2+\\delta}^2$ is an isomorphism by classical elliptic theory on cylinders (see \\cite{biquard_boalch_2004}), and the weighted spaces on $B_{0}(\\kappa)$ are defined similarly as in \\eqref{SobolevSpace_eq}, using the Euclidean distance from $0$ as the weight function, which is equivalent to ${(\\tau^{-1})}^\\ast r_t$ uniformly in $t$. Then $\\lVert  \\tilde{v}\\rVert_{L_{-2+\\delta}^2}\\geqslant c^{-1}\\lVert t^{2\/j(x)} \\tilde{u}\\rVert_{L_{-2+\\delta}^{2,2}}$, which implies that $\\lVert v\\rVert_{L_{-2+\\delta}^2}\\geqslant c^{-1}\\lVert u\\rVert_{L_{-2+\\delta}^{2,2}}$.\n\\end{proof}\n\nNext we study the behavior of $L_t$ around $x\\in I_t$. Define $L_\\delta^{k,p}(D_{x,t}),\\hat{L}_\\delta^{k,p}(D_{x,t})$ as in \\eqref{SobolevSpace_eq} using $\\tilde{r}_t=\\sigma_{x,t}^\\ast r_t$, $\\widetilde{\\Phi}_t$, $\\tilde{h}_t^{\\mathrm{app}}$, the Euclidean metric on $D_{x,t}$ (which is $\\sigma_{x,t}^{\\ast}g_t$), and $\\widetilde{A}_t=D(\\tilde{\\bar{\\partial}}_E,\\tilde{h}_t^{\\mathrm{app}})$. For $w\\in \\Omega^q(\\mathrm{End}\\,E)$ we have \\begin{equation}\\label{NormRelation_eq}\n     \\lVert \\sigma_{x,t}^\\ast w\\rVert_{L_\\delta^{k,p}(D_{x,t})}=2\\lVert w\\rVert_{L_\\delta^{k,p}(\\widetilde{B}_{x,t})}.\n   \\end{equation}\nLet $\\widetilde{L}_t(\\gamma)=\\Delta_{\\widetilde{A}_t}\\gamma-\\mathrm{i}\\star M_{\\widetilde{\\Phi}_t}\\gamma$, then $\\widetilde{L}_t(\\sigma^\\ast\\gamma)=\\sigma^\\ast (L_t\\gamma)$. The previous analysis near $x\\in I_u$ yields the following.\n\n\\begin{corollary}\\label{analysisIt_cor}\n  On $\\widetilde{B}_{x,t}$ for $x\\in I_t$, $L_t:L_{-2+\\delta}^{2,2}(i\\mathfrak{su}(E))\\to L_{-2+\\delta}^{2}(i\\mathfrak{su}(E))$ is an isomorphism with zero Dirichlet condition on the boundary for small weights $\\delta>0$. Moreover, the norm of $L_t^{-1}$ is uniformly bounded in $t$.\n\\end{corollary}\n\\begin{proof}\n Let $v\\in L_{-2+\\delta}^{2}(i\\mathfrak{su}(E))(\\widetilde{B}_{x,t})$ then $\\tilde{v}=\\sigma_{x,t}^\\ast v\\in L_{-2+\\delta}^2(i\\mathfrak{su}(E))(D_{x,t})$, and by Proposition \\ref{analysisIu_prop} with $m_x$ and $j(x)$ both replaced by $2(m_x-1)$, there exists $\\tilde{u}\\in L_{-2+\\delta}^{2,2}(i\\mathfrak{su}(E))(D_{x,t})$ such that $\\widetilde{L}_t \\tilde{u}=\\tilde{v}$ and $\\tilde{u}=0$ on $\\partial D_{x,t}$. Note that $\\widetilde{L}_t$ is $\\mathbb{Z}_2$-equivariant, meaning that $\\iota^\\ast(\\widetilde{L}_t (\\tilde{u}))=\\widetilde{L}_t (\\iota^\\ast \\tilde{u})$ for the involution $\\iota:\\xi_{x,t}\\mapsto -\\xi_{x,t}$. Then $\\widetilde{L}_t(\\iota^\\ast \\tilde{u}-\\tilde{u})=\\iota^\\ast \\tilde{v}-\\tilde{v}=0$, and $\\iota^\\ast \\tilde{u}=\\tilde{u}$ since $\\widetilde{L}_t$ is injective. $\\tilde{u}$ descends to $\\widetilde{B}_{x,t}$ to give $u\\in L_{-2+\\delta}^{2,2}(i\\mathfrak{su}(E))(\\widetilde{B}_{x,t})$ with $L_t u=v$ and $u=0$ on $\\partial \\widetilde{B}_{x,t}$. If $L_t u=0$, then $\\widetilde{L}_t(\\sigma_{x,t}^\\ast u)=0$ and $\\sigma_{x,t}^\\ast u=0$ which implies that $u=0$. Therefore $L_t$ is an isomorphism. The uniform boundedness of $L_t^{-1}$ follows from that of $\\widetilde{L}_t^{-1}$ and \\eqref{NormRelation_eq}.\n\\end{proof}\n The local behavior of $L_t$ around $x\\in T$ has been studied in \\cite{fredrickson2022asymptotic}, and we have the following result.\n\\begin{proposition}\\label{analysisT_prop}\n  On $\\hat{B}_{x,t}$, for $x\\in T$, $L_t:L_{-2+\\delta}^{2,2}(i\\mathfrak{su}(E))\\to L_{-2+\\delta}^{2}(i\\mathfrak{su}(E))$ is an isomorphism with zero Dirichlet condition on the boundary for small weights $\\delta>0$. Moreover, the norm of $L_t^{-1}$ is uniformly bounded in $t$.\n\\end{proposition}\n\\begin{proof}\nConsider $\\tau:\\hat{B}_{x,t}\\to B_0(1):=\\{\\,|\\tilde{z}|<1\\,\\},~\\zeta_{x,t}\\to \\lambda_x(t)^2\\zeta_{x,t}:=\\tilde{z}$. Let $\\widetilde{\\Phi}_t=(\\tau^{-1})^\\ast\\Phi_t$, and $\\tilde{h}_t^{\\mathrm{app}}=(\\tau^{-1})^\\ast \\tilde{h}_t^{\\mathrm{app}}$. Then by \\eqref{NormalFormT_eq} and \\eqref{approxMetricT_eq},\n\\[\\widetilde{\\Phi}_t=\\begin{pmatrix}\n  0&\\frac{1}{\\lambda_x(t)}\\\\ \\frac{\\lambda_x(t)}{\\tilde{z}}&0\n\\end{pmatrix}\\,\\mathrm{d}\\tilde{z},\\quad \\tilde{h}_t^{\\mathrm{app}}=\\begin{pmatrix}\n  \\frac{1}{\\lambda_x(t)}\\tilde{r}^{1\/2}\\mathrm{e}^{\\psi_{2,x} (8\\tilde{r}^{1\/2})}&0\\\\0&\\frac{1}{\\lambda_x(t)^3}\\tilde{r}^{3\/2}\\mathrm{e}^{-\\psi_{2,x}(8\\tilde{r}^{1\/2})}\n\\end{pmatrix},\\]\nwhere $\\tilde{r}=|\\tilde{z}|$. By the gauge transformation $g=\\left(\\begin{smallmatrix}\n  \\lambda_x(t)^{-1\/2}&0\\\\0&\\lambda_x(t)^{1\/2}\n\\end{smallmatrix}\\right)$, we have\n\\[\\widetilde{\\Phi}_t=\\begin{pmatrix}\n  0&1\\\\ \\frac{1}{\\tilde{z}}&0\n\\end{pmatrix}\\,\\mathrm{d}\\tilde{z},\\quad \\tilde{h}_t^{\\mathrm{app}}=\\frac{1}{\\lambda_x(t)^2}\\begin{pmatrix}\n  \\tilde{r}^{1\/2}\\mathrm{e}^{\\psi_{2,x} (8\\tilde{r}^{1\/2})}&0\\\\0&\\tilde{r}^{3\/2}\\mathrm{e}^{-\\psi_{2,x}(8\\tilde{r}^{1\/2})}\n\\end{pmatrix}.\\]\nThen in this trivialization, $\\widetilde{\\Phi}_t$ and $\\lambda_x(t)^2 \\tilde{h}_t^{\\mathrm{app}}$ are independent of $t$, which together with the Euclidean metric $g_e$ on $B_0(1)$ define an operator $\\tilde{L}$ as in \\eqref{LinOp_eq}. The indicial root analysis as in \\cite[Sec.~4.2,~Lem. 5.8]{fredrickson2022asymptotic} implies that $\\tilde{L}: L_{-2+\\delta}^{2,2}(i\\mathfrak{su}(E))\\to L_{-2+\\delta}^{2}(i\\mathfrak{su}(E))$ (these spaces are defined on $B_0(1)$ as in \\eqref{SobolevSpace_eq} using $g_e$, $\\widetilde{\\Phi}_t$, $\\lambda_x(t)^2 \\tilde{h}_t^{\\mathrm{app}}$) is an isomorphism for $\\delta>0$ small. Therefore $L_t$ is also an isomorphism, and the uniform boundedness of $L_t^{-1}$ follows in the same way as in Proposition \\ref{analysisIu_prop} for the diagonal part.\n\\end{proof}\n\nLet $U_{\\mathrm{ext}}:=C\\backslash (\\bigcup_{x\\in I} \\frac{1}{2}\\widetilde{B}_{x,t}\\cup\\bigcup_{x\\in T}\\frac{1}{2}\\hat{B}_{x,t})$, where $\\frac{1}{2}\\widetilde{B}_{x,t}=\\{\\,|\\zeta_{x,t}|< \\kappa t^{-1\/(m_x-1)}\/2\\,\\}$ for $x\\in I_t$, $\\frac{1}{2}\\widetilde{B}_{x,t}=\\{\\,|z_x|< \\kappa t^{-1\/j(x)}\/2\\,\\}$ for $x\\in I_u$, and $\\frac{1}{2}\\hat{B}_{x,t}=\\{\\,|\\zeta_{x,t}|< \\lambda_x(t)^{-2}\/2\\,\\}$ for $x\\in T$. In $U_{\\mathrm{ext}}$, $c^{-1}\\leqslant r_t\\leqslant 1$, which is immaterial in the analysis, so we omit the subscript of $L^{k,2}_{-2+\\delta}$.\n\n\\begin{proposition}\\label{analysisUext_prop}\n  On $U_{\\mathrm{ext}}$, $L_t:L^{2,2}(i\\mathfrak{su}(E))\\to L^2(i\\mathfrak{su}(E))$ is an isomorphism when imposing the zero Dirchlet boundary condition. The norm of $L_t^{-1}$ is bounded by $ct^4$.\n\\end{proposition}\n\\begin{proof}\n  Invertibility is standard (see \\cite[Sec.~5.2]{mazzeo_swoboda_weiss_witt_2016}). Let $u\\in L^{2,2}(i\\mathfrak{su}(E))$ be vanishing on $\\partial U_{\\mathrm{ext}}$, then \\begin{align*}\n    \\lVert \\nabla_{A_t} u\\rVert_{L^2}^2+\\lVert [\\Psi_t,u]\\rVert_{L^2}^2 =\\langle D_t^\\ast D_t u,u\\rangle_{L^2}\\leqslant \\lVert L_t u\\rVert_{L^2}\\lVert u\\rVert_{L^2}.\n  \\end{align*}\nLet $\\check{U}_{\\mathrm{ext}}=\\{|w|<2\\kappa^{-1}t^{1\/(m_0+N-4)}\\}$, where $w=1\/z$. Then $\\check{U}_{\\mathrm{ext}}$ is a disk centered at $\\infty$ and contains $U_{\\mathrm{ext}}$. Let $L_\\mathrm{e}^2$ be the $L^2$ norm on $\\check{U}_{\\mathrm{ext}}$ defined using the Euclidean metric $g_e$ on this disk. By Kato's inequality and Poincar\u00e9 inequality, $\\lVert u\\rVert_{L_\\mathrm{e}^2}\\leqslant ct^{1\/(m_0+N-4)}\\lVert \\nabla_{A_t} u\\rVert_{L_\\mathrm{e}^2}\\leqslant ct^{1\/2}\\lVert \\nabla_{A_t} u\\rVert_{L_\\mathrm{e}^2}$. By the Definition \\ref{Metricgt_def} of $g_t$, we have $c^{-1}t^{-1}g_t\\leqslant g_e\\leqslant ct^{4\/(m_0+N-4)} g_t\\leqslant ct^2g_t$ on $U_{\\mathrm{ext}}$. Therefore,\n\\[\\lVert u\\rVert_{L^2}^2\\leqslant ct\\lVert u\\rVert_{L_\\mathrm{e}^2}^2\\leqslant ct^2\\lVert \\nabla_{A_t}u\\rVert_{L_\\mathrm{e}^2}^2= ct^2\\lVert \\nabla_{A_t}u\\rVert_{L^2}^2,\\]\nwhere the last equality follows from the conformal invariance of the $L^2$ norm of a $1$-form. Then we have\n  \\begin{align*}\n    \\lVert u\\rVert_{L^2}^2&\\leqslant ct^2\\lVert L_t u\\rVert_{L^2}\\lVert u\\rVert_{L^2}\\,\\Rightarrow\\, \\lVert u\\rVert_{L^2}\\leqslant ct^2\\lVert L_t u\\rVert_{L^2},\\\\\n    \\lVert \\nabla_{A_t}u\\rVert_{L^2}&\\leqslant ct\\lVert L_tu\\rVert_{L^2},~\\lVert \\Psi_t\\otimes u\\rVert_{L^2}\\leqslant ct\\lvert L_t u\\rVert_{L^2}.\n  \\end{align*}\nWe deduce that \\begin{align*}\n    \\lVert (\\Psi_t\\otimes\\cdot)^2 u\\rVert_{L^2}&\\leqslant c\\sup |\\Phi_t|\\lVert \\Psi_t\\otimes u\\rVert_{L^2}\\leqslant ct\\sup |\\Phi_t|\\lVert L_tu\\rVert_{L^2}\\\\\n    \\lVert \\Psi_t\\otimes \\nabla_{A_t}u\\rVert_{L^2}&\\leqslant c\\sup |\\Phi_t|\\lVert  \\nabla_{A_t}u\\rVert_{L^2}\\leqslant ct\\sup |\\Phi_t|\\lVert L_tu\\rVert_{L^2}.\n    \\end{align*}\n  By Lemma \\ref{weitz_lemma} (\\romannumeral1), and the conformal change formula for $\\mathscr{R}$,\n  \\begin{align*}\n    \\lVert \\nabla_{A_t}^2 u\\rVert_{L^2}&\\leqslant c\\left(\\lVert D_t\\nabla_{A_t} u\\rVert_{L^2}+\\lVert D_t^\\ast \\nabla_{A_t} u\\rVert_{L^2}+t^3\\lVert\\nabla_{A_t} u\\rVert_{L^2}+\\lVert \\mathscr{F}(\\nabla_{A_t} u)\\rVert_{L^2}\\right)\\\\\n    &\\leqslant c\\left(t\\sup|\\Phi_t|+t^4+t\\max\\sup |E_{x(,j),t}|\\right)\\lVert L_t u\\rVert_{L^2},\n  \\end{align*}\n  where the error terms $E_{x(,j),t}$ decay exponentially in $t$. In $U_{\\mathrm{ext}}\\backslash \\big(\\bigcup_{x(,j)} \\widetilde{B}_{x(,j),t}\\big)$, $\\Phi_t$ can be diagonalized as $\\sqrt{-\\nu_t(z)}\\sigma_3\\,\\mathrm{d}z$, and $h_t^{\\mathrm{app}}$ is also diagonal so that (recall that the norm $|\\mathrm{d}z|$ is taken with respect to $g_t$) \\[|\\Phi_t|^2=2|\\nu_t(z)||\\mathrm{d}z|^2\\leqslant ct^2.\\]\n  In $\\widetilde{B}_{x,j,t}$, $\\Phi_t$ and $h_t^\\mathrm{app}$ are given by \\eqref{NormalFormZ_eq} and \\eqref{approxMetricZ_eq} in some holomorphic frame, so we have\n  \\begin{align*}\n    |\\Phi_t|^2&=2\\lambda_{x,j}(t)^2r\\cosh\\left(2l_{x,j,t}(r)\\chi\\left(r\\kappa^{-1}t^{1\/j(x)}\\right)\\right)|\\mathrm{d}\\zeta_{x,j,t}|^2\\\\\n    &\\leqslant c\\lambda_{x,j}(t)^{4\/3} \\rho^{2\/3}\\cosh(2\\psi_1(\\rho)\\chi)|\\mathrm{d}\\zeta_{x,j,t}|^2\\leqslant ct^2,\n  \\end{align*}\nwhere $r=|\\zeta_{x,j,t}|, \\rho=8\\lambda_{x,j}(t)r^{3\/2}\/3$. Similarly, in $\\widetilde{B}_{x,t}\\cap U_{\\mathrm{ext}}$,  $\\Phi_t$ and $h_t^\\mathrm{app}$ are given by \\eqref{NormalFormT_eq} and \\eqref{approxMetricT_eq} in some holomorphic frame, then\n\\begin{align*}\n  |\\Phi_t|^2&=2\\lambda_{x}(t)^2r\\cosh\\left(2m_{x,t}(r)\\chi\\left(r\\kappa^{-1}\\right)\\right)|\\mathrm{d}\\zeta_{x,t}|^2\\\\\n  &\\leqslant c \\rho^2\\cosh(2\\psi_{2,x}(\\rho)\\chi)|\\mathrm{d}\\zeta_{x,t}|^2\\leqslant ct,\n\\end{align*}\nwhere $r=|\\zeta_{x,t}|, \\rho=8\\lambda_{x}(t)r^{1\/2}$. Therefore, $\\lVert \\nabla_{A_t}^2 u\\rVert_{L^2}\\leqslant ct^4\\lVert L_t u\\rVert_{L^2}$. Similarly, applying the Weitzenb\u00f6ck formula for $w=[\\Psi_t,u]$ gives $\\lVert \\nabla_{A_t} [\\Psi_t,u] \\rVert_{L^2}\\leqslant ct^4\\lVert L_t u\\rVert_{L^2}$. Combining all the estimates above, $\\lVert u\\rVert_{L^{2,2}(U_{\\mathrm{ext}})}\\leqslant ct^4\\lVert L_t u\\rVert_{L^2}$.\n\\end{proof}\nFinally, we consider the global behavior of $L_t$.\n\\begin{lemma}\\label{GlobIso_lem}\n  $L_t:L_{-2+\\delta}^{2,2}(i\\mathfrak{su}(E))\\to L_{-2+\\delta}^2(i\\mathfrak{su}(E))$ is an isomorphism for small $\\delta>0$.\n\\end{lemma}\n\\begin{proof}\n  By \\cite[Lemma 5.1]{biquard_boalch_2004}, $L_t$ is Fredholm of index zero (the proof still work for $x\\in I_t$ by considering the local lifted problem as in Corollary \\ref{analysisIt_cor} and for $x\\in T$ by \\cite[Lem. 5.8]{fredrickson2022asymptotic}). For $u\\in\\ker L_t$, near $x\\in I_u\\cup T$ we have $|D_t u|=O(r^{\\delta-1})$ ($r=|\\zeta_{x,t}|$ for $x\\in T$, and $r=|z_x|$ for $x\\in I_u$) by the proof of \\cite[Lem.~4.6]{biquard_boalch_2004}. The boundary term in the integration by parts is $\\lim_{\\epsilon\\to 0^+}\\int_{r=\\epsilon} \\langle \\partial_r u,u\\rangle r\\,\\mathrm{d}\\theta$ \\cite[Lem.~5.5]{fredrickson2022asymptotic} which vanishes for $x\\in I_u\\cup T$ since $\\langle[\\Phi_t,u],u\\rangle=0$ and $\\langle \\partial_r u,u\\rangle=O(r^{\\delta-1})$, and similarly vanishes for $x\\in I_t$ by considering the double lifting. Therefore \\begin{equation}\\label{intbyParts_eq}\n  \\langle L_t u,u\\rangle_{L^2}=\\lVert \\mathrm{d}_{A_t}u\\rVert_{L^2}^2+\\lVert [\\Psi_t,u]\\rVert_{L^2}^2.\n\\end{equation}\nNow we have $\\mathrm{d}_{A_t}u=[\\Phi_t,u]=0$, which implies that $u\\equiv 0$ as in the proof of \\cite[Lem.~5.6]{fredrickson2022asymptotic} since $Z_t\\backslash (I_t\\cup T)\\neq \\varnothing$. $L_t$ has trivial kernel, and then it is an isomorphism.\n\\end{proof}\n\n\\begin{lemma}\\label{PoincareIneq_lem}Let $u\\in L_{-2+\\delta}^{2,2}(i\\mathfrak{su}(E))$, then for $t$ large enough,\n  \\begin{equation}\\label{PoincareIneq_eq}\n    \\lVert u\\rVert_{L_{-\\delta}^2}^2\\leqslant ct^2\\left(\\lVert \\nabla_{A_t}u\\rVert_{L^2}^2+\\lVert [\\Phi_t,u]\\rVert_{L^2}^2\\right).\n  \\end{equation}\n\\end{lemma}\n\\begin{proof}\nThe idea is to use $[\\Phi_t,u]$ to control $u$ around a zero of $\\tilde{\\nu}_t(z)$ near each $x\\in I$ (we choose $z_{x,0}(t)$, which has asymptotics given in Lemma \\ref{HiggsDetRoot_lem}), and use $\\nabla_{A_t}u$ to control $u$ elsewhere. The inequality is obtained by gluing these estimates together.\n\nFor $t$ large, $x\\in I_u$, define $W_x:=\\{\\,|\\zeta_{x,0,t}|<t^{-(2m_x+1)\/3j(x)}\\}\\subset \\frac{1}{2}\\widetilde{B}_{x,0,t}$, a disk centered at $z_{x,0}(t)$. Then $\\chi(r\\kappa^{-1}t^{1\/j(x)})=1$ on $W_x$, where $r=|\\zeta_{x,0,t}|$, and \\[\\rho=8\\lambda_{x,0}(t)r^{3\/2}\/3\\leqslant c\\text{ on }W_x.\\]  Recall that $\\Phi_t$ is given by \\eqref{NormalFormZ_eq} in some holomorphic frame over $W_x$. Since $u$ is a section of $i\\mathfrak{su}(E)$, it has form\n\\[u=\\begin{pmatrix}\n  p& \\bar{s} r^{-1}\\mathrm{e}^{-2l_{x,0,t}(r)}\\\\ s&-p\n\\end{pmatrix},\\]\nwhere $p$ is real valued. $\\Phi_t$ acts on $u$ as\n\\[[\\Phi_t,u]=\\lambda_{x,0}(t)\\begin{pmatrix}\n  s-\\bar{s}\\zeta_{x,0,t} r^{-1}\\mathrm{e}^{-2l_{x,0,t}(r)}&-2p\\\\\n  2\\zeta_{x,0,t} p&-s+\\bar{s}\\zeta_{x,0,t} r^{-1}\\mathrm{e}^{-2l_{x,0,t}(r)}\n\\end{pmatrix}\\,\\mathrm{d}\\zeta_{x,0,t}.\\]\nThen on $W_x$ we have \\begin{align}\n  |u|^2&=2p^2+2|s|^2 r^{-1}\\mathrm{e}^{-2l_{x,0,t}(r)}\\leqslant 2p^2+c|s|^2t^{(2m_x+1)\/3j(x)}\\rho^{-2\/3}\\mathrm{e}^{-2\\psi_1(\\rho)}\\notag\\\\&\\leqslant 2p^2+ct^{(2m_x+1)\/3j(x)}|s|^2,\\notag  \\\\\n  |[\\Phi_t,u]|^2&= \\lambda_{x,0}(t)^2\\left(8p^2 r\\cosh\\left(2l_{x,0,t}(r)\\right)+2\\left|s-\\bar{s}\\zeta_{x,0,t} r^{-1}\\mathrm{e}^{-2l_{x,0,t}(r)}\\right|^2\\right)\\notag\\\\\n  &\\geqslant c^{-1}t^{(2m_x+1)\/j(x)}\\left(p^2t^{-(2m_x+1)\/3j(x)}\\rho^{2\/3}\\cosh\\left(2\\psi_1(\\rho)\\right)+|s|^2\\left(1-\\mathrm{e}^{-2\\psi_1(\\rho)}\\right) \\right)\\notag\\\\&\\geqslant c^{-1}t^{(4m_x+2)\/3j(x)}|u|^2.\\label{HiggsIuEst_eq}\n\\end{align}\n\nConsider $\\chi_x(r):=\\chi(t^{(2m_x+1)\/3j(x)}r)$, then $\\chi_x=0$ in $C\\backslash W_x$ and $\\chi_x=1$ in $\\hat{W}_x:=\\frac{1}{2}W_x$. Let $r_{t,x}$ be the function obtained by extending $r_t|_{\\widetilde{B}_{x,t}}$ by $1$, and $\\hat{g}_t=\\hat{f}_tg_C$ where $\\hat{f}_t$ is the extension of $f_t|_{C\\backslash \\bigcup_{y\\in I_t}B_y}$ by $1$. We claim that for all $v\\in L_{-2+\\delta}^{2,2}$ with $v$ vanishing in $\\hat{W}_x$,\n\\begin{equation}\\label{hardyineqIu_eq}\n  \\lVert v\\rVert_{L_{-\\delta,x}^2}^2\\leqslant ct^{(6+(4m_x-4)\\delta)\/3j(x)}\\lVert \\nabla_{A_t}v\\rVert_{L_{-\\delta-1,x}^2}^2,\n\\end{equation}\nwhere the subscripts indicate the norms are defined using $r_{t,x},\\hat{g}_t$ instead of $r_t,g_t$. By Kato's inequality, we may assume that $v$ is a function, and prove the inequality with $\\nabla_{A_t}v$ replaced by $\\mathrm{d}v$. First consider the biholomorphism $\\tau_x:C\\to C_1=\\mathbb{CP}^1$, defined by $z_x\\mapsto c_xt^{1\/j(x)}z_x$, where $c_x$ is the constant satisfying $\\lim_{t\\to\\infty}\\tau_x(z_{x,0}(t))=x+1$. Define weighted spaces on $C_1$ using the round metric on $C_1$ and the weight function $r_x$ given by the geodesic distance from $x$, then $r_x$ is uniformly equivalent to $(\\tau_x^{-1})^\\ast r_{t,x}$, and $c^{-1}t^{-2\/j(x)}(\\tau_x^{-1})^\\ast \\hat{g}_t\\leqslant g_{C_1}$. Then for $v_1=(\\tau_x^{-1})^\\ast v$ we have\n\\begin{align*}\n  \\lVert v\\rVert_{L_{-\\delta,x}^2}^2&=\\int_{C} r_{t,x}^{2\\delta-2}|v|^2\\,\\mathrm{dvol}_{\\hat{g}_t}\\leqslant c \\int_{C_1} r_x^{2\\delta-2}|v_1|^2\\,(\\tau_x^{-1})^\\ast(\\mathrm{dvol}_{\\hat{g}_t})\\\\&\\leqslant ct^{2\/j(x)}\\int_{C_1} r_x^{2\\delta-2}|v_1|^2\\,\\mathrm{dvol}_{C_1}=ct^{2\/j(x)}\\lVert v_1\\rVert_{L_{-\\delta}^2(C_1)}^2\\\\\n  \\lVert \\mathrm{d}v_1\\rVert_{L_{-\\delta-1}^2(C_1)}^2&=\\int_{C_1}r_x^{2\\delta}|\\mathrm{d}v_1|_{g_{C_1}}^2\\,\\mathrm{dvol}_{C_1}\\leqslant c\\int_C r_{t,x}^{2\\delta}|\\mathrm{d}v|_{\\hat{g}_t}^2\\,\\mathrm{dvol}_{\\hat{g}_t}=c\\lVert \\mathrm{d}v\\rVert_{L_{-\\delta-1,x}^2}^2.\n\\end{align*}\n\\eqref{hardyineqIu_eq} will follow from\n\\begin{equation}\\label{hardyineqIu_eq1}\n  \\lVert v_1\\rVert_{L_{-\\delta}^2(C_1)}^2\\leqslant ct^{(4m_x-4)\\delta\/3j(x)}\\lVert \\mathrm{d}v_1\\rVert_{L_{-\\delta-1}^2(C_1)}^2.\n\\end{equation}\nSuppose $C_1$ is embedded in $\\mathbb{R}^3$ so that $x=(0,1,0)$ and $x+1=(0,0,1)$. Consider the stereographic projection $\\hat{\\tau}_x:C_1\\backslash \\{x+1\\}\\to \\mathbb{R}^2$, which sends $x+1$ to $\\infty$ and $x$ to $(0,1)$. Note that as $t\\to\\infty$, $\\tau_x(z_{x,0}(t))\\to x+1$ and the diameter of $\\tau_x(\\hat{W}_x)$ is $O(t^{-(2m_x-2)\/3j(x)})$, then for some constant $\\kappa_x$, we have $\\hat{\\tau}_x^{-1}\\left(\\mathbb{R}^2\\backslash C_2\\right)\\subset \\tau_x(\\hat{W}_x)$, where $C_2:=B_{(0,1)}(\\kappa_xt^{(2m_x-2)\/3j(x)})$. Define weighted spaces on $C_2$ using the Euclidean metric and the weight function $\\hat{r}$ given by the Euclidean distance from $(0,1)$. Then $v_2=(\\hat{\\tau}_x^{-1})^\\ast v_1$ vanishes on $\\partial C_2$, $c^{-1}(\\hat{\\tau}_x^{-1})^\\ast r_x\\leqslant \\hat{r}\\leqslant ct^{(2m_x-2)\/3j(x)}(\\hat{\\tau}_x^{-1})^\\ast r_x$, and $c^{-1}(\\hat{\\tau}_x^{-1})^\\ast g_{C_1}\\leqslant g_{C_2}$ on $C_2$. We deduce that\n\\begin{align*}\n\\lVert v_1\\rVert_{L_{-\\delta}^2(C_1)}^2&=\\int_{C_1}r_x^{2\\delta-2}|v_1|^2\\,\\mathrm{dvol}_{C_1}\\leqslant c\\int_{C_2}\\hat{r}^{2\\delta-2}|v_2|^2\\,\\mathrm{dvol}_{C_2}=c\\lVert v_2\\rVert_{L_{-\\delta}^2(C_2)}^2\\\\\n\\lVert \\mathrm{d}v_2\\rVert_{L_{-\\delta-1}^2(C_2)}^2&=\\int_{C_2}\\hat{r}^{2\\delta}|\\mathrm{d}v_2|_{g_{C_2}}^2\\,\\mathrm{dvol}_{C_2}\\leqslant ct^{(4m_x-4)\\delta\/3j(x)}\\lVert \\mathrm{d}v_1\\rVert_{L_{-\\delta-1}^2(C_1)}^2.\n\\end{align*}\nAn analogue of Lemma \\ref{SobEmbed_lem} (\\romannumeral2) implies that \\[\\lVert v_2\\rVert_{L_{-\\delta}^2(C_2)}^2\\leqslant c\\lVert \\mathrm{d}v_2\\rVert_{L_{-\\delta-1}^2(C_2)}^2,\\]\nsince $v_2$ vanishes on $\\partial C_2$. Therefore\n\\begin{align*}\n  \\lVert v_1\\rVert_{L_{-\\delta}^2(C_1)}^2\\leqslant c\\lVert v_2\\rVert_{L_{-\\delta}^2(C_2)}^2\\leqslant c\\lVert \\mathrm{d}v_2\\rVert_{L_{-\\delta-1}^2(C_2)}^2\\leqslant ct^{(4m_x-4)\\delta\/3j(x)}\\lVert \\mathrm{d}v_1\\rVert_{L_{-\\delta-1}^2(C_1)}^2,\n\\end{align*}\nverifying \\eqref{hardyineqIu_eq1}, and \\eqref{hardyineqIu_eq} holds.\n\nFor $t$ large, $x\\in I_t$, $W_x:=\\{\\,|\\zeta_{x,0,t}|<t^{-2m_x\/3j(x)}\\,\\}\\subset \\frac{1}{2}\\widetilde{B}_{x,0,t}$, so that $\\chi(r\\kappa^{-1}t^{1\/j(x)})=1$ and $\\rho\\leqslant c$ on $W_x$, where $r=|\\zeta_{x,0,t}|$ and $\\rho=8\\lambda_{x,0}(t)r^{3\/2}\/3$. Similar to \\eqref{HiggsIuEst_eq},\n\\begin{equation}\n  |[\\Phi_t,u]|^2\\geqslant c^{-1}t^{4m_x\/3j(x)} |u|^2.\\label{HiggsItEst_eq}\n\\end{equation}\nConsider $\\chi_x(r):=\\chi(t^{2m_x\/3j(x)}r)$, then $\\chi_x=0$ in $C\\backslash W_x$ and $\\hat{\\chi}=1$ in $\\hat{W}_x:=\\frac{1}{2}W_x$. Let $r_{t,x}$ be the function on $C$ which is the extension of $r_t|_{\\widetilde{B}_{x,t}}$ by $1$, and $g_{t,x}=f_{t,x}g_C$ where $f_{t,x}$ is the extension of $f_t|_{C\\backslash \\bigcup_{y\\in I_t\\backslash\\{x\\}}B_y}$ by $1$. We claim that for all $v\\in L_{-2+\\delta}^{2,2}$ with $v$ vanishing in $\\hat{W}_x$,\n\\begin{equation}\\label{hardyineqIt_eq}\n  \\lVert v\\rVert_{L_{-\\delta,x}^2}^2\\leqslant ct^{(6+(2m_x-3)\\delta)\/3j(x)}\\lVert \\nabla_{A_t}v\\rVert_{L_{-\\delta-1,x}^2}^2,\n\\end{equation}\nwhere the subscripts indicate the norms are defined using $r_{t,x},g_{t,x}$ instead of $r_t$, $g_t$. As above we prove the inequality for $v$ being a function. Consider the biholomorphism $\\tau_x: C\\to C_1$, defined by $z_x\\mapsto c_xt^{1\/j(x)}z_x$, where $c_x$ is the constant chosen so that $\\lim_{t\\to\\infty}\\tau_x(z_{x,0}(t))=x+1$.  Define weighted spaces on $C_1$ using the singular metric $g_{C_1}=r_x^{-1}g_{\\mathrm{FS}(C_1)}$ where $g_{\\mathrm{FS}(C_1)}$ is the Fubini-Study metric on $C_1$, and the weight function $r_x^{1\/2}$ where $r_x$ is the geodesic distance from $x$. Note that $r_x^{1\/2}$ is uniformly equivalent to $(\\tau_x^{-1})^\\ast r_{t,x}$, and $c^{-1}t^{-2\/j(x)}(\\tau_x^{-1})^\\ast g_{t,x}\\leqslant g_{C_1}$. Then for $v_1=(\\tau^{-1})^\\ast v$ we have\n\\[\\lVert v\\rVert_{L_{-\\delta,x}^2}^2\\leqslant ct^{2\/j(x)}\\lVert v_1\\rVert_{L_{-\\delta}^2(C_1)}^2,\\quad \\lVert \\mathrm{d}v_1\\rVert_{L_{-\\delta-1}^2(C_1)}^2\\leqslant c\\lVert \\mathrm{d}v\\rVert_{L_{-\\delta-1,x}^2}^2.\\]\n\\eqref{hardyineqIt_eq} follows once we prove\n\\begin{equation}\\label{hardyineqIt_eq1}\n  \\lVert v_1\\rVert_{L_{-\\delta}^2(C_1)}^2\\leqslant ct^{(2m_x-3)\\delta\/3j(x)}\\lVert \\mathrm{d}v_1\\rVert_{L_{-\\delta-1}^2(C_1)}^2.\n\\end{equation}\nSuppose $C_1$ is embedded in $\\mathbb{R}^3$ as above and consider the stereographic projection $\\hat{\\tau}_x:C_1\\backslash \\{x+1\\}\\to \\mathbb{R}^2$, which sends $x+1$ to $\\infty$ and $x$ to $(0,1)$. Note that the diameter of $\\tau_x(\\hat{W}_x)$ is $O(t^{-(2m_x-3)\/3j(x)})$, then for some constant $\\kappa_x$, we have $\\hat{\\tau}_x^{-1}\\left(\\mathbb{R}^2\\backslash C_2\\right)\\subset \\tau_x(W_2)$, where $C_2:=B_{(0,1)}(\\kappa_xt^{(2m_x-3)\/3j(x)})$. Define weighted spaces on $C_2$ using the singular metric $g_{C_2}=\\hat{r}^{-1}g_{e(C_2)}$ where $g_{e(C_2)}$ is the Euclidean metric, and the weight function $\\hat{r}^{1\/2}$ where $\\hat{r}$ is the Euclidean distance from $(0,1)$. Then $v_2=(\\hat{\\tau}_x^{-1})^\\ast v_1$ vanishes on $\\partial C_2$, $c^{-1}(\\hat{\\tau}_x^{-1})^\\ast r_x\\leqslant \\hat{r}\\leqslant ct^{(2m_x-3)\/3j(x)}(\\hat{\\tau}_x^{-1})^\\ast r_x$, and $c^{-1}(\\hat{\\tau}_x^{-1})^\\ast g_{C_1}\\leqslant g_{C_2}$ on $C_2$. We have\n\\[\\lVert v_1\\rVert_{L_{-\\delta}^2(C_1)}^2\\leqslant c\\lVert v_2\\rVert_{L_{-\\delta}^2(C_2)}^2,\\quad \\lVert \\mathrm{d}v_2\\rVert_{L_{-\\delta-1}^2(C_2)}^2\\leqslant ct^{(2m_x-3)\\delta\/3j(x)}\\lVert \\mathrm{d}v_1\\rVert_{L_{-\\delta-1}^2(C_1)}^2.\\]\nLet $\\check{L}_{\\delta}^{k,p}(C_2)$ be defined using the Euclidean metric $g_{e(C_2)}$ and the weight function $\\hat{r}$. An analogue of Lemma \\ref{SobEmbed_lem} (\\romannumeral2) implies that \\[\\lVert v_2\\rVert_{\\check{L}_{-\\delta\/2}^2(C_2)}^2\\leqslant c\\lVert \\mathrm{d}v_2\\rVert_{\\check{L}_{-\\delta\/2-1}^2(C_2)}^2,\\]\nsince $v_2$ vanishes on $\\partial C_2$. This is equivalent to $\\lVert v_2\\rVert_{L_{-\\delta}^2(C_2)}^2\\leqslant c\\lVert \\mathrm{d}v_2\\rVert_{L_{-\\delta-1}^2(C_2)}^2$. Therefore \\eqref{hardyineqIt_eq1} and \\eqref{hardyineqIt_eq} are verified.\n\nFor $x\\in T$, let $\\hat{\\chi}_x(r):=\\chi(\\lambda_x(t)^2 r)$, where $r=|\\zeta_{x,t}|$. In $\\hat{B}_{x,t}$, similar to \\eqref{HiggsIuEst_eq}, we have \\begin{equation}\\label{HiggsTEst_eq}\n|[\\Phi_t,u]|^2\\geqslant c^{-1}t^2|u|^2.\n\\end{equation}\nLet $v$ be vanishing on $\\partial\\hat{B}_{x,t}$, then as before,\n\\begin{equation}\\label{hardyineqT_eq}\n  \\lVert v\\rVert_{L_{-\\delta}^2(\\hat{B}_{x,t})}^2\\leqslant ct^{-2}\\lVert \\nabla_{A_t}u\\rVert_{L_{-\\delta-1}^2(\\hat{B}_{x,t})}^2.\n\\end{equation}\n\nApplying \\eqref{hardyineqIu_eq}, \\eqref{hardyineqIt_eq} to $v=(1-\\chi_x)u$ and \\eqref{hardyineqT_eq} to $v=\\hat{\\chi}_x u$, and using \\eqref{HiggsIuEst_eq}, \\eqref{HiggsItEst_eq}, \\eqref{HiggsTEst_eq},\n\\begin{align*}\n  \\lVert u\\rVert_{L_{-\\delta}^2}^2&\\leqslant c\\sum_{x\\in I}\\lVert u\\rVert_{L_{-\\delta,x}^2}^2+ c\\sum_{x\\in T} \\lVert\\hat{\\chi}_x u\\rVert_{L_{-\\delta}^2(\\hat{B}_{x,t})}^2\\\\& \\leqslant c\\sum_{x\\in I}\\lVert \\chi_xu\\rVert_{L^2(W_x)}^2+c\\sum_{x\\in I}\\lVert (1-\\chi_x)u\\rVert_{L_{-\\delta,x}^2}^2+ c\\sum_{x\\in T} \\lVert\\hat{\\chi}_x u\\rVert_{L_{-\\delta}^2(\\hat{B}_{x,t})}^2\\\\&\\leqslant c\\left(\\sum_{x\\in I_u}t^{-(4m_x+2)\/3j(x)}+\\sum_{x\\in I_t}t^{-4m_x\/3j(x)}\\right)\\lVert [\\Phi_t,u]\\rVert_{L^2}^2\\\\&+c\\sum_{x\\in I_u} t^{(6+(4m_x-4)\\delta)\/3j(x)}\\lVert \\nabla_{A_t}(1-\\chi_x)u\\rVert_{L_{-\\delta-1,x}^2}^2\\\\&+c\\sum_{x\\in I_t} t^{(6+(2m_x-3)\\delta)\/3j(x)}\\lVert \\nabla_{A_t}(1-\\chi_x)u\\rVert_{L_{-\\delta-1,x}^2}^2+ct^{-2}\\sum_{x\\in T}\\lVert \\nabla_{A_t}\\hat{\\chi}_x u\\rVert_{L_{-\\delta-1,x}^2}^2\\\\\n  &\\leqslant ct^2\\left(\\lVert \\nabla_{A_t}u\\rVert_{L^2}^2+\\lVert [\\Phi_t,u]\\rVert_{L^2}^2\\right),\n\\end{align*}\nwhere in the last inequality we used $\\delta<1$, $(4m_x+2)\/3j(x)<4\/3+2\/3=2$, and the $L^2$ norm is uniformly stronger than various $L_{-\\delta-1}^2$ norms.\n\\end{proof}\n\n\\begin{proposition}\\label{InvNormBd_prop}\n  The norm of $L_t^{-1}:L_{-2+\\delta}^2(i\\mathfrak{su}(E))\\to L_{-2+\\delta}^{2,2}(i\\mathfrak{su}(E))$ is bounded by $ct^8$.\n\\end{proposition}\n\\begin{proof}\nLet $v\\in C_0^\\infty(C\\backslash S)$, then $u=L_t^{-1}(v)\\in L_{-2+\\delta}^{2,2}$ and the boundary terms leading to the integration by parts vanish as above, so \\eqref{intbyParts_eq} holds. By the density of $C_0^\\infty(C\\backslash S)$ in $L_{-2+\\delta}^{2}$ and the continuity of $L_t^{-1}, \\mathrm{d}_{A_t}, [\\Psi_t,\\cdot]$, \\eqref{intbyParts_eq} holds for all $u\\in L_{-2+\\delta}^{2,2}$. Using \\eqref{intbyParts_eq} and \\eqref{PoincareIneq_eq},\n  \\begin{align*}\n   \\lVert u\\rVert_{L_{-\\delta}^2}^2&\\leqslant ct^2\\left(\\lVert \\nabla_{A_t} u\\rVert_{L^2}^2+\\lVert [\\Phi_t,u]\\rVert_{L^2}^2\\right)\\leqslant ct^2\\langle L_t u,u\\rangle_{L^2}\\\\&\\leqslant ct^2\\lVert u\\rVert_{L_{-\\delta}^2}\\lVert L_t u\\rVert_{L_{-2+\\delta}^2},\\\\\n   \\lVert u\\rVert_{L_{-2+\\delta}^2}&\\leqslant \\lVert u\\rVert_{L_{-\\delta}^2}\\leqslant ct^2\\lVert L_t u\\rVert_{L_{-2+\\delta}^2},\\\\\n  \\lVert \\nabla_{A_t}u\\rVert_{L_{-2+\\delta}^2}&\\leqslant \\lVert\\nabla_{A_t}u\\rVert_{L^2}\\leqslant ct\\lVert L_t u\\rVert_{L_{-2+\\delta}^2},\\\\\\lVert [\\Phi_t,u]\\rVert_{L^2}&\\leqslant ct\\lVert L_t u\\rVert_{L_{-2+\\delta}^2}.\n \\end{align*}\nFor $x\\in I$, let $\\hat{\\chi}_x$ be the function supported in $\\widetilde{B}_{x,t}$ where it satisfies $\\hat{\\chi}_x(r)=\\chi(\\kappa^{-1} t^{1\/j(x)} r)$ ($r=|z_x|$ for $x\\in I_u$ and $r=|\\zeta_{x,t}|$ for $x\\in I_t$). Recall that we have defined $\\hat{\\chi}_x$ for $x\\in T$ in a similar way. Using Propositions \\ref{analysisIu_prop}, \\ref{analysisIt_cor}, \\ref{analysisT_prop}, \\ref{analysisUext_prop} we have\n \\begin{align}\n   \\lVert u\\rVert_{L_{-2+\\delta}^{2,2}}&\\leqslant \\sum_{x\\in S}\\lVert \\hat{\\chi}_x u\\rVert_{L_{-2+\\delta}^{2,2}}+\\lVert (1-\\sum_{x\\in S} \\hat{\\chi}_x)u \\rVert_{L^{2,2}}\\\\&\\leqslant c\\sum_{x\\in S}\\lVert L_t\\hat{\\chi}_x u\\rVert_{L_{-2+\\delta}^2}+ct^4\\lVert L_t(1-\\sum_{x\\in S}\\hat{\\chi}_x) u\\rVert_{L^2}\\notag\\\\\n   &\\leqslant ct^4\\left(\\lVert  L_t u\\rVert_{L_{-2+\\delta}^2}+t^2\\lVert u\\rVert_{L_{-\\delta}^2}+t\\lVert  \\nabla_{A_t} u\\rVert_{L^2}+t\\lVert [\\Phi_t,u]\\rVert_{L^2}\\right)\\notag\\\\\n   &\\leqslant ct^8\\lVert L_t u\\rVert_{L_{-2+\\delta}^2}.\\label{InverseBound_eq}\n \\end{align}\n\\end{proof}\n\\section{Deforming to a Solution}\nNow we find a solution to Hitchin's equation by deforming the approximate solution $(A_t,\\Phi_t)$ via a complex gauge transformation $\\exp(\\gamma)$ for $\\gamma\\in\\Omega^0(i\\mathfrak{su}(E))$, which satisfies\n\\[F_t(\\gamma)=F_{A_t^{\\exp(\\gamma)}}+\\left[\\mathrm{e}^{-\\gamma}\\Phi_t\\mathrm{e}^{\\gamma},\\mathrm{e}^{\\gamma}\\Phi_t^\\ast \\mathrm{e}^{-\\gamma}\\right]=:F_{A_t}+[\\Phi_t,\\Phi_t^\\ast]+L_t\\gamma +Q_t\\gamma=0,\\]\nwhere by abuse of notation we still denote by $L_t$ the linear operator in the previous section composed with the isomorphism $\\mathrm{i}\\star$. We will analyze the nonlinear term $Q_t$ in a similar way as that in \\cite{mazzeo_swoboda_weiss_witt_2016}. Denote\n\\begin{align*}\n  A_t^{\\exp \\gamma}&=A_t+(\\bar{\\partial}_{A_t}-\\partial_{A_t})\\gamma+R_{A_t}(\\gamma),\\\\ \\Phi_t^{\\exp \\gamma}&=\\Phi_t+[\\Phi_t,\\gamma]+R_{\\Phi_t}(\\gamma),\\\\\n  R_{A_t}(\\gamma)&=\\mathrm{e}^{-\\gamma}(\\bar{\\partial}_{A_t}\\mathrm{e}^{\\gamma})-(\\partial_{A_t}\\mathrm{e}^{\\gamma})\\mathrm{e}^{-\\gamma}-(\\bar{\\partial}_{A_t}-\\partial_{A_t})\\gamma,\\\\\n  R_{\\Phi_t}(\\gamma)&=\\mathrm{e}^{-\\gamma}\\Phi_t \\mathrm{e}^{\\gamma}-[\\Phi_t,\\gamma]-\\Phi_t.\n\\end{align*}\nThen we have \\begin{align}\n  Q_t(\\gamma)&=\\mathrm{d}_{A_t}(R_{A_t}(\\gamma))+[R_{\\Phi_t}(\\gamma),\\Phi_t^\\ast]+[\\Phi_t,R_{\\Phi_t}(\\gamma)^\\ast]\\notag\\\\&\\quad+\\frac{1}{2}[(\\bar{\\partial}_{A_t}-\\partial_{A_t})\\gamma+R_{A_t}(\\gamma),(\\bar{\\partial}_{A_t}-\\partial_{A_t})\\gamma+R_{A_t}(\\gamma)]\\notag\\\\\n  &\\quad+[[\\Phi_t,\\gamma]+R_{\\Phi_t}(\\gamma),([\\Phi_t,\\gamma]+R_{\\Phi_t}(\\gamma))^\\ast].\\label{NonlinTerm_eq}\n\\end{align}\nFor $u\\in \\Omega^p(\\mathrm{End}\\, E)$, define \\[\\lVert u\\rVert_{\\widetilde{L}_{\\delta}^{k,q}}:=\\left(\\sum_{j=0}^k\\lVert \\nabla_{A_t}^j u\\rVert_{L_{\\delta}^q}^q\\right)^{1\/q}.\\]\nClearly $\\lVert u\\rVert_{\\widetilde{L}_{\\delta}^{k,q}}\\leqslant \\lVert u\\rVert_{L_{\\delta}^{k,q}}$.\n\n\\begin{lemma}\n  $\\widetilde{L}_{-2+\\delta}^{2,2}(\\mathrm{End}\\,E)$ is an algebra, and $\\widetilde{L}_{-2+\\delta}^{1,2}(\\mathrm{End}\\,E)$ is a module over this algebra. Moreover,\n  \\begin{align}\n    \\lVert uv\\rVert_{\\widetilde{L}_{-2+\\delta}^{2,2}}&\\leqslant ct^5 \\lVert u\\rVert_{\\widetilde{L}_{-2+\\delta}^{2,2}}\\lVert v\\rVert_{\\widetilde{L}_{-2+\\delta}^{2,2}},\\\\\n    \\lVert uv\\rVert_{\\widetilde{L}_{-2+\\delta}^{1,2}}&\\leqslant ct^{9\/2} \\lVert u\\rVert_{\\widetilde{L}_{-2+\\delta}^{2,2}}\\lVert v\\rVert_{\\widetilde{L}_{-2+\\delta}^{1,2}},\\\\\n    \\lVert u\\rVert_{L^\\infty}&\\leqslant ct^3 \\lVert u\\rVert_{\\widetilde{L}_{-2+\\delta}^{2,2}}.\n  \\end{align}\n\\end{lemma}\n\\begin{proof}\n  Let $\\tau_x$ and $\\hat{\\chi}_x$ be defined as in the previous section, and denote $u_x:=\\hat{\\chi}_xu$, $u_{\\mathrm{ext}}:=(1-\\sum_{x\\in S}\\hat{\\chi}_x)u$. For $x\\in I_u$, $\\tau_x$ maps $\\widetilde{B}_{x,t}$ to $B_0(\\kappa |c_x|):=\\{\\,|z|<\\kappa |c_x|\\,\\}$. Denote the Euclidean metric on this disk by $g_e$, then $c^{-1}t^{2\/j(x)}(\\tau_x^{-1})^\\ast g_t\\leqslant g_e\\leqslant ct^{2\/j(x)}(\\tau_x^{-1})^\\ast g_t$. Define the Sobolev spaces on $B_0(\\kappa |c_x|)$ using the weight function $r=|z|$ and the metric $g_e$. For $w$ a $p$-form with values in $\\mathrm{End} E$, using the Sobolev embedding $L_{-2+\\delta}^{1,2}\\hookrightarrow L_{-1+\\delta}^q$ on $B_0(\\kappa |c_x|)$ for any fixed $q\\in[2,\\infty)$,\n  \\begin{align*}\n      \\lVert w_x\\rVert_{L_{-1+\\delta}^q}&\\leqslant ct^{-2\/qj(x)}\\lVert (\\tau_x^{-1})^{\\ast}|w_x|\\rVert_{L_{-1+\\delta}^q(B_0(\\kappa |c_x|))}\\\\&\\leqslant ct^{-2\/qj(x)}\\left(\\lVert (\\tau_x^{-1})^\\ast |w_x|\\rVert_{L_{-2+\\delta}^2(B_0(\\kappa |c_x|))}+\\lVert (\\tau_x^{-1})^\\ast \\mathrm{d}|w_x|\\rVert_{L_{-2+\\delta}^2(B_0(\\kappa |c_x|))}\\right)\\\\\n      &\\leqslant ct^{-2\/qj(x)}\\left(t^{1\/j(x)}\\lVert w_x\\rVert_{L_{-2+\\delta}^2}+\\lVert \\nabla_{A_t}w_x\\rVert_{L_{-2+\\delta}^2}\\right)\\leqslant ct^{(q-2)\/qj(x)}\\lVert w_x\\rVert_{\\widetilde{L}_{-2+\\delta}^{1,2}}.\n  \\end{align*}\n  Similarly for $x\\in T$, we have $\\lVert w_x\\rVert_{L_{-1+\\delta}^q}\\leqslant ct^{(q-2)\/q}\\lVert w_x\\rVert_{\\widetilde{L}_{-2+\\delta}^{1,2}}$. For $x\\in I_t$, let $\\check{L}_{\\delta}^{k,m}(B_0(\\kappa |c_x|))$ be defined using $r^{-1}g_e$ and the weight function $r^{1\/2}$, then $\\check{L}_{\\delta}^{q}$ is equivalent to $L_{\\delta\/2}$ for functions on $B_0(\\kappa |c_x|)$, and we have\n  \\begin{align*}\n    \\lVert w_x\\rVert_{L_{-1+\\delta}^q}&\\leqslant ct^{-2\/qj(x)}\\lVert (\\tau_x^{-1})^{\\ast}|w_x|\\rVert_{\\check{L}_{-1+\\delta}^q(B_0(\\kappa |c_x|))}\\leqslant ct^{-2\/qj(x)}\\lVert (\\tau_x^{-1})^{\\ast}|w_x|\\rVert_{L_{(-1+\\delta)\/2}^q(B_0(\\kappa |c_x|))}\n    \\\\&\\leqslant ct^{-2\/qj(x)}\\left(\\lVert (\\tau_x^{-1})^\\ast |w_x|\\rVert_{L_{(-1+\\delta)\/2-1}^2(B_0(\\kappa |c_x|))}+\\lVert (\\tau_x^{-1})^\\ast \\mathrm{d}|w_x|\\rVert_{L_{(-1+\\delta)\/2-1}^2(B_0(\\kappa |c_x|))}\\right)\n    \\\\&\\leqslant ct^{-2\/qj(x)}\\left(\\lVert (\\tau_x^{-1})^\\ast |w_x|\\rVert_{\\check{L}_{-2+\\delta}^2(B_0(\\kappa |c_x|))}+\\lVert (\\tau_x^{-1})^\\ast \\mathrm{d}|w_x|\\rVert_{\\check{L}_{-2+\\delta}^2(B_0(\\kappa |c_x|))}\\right)\\\\\n    &\\leqslant ct^{-2\/qj(x)}\\left(t^{1\/j(x)}\\lVert w_x\\rVert_{L_{-2+\\delta}^2}+\\lVert \\nabla_{A_t}w_x\\rVert_{L_{-2+\\delta}^2}\\right)\\leqslant ct^{(q-2)\/qj(x)}\\lVert w_x\\rVert_{\\widetilde{L}_{-2+\\delta}^{1,2}}.\n\\end{align*}\nOn $U_{\\mathrm{ext}}$, $c^{-1}g_C\\leqslant g_t\\leqslant ctg_C$, and $r_t$ is uniformly bounded, then\n\\begin{align}\n  \\lVert w_{\\mathrm{ext}}\\rVert_{L^q}&\\leqslant ct^{1\/q}\\lVert w_{\\mathrm{ext}}\\rVert_{L^q(g_C)}\\leqslant ct^{1\/q}\\left(\\lVert w_{\\mathrm{ext}}\\rVert_{L^2(g_C)}+\\lVert \\nabla_{A_t}w_{\\mathrm{ext}}\\rVert_{L^2(g_C)}\\right)\\notag\\\\\n  &\\leqslant ct^{1\/q}\\left(t^{\\max(p-1,0)\/2}\\lVert w_{\\mathrm{ext}}\\rVert_{L^2}+t^{p\/2}\\lVert \\nabla_{A_t}w_{\\mathrm{ext}}\\rVert_{L^2}\\right).\\label{SobineqExt_eq}\n\\end{align}\nCombining the above inequalities,\n  \\[\\lVert w\\rVert_{L_{-1+\\delta}^q}\\leqslant \\sum_{x\\in S}\\lVert w_x\\rVert_{L_{-1+\\delta}^q}+\\lVert w_{\\mathrm{ext}}\\rVert_{L_{-1+\\delta}^q}\\leqslant ct^{2+p\/2}\\lVert w\\rVert_{\\widetilde{L}_{-2+\\delta}^{1,2}}.\\]\nFor $w_i\\in \\Omega^{p_i}(\\mathrm{End} E)$ ($i=1,2$),\n  \\begin{align}\n    \\lVert w_1w_2\\rVert_{L_{-2+\\delta}^2}&=\\lVert r_t^{1-\\delta}w_1w_2\\rVert_{L^2}\\leqslant \\lVert r_t^{1-2\\delta}w_1w_2\\rVert_{L^2}\\leqslant \\lVert r_t^{1\/2-\\delta}w_1\\rVert_{L^4}\\lVert r_t^{1\/2-\\delta}w_2\\rVert_{L^4}\\notag\\\\&=\\lVert w_1\\rVert_{L_{-1+\\delta}^4}\\lVert w_2\\rVert_{L_{-1+\\delta}^4}\\leqslant ct^{4+(p_1+p_2)\/2}\\lVert w_1\\rVert_{\\widetilde{L}_{-2+\\delta}^{1,2}}\\lVert w_2\\rVert_{\\widetilde{L}_{-2+\\delta}^{1,2}}.\\label{SobMultIneq_eq}\n  \\end{align}\nFor $2<q<2\/(1-\\delta)$, $x\\in I_u$, using $L_{-1+\\delta}^q\\subset L^q$, $ L^{1,q}\\subset L^\\infty$ on $B_0(\\kappa|c_x|)$,\n  \\begin{align*}\n    \\lVert u_x\\rVert_{L^\\infty}&=\\lVert (\\tau_x^{-1})^\\ast |u_x|\\rVert_{L^\\infty(B_0(\\kappa|c_x|))}\\\\&\\leqslant c\\left(\\lVert (\\tau_x^{-1})^{\\ast}|u_x|\\rVert_{L_{-1+\\delta}^q(B_0(\\kappa|c_x|)}+\\lVert (\\tau_x^{-1})^\\ast \\mathrm{d}|u_x|\\rVert_{L_{-1+\\delta}^q(B_0(\\kappa|c_x|)}\\right)\\\\\n    &\\leqslant c\\left(t^{2\/qj(x)}\\lVert u_x\\rVert_{L_{-1+\\delta}^q}+t^{(2-q)\/qj(x)}\\lVert\\nabla_{A_t}u_x\\rVert_{L_{-1+\\delta}^q}\\right)\\\\\n    &\\leqslant c\\left(t^{2\/qj(x)}t^{(q-2)\/qj(x)}\\lVert u_x\\rVert_{\\widetilde{L}_{-2+\\delta}^{1,2}}+\\lVert \\nabla_{A_t}u_x\\rVert_{\\widetilde{L}_{-2+\\delta}^{1,2}}\\right)\\leqslant ct^{1\/j(x)}\\lVert u\\rVert_{\\widetilde{L}_{-2+\\delta}^{2,2}}.\n  \\end{align*}\n  Similarly, we have $\\lVert u_x\\rVert_{L^\\infty}\\leqslant ct\\lVert u_x\\rVert_{\\widetilde{L}_{-2+\\delta}^{2,2}}$ for $x\\in T$, and $\\lVert u_x\\rVert_{L^\\infty}\\leqslant ct^{1\/j(x)}\\lVert u_x\\rVert_{\\widetilde{L}_{-2+\\delta}^{2,2}}$ for $x\\in I_t$. On $U_{\\mathrm{ext}}$, by \\eqref{SobineqExt_eq},\n  \\[  \\lVert u_{\\mathrm{ext}}\\rVert_{L^\\infty}\\leqslant c\\left(\\lVert u_{\\mathrm{ext}}\\rVert_{L^q(g_C)}+\\lVert \\nabla_{A_t}u_{\\mathrm{ext}}\\rVert_{L^q(g_C)}\\right)\\leqslant ct^{1\/2}\\lVert u_{\\mathrm{ext}}\\rVert_{\\widetilde{L}_{-2+\\delta}^{2,2}}.\\]\nTherefore,\n  \\[\\lVert u\\rVert_{L^{\\infty}}\\leqslant \\sum_{x\\in S}\\lVert u_x\\rVert_{L^{\\infty}}+\\lVert u_{\\mathrm{ext}}\\rVert_{L^{\\infty}}\\leqslant ct^3\\lVert u\\rVert_{\\widetilde{L}_{-2+\\delta}^{2,2}}.\\]\n  Then by \\eqref{SobMultIneq_eq},\n  \\begin{align*}\n    \\lVert \\nabla_{A_t}(uv)\\rVert_{L_{-2+\\delta}^2}&\\leqslant \\lVert (\\nabla_{A_t}u)v\\rVert_{L_{-2+\\delta}^2}+\\lVert u\\nabla_{A_t}v\\rVert_{L_{-2+\\delta}^2}\\\\&\\leqslant ct^{9\/2}\\lVert u\\rVert_{\\widetilde{L}_{-2+\\delta}^{2,2}}\\lVert v\\rVert_{\\widetilde{L}_{-2+\\delta}^{1,2}}+\\lVert u\\rVert_{L^\\infty}\\lVert v\\rVert_{\\widetilde{L}_{-2+\\delta}^{1,2}}\n  \\leqslant ct^{9\/2}\\lVert u\\rVert_{\\widetilde{L}_{-2+\\delta}^{2,2}}\\lVert v\\rVert_{\\widetilde{L}_{-2+\\delta}^{1,2}}.\n  \\end{align*}\n  Finally we estimate\n  \\begin{align*}\n    \\lVert\\nabla_{A_t}^2 uv\\rVert_{L_{-2+\\delta}^2}&\\leqslant \\lVert u\\rVert_{L^\\infty}\\lVert \\nabla_{A_t}^2 v\\rVert_{L_{-2+\\delta}^2}+ct^5\\lVert \\nabla_{A_t}u\\rVert_{\\widetilde{L}_{-2+\\delta}^{1,2}}\\lVert \\nabla_{A_t}v\\rVert_{\\widetilde{L}_{-2+\\delta}^{1,2}}+\\lVert v\\rVert_{L^\\infty}\\lVert \\nabla_{A_t}^2 u\\rVert_{L_{-2+\\delta}^2}\\\\\n    &\\leqslant ct^5\\lVert u\\rVert_{\\widetilde{L}_{-2+\\delta}^{2,2}}\\lVert v\\rVert_{\\widetilde{L}_{-2+\\delta}^{2,2}}.\n  \\end{align*}\n\\end{proof}\n\\begin{theorem}\\label{DefSol_thm}\n There exists $t_1,c,c'$ such that for every $t\\geqslant t_1$, there exists a unique $h_t^{\\mathrm{app}}$-Hermitian $\\gamma_t\\in L_{-2+\\delta}^{2,2}(i\\mathfrak{su}(E))$ with $\\lVert \\gamma_t\\rVert_{L_{-2+\\delta}^{2,2}}\\leqslant c\\exp(-c't^\\sigma)$, such that $F_t(\\gamma_t)=0$. Hence $h_t$ defined by $h_t(v,w)=h_t^{\\mathrm{app}}(\\mathrm{e}^{-\\gamma_t}v,\\mathrm{e}^{-\\gamma_t}w)$ is a harmonic metric associated to $(\\bar{\\partial}_E,\\Phi_t)$ and is compatible with the parabolic structure at each $x\\in S$.\n\\end{theorem}\n\\begin{proof}\nLet $0<\\rho\\leqslant 1$ and $\\gamma_1,\\gamma_2\\in B_\\rho:=\\{\\gamma\\in L_{-2+\\delta}^{2,2}(i\\mathfrak{su}(E)):\\lVert \\gamma\\rVert_{L_{-2+\\delta}^{2,2}}<\\rho\\}$. We estimate each of the five terms in the difference $Q_t(\\gamma_1)-Q_t(\\gamma_2)$, with $Q_t$ given by \\eqref{NonlinTerm_eq}. First, \\begin{align}\n  \\lVert R_{A_t}(\\gamma_1)-R_{A_t}(\\gamma_2))\\rVert_{\\widetilde{L}_{-2+\\delta}^{1,2}}&\\leqslant\\lVert \\left(\\mathrm{e}^{-\\gamma_1}(\\bar{\\partial}_{A_t}\\mathrm{e}^{\\gamma_1})-\\mathrm{e}^{-\\gamma_2}(\\bar{\\partial}_{A_t}\\mathrm{e}^{\\gamma_2})-\\bar{\\partial}_{A_t}(\\gamma_1-\\gamma_2)\\right)\\rVert_{\\widetilde{L}_{-2+\\delta}^{1,2}}\\notag\\\\\n  &\\quad +\\lVert \\left((\\partial_{A_t}\\mathrm{e}^{\\gamma_1})\\mathrm{e}^{-\\gamma_1}-(\\partial_{A_t}\\mathrm{e}^{\\gamma_2})\\mathrm{e}^{-\\gamma_2}+\\partial_{A_t}(\\gamma_1-\\gamma_2)\\right)\\rVert_{\\widetilde{L}_{-2+\\delta}^{1,2}}.\\label{NonlinearAaDiff_eq}\n\\end{align}\nThe first term is bounded by\n\\begin{align*}\n  \\lVert (\\mathrm{e}^{-\\gamma_1}-\\mathrm{e}^{-\\gamma_2})\\bar{\\partial}_{A_t} \\mathrm{e}^{\\gamma_1}\\rVert_{\\widetilde{L}_{-2+\\delta}^{1,2}}+\\lVert \\mathrm{e}^{-\\gamma_2}(\\bar{\\partial}_{A_t}(\\mathrm{e}^{\\gamma_1}-\\mathrm{e}^{\\gamma_2}))-\\bar{\\partial}_{A_t}(\\gamma_1-\\gamma_2) \\rVert_{\\widetilde{L}_{-2+\\delta}^{1,2}}:=\\uppercase\\expandafter{\\romannumeral1}+\\uppercase\\expandafter{\\romannumeral2}.\n\\end{align*}\nWrite $\\mathrm{e}^\\gamma=1+\\gamma+S(\\gamma)$, then\n\\begin{align*}\n  \\uppercase\\expandafter{\\romannumeral1}&\\leqslant c t^{9\/2}\\lVert \\mathrm{e}^{-\\gamma_1}-\\mathrm{e}^{-\\gamma_2}\\rVert_{\\widetilde{L}_{-2+\\delta}^{2,2}}\\lVert \\bar{\\partial}_{A_t}\\mathrm{e}^{\\gamma_1}\\rVert_{\\widetilde{L}_{-2+\\delta}^{1,2}}\\leqslant c t^{9\/2}\\lVert \\mathrm{e}^{-\\gamma_1}- \\mathrm{e}^{-\\gamma_2}\\rVert_{\\widetilde{L}_{-2+\\delta}^{2,2}}\\lVert \\gamma_1+S(\\gamma_1)\\rVert_{\\widetilde{L}_{-2+\\delta}^{2,2}}\n  \\\\&\\leqslant c t^{9\/2}\\lVert \\gamma_1-\\gamma_2\\rVert_{\\widetilde{L}_{-2+\\delta}^{2,2}}\\left(\\sum_{j=0}^\\infty \\frac{(\\rho ct^5)^j}{j!}\\right)\\rho\\left(\\sum_{j=0}^\\infty \\frac{(\\rho ct^5)^j}{(j+1)!}\\right)\\\\\n  &\\leqslant ct^{9\/2}\\rho\\lVert \\gamma_1-\\gamma_2\\rVert_{\\widetilde{L}_{-2+\\delta}^{2,2}},\n\\end{align*}\nfor $\\rho<t^{-5}$. Similarly,\n\\begin{align*}\n  \\uppercase\\expandafter{\\romannumeral2}&=\\lVert (1-\\gamma_2+S(-\\gamma_2))\\left(\\bar{\\partial}_{A_t}(\\gamma_1-\\gamma_2+S(\\gamma_1)-S(\\gamma_2))\\right)-\\bar{\\partial}_{A_t}(\\gamma_1-\\gamma_2)\\rVert_{\\widetilde{L}_{-2+\\delta}^{1,2}}\\\\\n  &\\leqslant \\lVert \\bar{\\partial}_{A_t}(S(\\gamma_1)-S(\\gamma_2))\\rVert_{\\widetilde{L}_{-2+\\delta}^{1,2}}+\\lVert (-\\gamma_2+S(-\\gamma_2))\\bar{\\partial}_{A_t}(\\gamma_1-\\gamma_2+S(\\gamma_1)-S(\\gamma_2))\\rVert_{\\widetilde{L}_{-2+\\delta}^{1,2}}\\\\\n  &\\leqslant \\lVert S(\\gamma_1)-S(\\gamma_2)\\rVert_{\\widetilde{L}_{-2+\\delta}^{2,2}}+  ct^{9\/2}\\lVert -\\gamma_2+S(-\\gamma_2)\\rVert_{\\widetilde{L}_{-2+\\delta}^{2,2}}\\lVert \\gamma_1-\\gamma_2+S(\\gamma_1)-S(\\gamma_2)\\rVert_{\\widetilde{L}_{-2+\\delta}^{2,2}}\\\\\n  &\\leqslant ct^{9\/2}\\rho \\lVert \\gamma_1-\\gamma_2\\rVert_{\\widetilde{L}_{-2+\\delta}^{2,2}}.\n\\end{align*}\nThe estimate for the other term in \\eqref{NonlinearAaDiff_eq} is analogous, and we obtain \\begin{align}\n    \\lVert R_{A_t}(\\gamma_1)-R_{A_t}(\\gamma_2))\\rVert_{\\widetilde{L}_{-2+\\delta}^{1,2}}&\\leqslant ct^{9\/2}\\rho\\lVert \\gamma_1-\\gamma_2\\rVert_{\\widetilde{L}_{-2+\\delta}^{2,2}}\\leqslant ct^{9\/2}\\rho\\lVert \\gamma_1-\\gamma_2\\rVert_{L_{-2+\\delta}^{2,2}},\\label{RAtIneq_eq}\\\\\n    \\lVert \\mathrm{d}_{A_t}(R_{A_t}(\\gamma_1)-R_{A_t}(\\gamma_2))\\rVert_{L_{-2+\\delta}^2}&\\leqslant ct^{9\/2}\\rho\\lVert \\gamma_1-\\gamma_2\\rVert_{L_{-2+\\delta}^{2,2}}.\\notag\n\\end{align}\nLet $\\beta_1,\\ldots,\\beta_n\\in L_{-2+\\delta}^{2,2}$, where $n\\geqslant 2$. By the Jacobi identity, $\\left|[\\Phi_t^\\ast,[\\beta_1,\\cdots,[\\beta_n,\\Phi_t]\\cdots]]\\right|$ is bounded by\n\\begin{align*}\n  & \\left|[[\\Phi_t^\\ast,\\beta_1],[\\beta_2,\\cdots,[\\Phi_t,\\beta_n]\\cdots]]\\right|+\\left|[\\beta_1,[[\\Phi_t^\\ast,\\beta_2],[\\beta_3,\\cdots[\\Phi_t,\\beta_n]\\cdots]]]\\right|\\\\\n  +\\cdots&+\\left|[\\beta_1,\\cdots[[\\Phi_t^\\ast,\\beta_{n-1}],[\\Phi_t,\\beta_n]]\\cdots]\\right|+\\left|[\\beta_1,\\cdots[\\beta_{n-1},[\\Phi_t^\\ast,[\\Phi_t,\\beta_n]]]\\cdots]\\right|\\\\\n  &\\leqslant 2^n\\left(\\sum_{j=1}^{n-1}\\left(|[\\Phi_t^\\ast,\\beta_j]||[\\Phi_t,\\beta_n]|\\prod_{m\\neq j,n}\\lVert \\beta_m\\rVert_{L^\\infty}\\right)+|[\\Phi_t^\\ast,[\\Phi_t,\\beta_n]]|\\prod_{m=1}^{n-1}\\lVert \\beta_m\\rVert_{L^\\infty}\\right).\n\\end{align*}\nIntegrate both sides, then $\\left\\lVert[\\Phi_t^\\ast,[\\beta_1,\\cdots,[\\beta_n,\\Phi_t]\\cdots]]\\right\\rVert_{L_{-2+\\delta}^2}$ is bounded by\n\\begin{align*}\n    & 2^n\\left(\\sum_{j=1}^{n-1}ct^5\\lVert [\\Phi_t^\\ast,\\beta_j]\\rVert_{\\widetilde{L}_{-2+\\delta}^{1,2}}\\lVert [\\Phi_t,\\beta_n]\\rVert_{\\widetilde{L}_{-2+\\delta}^{1,2}}(ct^3)^{n-2}\\prod_{m\\neq j,n}\\lVert \\beta_m\\rVert_{\\widetilde{L}_{-2+\\delta}^{2,2}} \\right.\\\\\n    &\\qquad+\\left. \\lVert[\\Phi_t^\\ast,[\\Phi_t,\\beta_n]]\\rVert_{L_{-2+\\delta}^2}(ct^3)^{n-1}\\prod_{m=1}^{n-1}\\lVert \\beta_m\\rVert_{\\widetilde{L}_{-2+\\delta}^{2,2}} \\right)\\\\\n    &\\leqslant (ct^{9\/2})^{n-1}t^{1\/2}\\prod_{m=1}^n\\lVert \\beta_m\\rVert_{L_{-2+\\delta}^{2,2}}.\n\\end{align*}\nThen for $\\gamma_1,\\gamma_2\\in B_\\rho$,\n\\begin{align*}\n\\left\\lVert [\\Phi_t^\\ast,(\\mathrm{ad}_{\\gamma_1})^n\\Phi_t-(\\mathrm{ad}_{\\gamma_2})^n\\Phi_t]\\right\\rVert_{L_{-2+\\delta}^2}&\\leqslant \\left\\lVert[\\Phi_t^\\ast,[\\gamma_1-\\gamma_2,[\\gamma_1,\\cdots,[\\gamma_1,\\Phi_t]\\cdots]]]\\right\\rVert_{L_{-2+\\delta}^2}\\\\&\\quad+\\left\\lVert[\\Phi_t^\\ast,[\\gamma_2,[\\gamma_1-\\gamma_2,[\\gamma_1,\\cdots,[\\gamma_1,\\Phi_t]\\cdots]]]]\\right\\rVert_{L_{-2+\\delta}^2}\\\\&\\quad+\\cdots+\\left\\lVert[\\Phi_t^\\ast,[\\gamma_2,[\\gamma_2,\\cdots,[\\gamma_1-\\gamma_2,\\Phi_t]\\cdots]]]\\right\\rVert_{L_{-2+\\delta}^2}\\\\\n  &\\leqslant  n(ct^{9\/2}\\rho)^{n-1}t^{1\/2}\\lVert \\gamma_1-\\gamma_2\\rVert_{L_{-2+\\delta}^{2,2}}.\n\\end{align*}\nThen we compute that\n\\begin{align*}\n  \\lVert [R_{\\Phi_t}(\\gamma_1),\\Phi_t^\\ast]-[R_{\\Phi_t}(\\gamma_2),\\Phi_t^\\ast]\\rVert_{L_{-2+\\delta}^2}&=\\lVert[\\Phi_t^\\ast,\\sum_{j=2}^\\infty \\frac{1}{j!}(\\mathrm{ad}_{-\\gamma_1})^j\\Phi_t-(\\mathrm{ad}_{-\\gamma_2})^j\\Phi_t]\\rVert_{L_{-2+\\delta}^2}\\\\\n  &\\leqslant \\sum_{j=2}^\\infty \\frac{1}{j!}j(c t^{9\/2}\\rho)^{j-1}t^{1\/2}\\lVert \\gamma_1-\\gamma_2\\rVert_{L_{-2+\\delta}^{2,2}}\\\\\n  &\\leqslant ct^{5}\\rho\\lVert \\gamma_1-\\gamma_2\\rVert_{L_{-2+\\delta}^{2,2}}.\n\\end{align*}\nSimilarly,\n\\[\\lVert [\\Phi_t,R_{\\Phi_t}(\\gamma_1)^\\ast]-[\\Phi_t,R_{\\Phi_t}(\\gamma_2)^\\ast]\\rVert_{L_{-2+\\delta}^2}\\leqslant ct^{5}\\rho\\lVert \\gamma_1-\\gamma_2\\rVert_{L_{-2+\\delta}^{2,2}}.\\]\nThe $L_{-2+\\delta}^2$ norm of the fourth term in $Q_t(\\gamma_1)-Q_t(\\gamma_2)$ is bounded by\n\\begin{align*}\n&\\,\\quad \\lVert[(\\bar{\\partial}_{A_t}-\\partial_{A_t})(\\gamma_1-\\gamma_2)+R_{A_t}(\\gamma_1)-R_{A_t}(\\gamma_2),(\\bar{\\partial}_{A_t}-\\partial_{A_t})\\gamma_1+R_{A_t}(\\gamma_1)]\\rVert_{L_{-2+\\delta}^2}\\\\\n  &+\\lVert[(\\bar{\\partial}_{A_t}-\\partial_{A_t})\\gamma_2+R_{A_t}(\\gamma_2),(\\bar{\\partial}_{A_t}-\\partial_{A_t})(\\gamma_1-\\gamma_2)+R_{A_t}(\\gamma_1)-R_{A_t}(\\gamma_2)]\\rVert_{L_{-2+\\delta}^2}:=\\uppercase\\expandafter{\\romannumeral1}'+\\uppercase\\expandafter{\\romannumeral2}'.\n\\end{align*}\nBy \\eqref{SobMultIneq_eq}, \\eqref{RAtIneq_eq},\n\\begin{align*}\n  \\uppercase\\expandafter{\\romannumeral1}'&\\leqslant ct^5\\lVert (\\bar{\\partial}_{A_t}-\\partial_{A_t})(\\gamma_1-\\gamma_2)+R_{A_t}(\\gamma_1)-R_{A_t}(\\gamma_2)\\rVert_{\\widetilde{L}_{-2+\\delta}^{1,2}}\\lVert (\\bar{\\partial}_{A_t}-\\partial_{A_t})\\gamma_1+R_{A_t}(\\gamma_1)\\rVert_{\\widetilde{L}_{-2+\\delta}^{1,2}}\\\\\n  &\\leqslant ct^5\\lVert \\gamma_1-\\gamma_2\\rVert_{L_{-2+\\delta}^{2,2}}(1+ct^{9\/2}\\rho)^2\\rho\\leqslant ct^5\\rho\\lVert \\gamma_1-\\gamma_2\\rVert_{L_{-2+\\delta}^{2,2}}.\n\\end{align*}\nSimilarly, $\\uppercase\\expandafter{\\romannumeral2}'$ is also bounded by $ct^5\\rho\\lVert \\gamma_1-\\gamma_2\\rVert_{L_{-2+\\delta}^{2,2}}$. Consider $\\beta_1,\\ldots,\\beta_n\\in L_{-2+\\delta}^{2,2}$, then\n\\begin{align*}\n  \\lVert [\\beta_1,[\\beta_2,\\cdots,[\\beta_n,\\Phi_t]\\cdots]]\\rVert_{\\widetilde{L}_{-2+\\delta}^{1,2}}&\\leqslant (  ct^{9\/2})^{n-1}\\prod_{m=1}^{n-1}\\lVert\\beta_m\\rVert_{L_{-2+\\delta}^{2,2}}\\lVert [\\Phi_t,\\beta_n]\\rVert_{\\widetilde{L}_{-2+\\delta}^{1,2}}\\\\\n  &\\leqslant (  ct^{9\/2})^{n-1}\\prod_{m=1}^{n}\\lVert\\beta_m\\rVert_{L_{-2+\\delta}^{2,2}}.\n\\end{align*}\nThis implies that\n\\[\n  \\lVert R_{\\Phi_t}(\\gamma_1)-R_{\\Phi_t}(\\gamma_2)\\rVert_{\\widetilde{L}_{-2+\\delta}^{1,2}}\\leqslant \\sum_{j=2}^\\infty \\frac{1}{j!}j(  ct^{9\/2}\\rho)^{j-1}\\lVert \\gamma_1-\\gamma_2\\rVert_{L_{-2+\\delta}^{2,2}}\\leqslant ct^{9\/2}\\rho\\lVert \\gamma_1-\\gamma_2\\rVert_{L_{-2+\\delta}^{2,2}}.\n\\]\nThe $L_{-2+\\delta}^2$ norm of the last term in $Q_t(\\gamma_1)-Q_t(\\gamma_2)$ is bounded by\n\\begin{align*}\n&\\,\\quad\\lVert [[\\Phi_t,\\gamma_1-\\gamma_2]+R_{\\Phi_t}(\\gamma_1)-R_{\\Phi_t}(\\gamma_2),([\\Phi_t,\\gamma_1]+R_{\\Phi_t}(\\gamma_1))^\\ast]\\rVert_{L_{-2+\\delta}^2}\\\\&+ \\lVert [[\\Phi_t,\\gamma_2]+R_{\\Phi_t}(\\gamma_2),([\\Phi_t,\\gamma_1-\\gamma_2]+R_{\\Phi_t}(\\gamma_1)-R_{\\Phi_t}(\\gamma_2))^\\ast]\\rVert_{L_{-2+\\delta}^2}:=\\uppercase\\expandafter{\\romannumeral1}''+\\uppercase\\expandafter{\\romannumeral2}'',\n\\end{align*}\nwhere $\\uppercase\\expandafter{\\romannumeral1}''$ can be estimated as\n\\begin{align*}\n  \\uppercase\\expandafter{\\romannumeral1}''&\\leqslant ct^5\\lVert[\\Phi_t,\\gamma_1-\\gamma_2]+R_{\\Phi_t}(\\gamma_1)-R_{\\Phi_t}(\\gamma_2) \\rVert_{\\widetilde{L}_{-2+\\delta}^{1,2}}\\lVert [\\Phi_t,\\gamma_1]+R_{\\Phi_t}(\\gamma_1) \\rVert_{\\widetilde{L}_{-2+\\delta}^{1,2}}\\\\\n  &\\leqslant ct^5\\lVert \\gamma_1-\\gamma_2\\rVert_{L_{-2+\\delta}^{2,2}}(1+ct^{9\/2}\\rho)^2\\rho\\leqslant ct^5\\rho\\lVert \\gamma_1-\\gamma_2\\rVert_{L_{-2+\\delta}^{2,2}},\n\\end{align*}\nsimilar for $\\uppercase\\expandafter{\\romannumeral2}''$. Therefore\n\\begin{equation}\\label{NonlinearEst_eq}\n  \\lVert Q_t(\\gamma_1)-Q_t(\\gamma_2)\\rVert_{L_{-2+\\delta}^2}\\leqslant ct^{5}\\rho\\lVert \\gamma_1-\\gamma_2\\rVert_{L_{-2+\\delta}^{2,2}}.\n\\end{equation}\nNow $F_t(\\gamma)=0$ is equivalent to \\[\\gamma=-L_t^{-1}(F_{A_t}+[\\Phi_t,\\Phi_t^\\ast]+Q_t(\\gamma)):=T_t(\\gamma).\\]\nBy \\eqref{InverseBound_eq}, \\eqref{NonlinearEst_eq}, $\\lVert T_t(\\gamma_1)-T_t(\\gamma_2)\\rVert_{L_{-2+\\delta}^{2,2}}\\leqslant ct^{13}\\rho \\lVert \\gamma_1-\\gamma_2\\rVert_{L_{-2+\\delta}^{2,2}}$. Furthermore, by \\eqref{ErrorEst_eq} (which is still valid when the norm is taken with respect to $g_t$), \\[\\lVert T_t(\\gamma)\\rVert_{L_{-2+\\delta}^{2,2}}\\leqslant ct^{8}\\mathrm{e}^{-c' t^\\sigma}+ct^{13}\\rho^2,\\]\nfor $\\gamma\\in B_\\rho$. Let $\\rho_t=\\mathrm{e}^{(-c'\/2+\\epsilon)t^\\sigma}$, where $\\epsilon<c'\/2$ is small. Then for $t$ large, $\\rho_t<t^{-5}$, $T_t$ maps $B_{\\rho_t}$ to $B_{\\rho_t}$, and is a contraction mapping, hence there is a unique fixed point $\\gamma_t$ in $B_{\\rho_t}$. In other words, the metric $h_t$ is harmonic, which is also compatible with the parabolic structure as $h_t^{\\mathrm{app}}$ is and $\\gamma_t\\in L_{-2+\\delta}^{2,2}$.\n\\end{proof}\n\n\\section{Asymptotic Metric}\nIn this section, we study the asymptotic geometry of $\\mathcal{M}$ using the framework provided by \\cite{fredrickson_2019, fredrickson2022asymptotic}.\n\nElements in $\\mathcal{M}$ are gauge equivalence classes of pairs $(\\bar{\\partial}_E,\\Phi)$, with infinitesimal deformations denoted by $(\\dot{\\eta},\\dot{\\Phi})$. Linearize $\\bar{\\partial}_E\\Phi=0$ and we get \\begin{equation}\\label{LinHol_eq}\n  \\bar{\\partial}_E\\dot{\\Phi}+[\\dot{\\eta},\\Phi]=0.\n\\end{equation}\nFor $[(\\bar{\\partial}_E,\\Phi)]\\in \\mathcal{M}$, let $h$ be the unique harmonic metric (if exists) which induces $h_{\\det E}$, and denote its infinitesimal deformation by $\\dot{h}$. The infinitesimal gauge transformation $\\dot{\\gamma}$ acts as\n\\[(\\dot{\\eta},\\dot{\\Phi},\\dot{h})\\mapsto (\\dot{\\eta}+\\bar{\\partial}_E\\dot{\\gamma},\\dot{\\Phi}+[\\Phi,\\dot{\\gamma}],\\dot{h}+\\dot{\\gamma}).\\]\n The linearized Hitchin equation and the Coulomb gauge condition can be combined to form a single equation \\cite[Prop.~2.2]{fredrickson_2019}\n\\begin{equation}\\label{LinHitCou_eq}\n  \\partial_E^h\\bar{\\partial}_E\\dot{h}-\\partial_E^h\\dot{\\eta}-[\\Phi^{\\ast_h},\\dot{\\Phi}+[\\dot{h},\\Phi]]=0.\n\\end{equation}\nRecall that $2\\partial_E^h\\bar{\\partial}_E=F_{A_h}+\\mathrm{i}\\star \\Delta_{A_h}=-[\\Phi,\\Phi^{\\ast_h}]+\\mathrm{i}\\star \\Delta_{A_h}$, the equation can be written as \\[-\\frac{1}{2}[[\\Phi,\\Phi^{\\ast_h}],\\dot{h}]+\\frac{1}{2}\\mathrm{i}\\star \\Delta_{A_h}\\dot{h}-[\\Phi^{\\ast_h},[\\dot{h},\\Phi]]=\\partial_{E}^h\\dot{\\eta}+[\\Phi^{\\ast_h},\\dot{\\Phi}].\\]\nComposing with $-2\\mathrm{i}\\star$ and simplifying, we obtain\n\\[L_{\\Phi}\\dot{h}:=\\Delta_{A_h}\\dot{h}-\\mathrm{i}\\star([\\Phi^{\\ast_h},[\\Phi,\\dot{h}]] -[\\Phi,[\\Phi^{\\ast_h},\\dot{h}]])=-2\\mathrm{i}\\star(\\partial_E^h\\dot{\\eta}+[\\Phi^{\\ast_h},\\dot{\\Phi}]).\\]\nNote that $L_{\\Phi}$ is exactly $L_t$ in \\eqref{LinOp_eq} with $(\\bar{\\partial}_E,\\Phi_t,h_{t}^{\\mathrm{app}})$ replaced by $(\\bar{\\partial}_E,\\Phi,h)$. Now we consider the solvability of \\eqref{LinHitCou_eq}. Let $(\\bar{\\partial}_E,\\Phi_t)$ and $L_{\\delta}^{k,p}$ (depending on $g_t$, $\\Phi_t$, and $h_t^{\\mathrm{app}}$) be defined as before, $\\nabla_t$ be the Chern connection determined by $\\bar{\\partial}_E$ and $h_t$.\n\\begin{lemma}\\label{InvLin_lem}\nFix a pair $[(\\bar{\\partial}_E,\\Phi_t)]\\in\\mathcal{M}$, $t\\geqslant t_1$ so that the harmonic metric $h_t$ exists. Then $L_{\\Phi_t}: L_{-2+\\delta}^{2,2}(\\mathfrak{sl}(E))\\to L_{-2+\\delta}^2(\\mathfrak{sl}(E))$ is an isomorphism.\n\\end{lemma}\n\\begin{proof}\nThat $L_t$ is Fredholm of index zero follows from Lemma \\ref{GlobIso_lem}, and the fact that $L_t$ preserves the decomposition $\\mathfrak{sl}(E)=\\mathfrak{su}(E)\\oplus i\\mathfrak{su}(E)$. Since $\\gamma_t\\in L_{-2+\\delta}^{2,2}$, $L_{\\Phi_t}$ differs from $L_t$ by a compact perturbation, it is also Fredholm of index zero. If $u\\in \\ker L_{\\Phi_t}$, then as before the integration by parts yields $\\nabla_t u=[\\Phi_t,u]=0$. By the proof of \\cite[Prop. 7.4]{fredrickson2022asymptotic}, we have $u=0$. Therefore $L_{\\Phi_t}$ is an isomorphism.\n\\end{proof}\n\\begin{lemma}\\label{InfsmRep_lem}\n  Let $[(\\bar{\\partial}_E,\\Phi)]\\in \\mathcal{M}$ and $(\\dot{\\eta}_1,\\dot{\\Phi}_1)$ be an infinitesimal variation. There exists $(\\dot{\\eta}_2,\\dot{\\Phi}_2)$ infinitesimally gauge equivalent to $(\\dot{\\eta}_1,\\dot{\\Phi}_1)$ such that the followings hold.\n\\begin{enumerate}[label=(\\roman*)]\n  \\item In a neighborhood of $x\\in T$ and a holomorphic frame where \\[\\bar{\\partial}_E=\\bar{\\partial},\\quad \\Phi=\\begin{pmatrix}\n    0&1\\\\\\zeta_x^{-1}&0\n  \\end{pmatrix}\\,\\mathrm{d}\\zeta_x,\\]\n  we have\n  \\[\\dot{\\eta}_2=0,\\quad\\dot{\\Phi}_2=\\begin{pmatrix}\n    0&0\\\\\\frac{\\dot{P}}{\\zeta_x}&0\n  \\end{pmatrix}\\,\\mathrm{d}\\zeta_x,\\quad\\dot{P}\\text{ holomorphic}.\\]\n  \\item In a neighborhood of $x\\in I_u$ and a holomorphic frame where $\\bar{\\partial}_E=\\bar{\\partial}$, $\\Phi=fz_x^{-2m_x} \\sigma_3\\,\\mathrm{d}z_x$ with $f$ being holomorphic and nowhere vanishing, we have $\\dot{\\eta}_2=0, \\dot{\\Phi}_2=\\dot{P}\\sigma_3\\,\\mathrm{d}z$, $\\dot{P}$ is holomorphic.\n  \\item In a neighborhood of $x\\in I_t$ and a holomorphic frame where \\[\\bar{\\partial}_E=\\bar{\\partial},\\quad \\Phi=\\begin{pmatrix}\n    0&\\frac{1}{\\zeta_{x}^{m_x-1}}\\\\ \\frac{1}{\\zeta_{x}^{m_x}}&0\n  \\end{pmatrix}\\,\\mathrm{d}\\zeta_{x},\\] we have \\[\\dot{\\eta}_2=0,\\quad \\dot{\\Phi}_2=\\begin{pmatrix}\n    0&\\dot{P}\\\\\\frac{\\dot{P}}{\\zeta_x}&0\n  \\end{pmatrix}\\,\\mathrm{d}\\zeta_x,\\quad \\dot{P}\\text{ is holomorphic}.\\]\n\\end{enumerate}\n\\end{lemma}\n\\begin{proof}\n  (\\romannumeral1) is just \\cite[Lem.~7.3]{fredrickson2022asymptotic}.\\\\\n  (\\romannumeral2) By the Poincar\u00e9 lemma we may assume that $\\dot{\\eta}_1=0$ in the disk. The leading term of $\\dot{\\Phi}_1$ preserves the flag at $x$, then we can write $\\dot{\\Phi}_1$ as \\[\\dot{\\Phi}_1=\\frac{1}{z_x^{2m_x}}\\begin{pmatrix}\n    \\dot{P}_1&\\dot{P_2}z_x\\\\ \\dot{P}_3&-\\dot{P}_1\n  \\end{pmatrix}.\\]\n  where $\\dot{P}_1,\\dot{P}_2,\\dot{P}_3$ are holomorphic by \\eqref{LinHol_eq}. Then $\\dot{\\Phi}_2=\\dot{\\Phi}_1+[\\Phi,\\dot{\\gamma}]=\\dot{P}_1z_x^{-2m_x}\\sigma_3\\,\\mathrm{d}z$ if we choose \\[\\dot{\\gamma}=\\begin{pmatrix}\n    0&-\\frac{\\dot{P}_2 z_x}{2f}\\\\\\frac{\\dot{P}_3}{2f}&0\n  \\end{pmatrix},\\]\n  which satisfies $\\bar{\\partial}\\dot{\\gamma}=0$ and preserves the flag. Higgs fields in $\\mathcal{M}$ have fixed polar parts, implying that $\\dot{P}_1z_x^{-2m_x}=\\dot{P}$ for some holomorphic $\\dot{P}$.\\\\\n  (\\romannumeral3) As above, assume that $\\dot{\\eta}_1=0$. The leading term of $\\dot{\\Phi}_1$ preserves the flag at $x$ and the leading term of the Higgs field is nilpotent, then we can write $\\dot{\\Phi}_1$ as \\[\\dot{\\Phi}_1=\\frac{1}{\\zeta_x^{2m_x-1}}\\begin{pmatrix}\n    \\dot{P}_1&\\dot{P_2}\\\\ \\frac{\\dot{P}_3}{\\zeta_x}&-\\dot{P}_1\n  \\end{pmatrix}.\\]\n  where $\\dot{P}_1,\\dot{P}_2,\\dot{P}_3$ are holomorphic by \\eqref{LinHol_eq}. We choose \\[\\dot{\\gamma}=\\begin{pmatrix}\n    \\frac{\\dot{P}_2-\\dot{P}_3}{4}&\\zeta_x\\dot{P}_1\\\\0&-\\frac{\\dot{P}_2-\\dot{P}_3}{4}\n  \\end{pmatrix},\\]\n  which satisfies $\\bar{\\partial}\\dot{\\gamma}=0$ and preserves the flag. Then \\[\\dot{\\Phi}_2=\\dot{\\Phi}_1+[\\Phi,\\dot{\\gamma}]=\\begin{pmatrix}\n    0&\\frac{\\dot{P}_2+\\dot{P}_3}{2\\zeta_x^{2m_x-1}}\\\\ \\frac{\\dot{P}_2+\\dot{P}_3}{2\\zeta_x^{2m_x}}&0\n  \\end{pmatrix}\\,\\mathrm{d}\\zeta_x.\\] Higgs fields in $\\mathcal{M}$ have fixed polar parts, implying that $(\\dot{P}_2+\\dot{P}_3)\/(2\\zeta_x^{2m_x-1})=\\dot{P}$ for $\\dot{P}$ holomorphic.\n\\end{proof}\n\nSuppose $[(\\dot{\\eta},\\dot{\\Phi})]\\in T_{[(\\bar{\\partial}_E,\\Phi_t)]}\\mathcal{M}$, by the above lemma we may assume that $\\dot{\\eta}=0$ and $\\dot{\\Phi}$ has the above forms near every $x\\in S$. Then $\\partial_E^{h_t}\\dot{\\eta}+[\\Phi_t^{\\ast_{h_t}},\\dot{\\Phi}]\\in L_{-2+\\delta}^2$, as $\\partial_E^{h_t}\\dot{\\eta}=0$ near $x\\in S$, $[\\Phi_t^{\\ast_{h_t^{\\mathrm{app}}}},\\dot{\\Phi}]\\in L_{-2+\\delta}^2$ for $\\delta$ small and $\\gamma_t\\in L_{-2+\\delta}^{2,2}$. By Lemma \\ref{InvLin_lem}, a unique $\\dot{h}\\in L_{-2+\\delta}^{2,2}$ solving \\eqref{LinHitCou_eq} can be found. Now the hyperk\u00e4hler metric is given by \\cite[(7.5)]{fredrickson2022asymptotic}\n\\begin{equation}\\label{HKahlerMetric_eq}\n\\lVert [(\\dot{\\eta},\\dot{\\Phi})]\\rVert_{g_{L^2}}^2=2\\int_C |\\dot{\\eta}-\\bar{\\partial}_E\\dot{h}|_{h_t}^2+|\\dot{\\Phi}+[\\dot{h},\\Phi_t]|_{h_t}^2=2\\int_C \\langle \\dot{\\eta}-\\bar{\\partial}_E\\dot{h},\\dot{\\eta}\\rangle_{h_t}+\\langle \\dot{\\Phi}+[\\dot{h},\\Phi_t],\\dot{\\Phi}\\rangle_{h_t},\n\\end{equation}\nwhere the second equality follows from \\eqref{LinHitCou_eq} and the integration by parts which is justified as in the proof of Lemma \\ref{GlobIso_lem} and \\cite[p.~43]{fredrickson2022asymptotic}.\n\nFor the pair $[(\\bar{\\partial}_E,\\Phi_t)]\\in\\mathcal{M}$, one can associate metrics $h_t^{\\mathrm{app}}$ and $h_{t,\\infty}$, with infinitesimal deformations denoted by $\\dot{h}_{\\mathrm{app}}$ and $\\dot{h}_{\\infty}$. The metric $g_{\\mathrm{app}}$ on $\\mathcal{M}$ is defined by\n\\begin{equation}\\label{ApproxMetric_eq}\n\\lVert [(\\dot{\\eta},\\dot{\\Phi})]\\rVert_{g_{\\mathrm{app}}}^2=2\\int_C \\langle \\dot{\\eta}-\\bar{\\partial}_E\\dot{h}_{\\mathrm{app}},\\dot{\\eta}\\rangle_{h_t^{\\mathrm{app}}}+\\langle \\dot{\\Phi}+[\\dot{h}_{\\mathrm{app}},\\Phi_t],\\dot{\\Phi}\\rangle_{h_t^{\\mathrm{app}}},\n\\end{equation}\nwhere $\\dot{h}_{\\mathrm{app}}$ satisfies\n\\begin{equation}\\label{AppLinHitCol_eq}\n  \\partial_E^{h_t^{\\mathrm{app}}}\\bar{\\partial}_E\\dot{h}_{\\mathrm{app}}-\\partial_E^{h_t^{\\mathrm{app}}}\\dot{\\eta}-[\\Phi_t^{\\ast_{h_t^{\\mathrm{app}}}},\\dot{\\Phi}+[\\dot{h}_{\\mathrm{app}},\\Phi_t]]=0.\n\\end{equation}\nSimilarly, the metric $g_\\infty$ is\n\\begin{equation}\\label{InftyMetric_eq}\n  \\lVert[(\\dot{\\eta},\\dot{\\Phi})]\\rVert_{g_\\infty}^2=2\\int_C \\langle \\dot{\\eta}-\\bar{\\partial}_E\\dot{h}_{\\mathrm{\\infty}},\\dot{\\eta}\\rangle_{h_{t,\\infty}}+\\langle \\dot{\\Phi}+[\\dot{h}_\\infty,\\Phi_t],\\dot{\\Phi}\\rangle_{h_{t,\\infty}},\n\\end{equation}\nwhere $\\dot{h}_{\\infty}$ satisfies the decoupled equations\n\\begin{equation}\\label{DecoupInftyDef_eq}\n  \\partial_E^{h_{t,\\infty}}\\bar{\\partial}_E\\dot{h}_\\infty-\\partial_{E}^{h_{t,\\infty}}\\dot{\\eta}=0,\\quad [\\Phi_t^{\\ast_{h_{t,\\infty}}},\\dot{\\Phi}+[\\dot{h}_\\infty,\\Phi_t]]=0.\n\\end{equation}\nRecall that for each $[(\\bar{\\partial}_E,\\Phi_t)]\\in \\mathcal{M}$, we can view the associated spectral data as a pair $(\\bar{\\partial}_{L,t},\\tau_t)$, where $L$ is a complex line bundle of degree $-|S|+N-2$ over a surface $S_t$ with genus $N-3$, $\\tau_t$ is the tautological eigenvalue of $\\pi^\\ast \\Phi_t$ which embeds $S_t$ in $\\pi^\\ast K(D)$. The tangent map induces an isomorphism\n\\[T_{[(\\bar{\\partial}_E,\\Phi_t)]}\\mathcal{M}\\to H^0(S_t,K_{S_t})^\\ast_{\\mathrm{odd}}\\oplus H^0(S_t,K_{S_t})_{\\mathrm{odd}},\\quad [(\\dot{\\eta},\\dot{\\Phi})]\\mapsto (\\dot{\\xi},\\dot{\\tau}),\\] where $H^0(S_t,K_{S_t})_{\\mathrm{odd}}$ consists of elements of $H^0(S_t,K_{S_t})$ which are odd under the exchange of two sheets of $\\pi:S_t\\to C$, and $\\dot{\\xi},\\dot{\\tau}$ are the infinitesimal deformations of $\\bar{\\partial}_{L,t}$ and $\\tau$. The metric $h_{t,\\infty}$ is the orthogonal pushforward of the harmonic metric $h_{\\mathcal{L}_t}$ on the spectral line bundle $\\mathcal{L}_t=(L,\\bar{\\partial}_{L,t})\\to S_t$, and the infinitesimal deformation $\\dot{h}_L$ of $h_{\\mathcal{L}_t}$ is determined by\n\\[\\partial_L^{h_{\\mathcal{L}_t}}\\bar{\\partial}_{L,t}\\dot{h}_L-\\partial_L^{h_{\\mathcal{L}_t}}\\dot{\\xi}=0.\\]\n\\begin{lemma}[{\\cite[Prop.~2.14]{fredrickson_2019}}]The semiflat metric is characterized by three properties:\n\\begin{enumerate}[label=(\\arabic*)]\n  \\item On horizontal deformations, the semiflat metric is $\\int_{S_t}2|\\dot{\\tau}|^2$.\n  \\item On vertical deformations, the semiflat metric is $\\int_{S_t} 2|\\dot{\\xi}-\\bar{\\partial}_L\\dot{h}_L|^2$.\n  \\item Horizontal and vertical deformations are orthogonal.\n\\end{enumerate}\n\\end{lemma}\n\\begin{lemma}[{\\cite[Cor.~7.9]{fredrickson2022asymptotic}}]\n  The semiflat metric $g_{\\mathrm{sf}}$ on $\\mathcal{M}$ coincides with $g_{\\infty}$.\n\\end{lemma}\n\\subsection{Comparing $g_{L^2}$ and $g_{\\mathrm{app}}$}\nThe difference $\\frac{1}{2}|\\lVert [(\\dot{\\eta},\\dot{\\Phi})]\\rVert_{g_{\\mathrm{app}}}^2-\\lVert [(\\dot{\\eta},\\dot{\\Phi})]\\rVert_{g_{L^2}}^2|$ is bounded by\n\\[  \\left|\\int_C | \\dot{\\eta}-\\bar{\\partial}_E\\dot{h}_{\\mathrm{app}}|_{h_t^{\\mathrm{app}}}^2-| \\dot{\\eta}-\\bar{\\partial}_E\\dot{h}|_{h_t}^2\\right|+\\left|\\int_C |\\dot{\\Phi}+[\\dot{h}_{\\mathrm{app}},\\Phi_t]|_{h_t^{\\mathrm{app}}}^2-| \\dot{\\Phi}+[\\dot{h},\\Phi_t]|_{h_t}^2\\right|:=\\uppercase\\expandafter{\\romannumeral1}+\\uppercase\\expandafter{\\romannumeral2}.\\]\nWe first estimate $\\uppercase\\expandafter{\\romannumeral1}$, \\begin{align}\n\\uppercase\\expandafter{\\romannumeral1}&\\leqslant \\int_C |\\langle (\\mathrm{Id}-\\mathrm{e}^{-\\gamma_t})(\\dot{\\eta}-\\bar{\\partial}_E\\dot{h}_{\\mathrm{app}}),\\dot{\\eta}-\\bar{\\partial}_E\\dot{h}_{\\mathrm{app}}\\rangle_{h_t^{\\mathrm{app}}}|\\notag\\\\&\\quad+|\\langle \\mathrm{e}^{-\\gamma_t}(\\dot{\\eta}-\\bar{\\partial}_E\\dot{h}_{\\mathrm{app}}),(\\mathrm{Id}-\\mathrm{e}^{-\\gamma_t})(\\dot{\\eta}-\\bar{\\partial}_E\\dot{h}_{\\mathrm{app}})\\rangle_{h_t^{\\mathrm{app}}}|\\notag\\\\\n&\\quad+2|\\langle \\mathrm{e}^{-\\gamma_t}\\bar{\\partial}_E(\\dot{h}_{\\mathrm{app}}-\\dot{h}),\\mathrm{e}^{-\\gamma_t}(\\dot{\\eta}-\\bar{\\partial}_E\\dot{h}_{\\mathrm{app}}\\rangle_{h_t^{\\mathrm{app}}}|+|\\mathrm{e}^{-\\gamma_t}\\bar{\\partial}_E(\\dot{h}_{\\mathrm{app}}-\\dot{h})|_{h_t^{\\mathrm{app}}}^2.\\label{L2AppCompTerm1_eq}\n\\end{align}\nSince the $L_{-2+\\delta}^{2,2}$ norm of $\\gamma_t$ is exponentially decaying, the first two terms are bounded by \\[c\\mathrm{e}^{-c't^\\sigma}\\lVert \\dot{\\eta}-\\bar{\\partial}_E\\dot{h}_{\\mathrm{app}}\\rVert_{L^2}^2.\\]\nHere and following, unless specified, the norms are taken with respect to $h_t^{\\mathrm{app}}$ as in \\eqref{SobolevSpace_eq}. It remains to estimate $\\lVert\\nabla_{A_t}\\dot{\\mathbf{v}}\\rVert_{L^2}$, where $\\dot{\\mathbf{v}}=\\dot{h}_{\\mathrm{app}}-\\dot{h}$. By \\eqref{LinHitCou_eq}, \\eqref{AppLinHitCol_eq}, we have\n\\begin{align*}\n  &\\partial_{E}^{h_t^{\\mathrm{app}}}\\bar{\\partial}_E \\dot{\\mathbf{v}}-[\\Phi_t^{\\ast_{h_t^{\\mathrm{app}}}},[\\dot{\\mathbf{v}},\\Phi_t]]+T_t^1(\\dot{\\eta}-\\bar{\\partial}_E\\dot{h})+T_t^2(\\dot{\\Phi}+[\\dot{h},\\Phi_t])=0,\\text{ where }\\\\\n  &T_t^1=\\partial_E^{h_t}-\\partial_E^{h_t^{\\mathrm{app}}}=-2[\\partial_E^{h_t^{\\mathrm{app}}}\\gamma_t,\\cdot]+R_t^1,\\quad T_t^2=[\\Phi_t^{\\ast_{h_t}}-\\Phi_t^{\\ast_{h_t^{\\mathrm{app}}}},\\cdot]=-2[[\\gamma_t,\\Phi_t^{\\ast_{h_t^{\\mathrm{app}}}}],\\cdot]+R_t^2,\n\\end{align*}\n$R_t^1,R_t^2$ are nonlinear in $\\gamma_t$. As before, by composing with $-2\\mathrm{i}\\star$, the equation becomes\n\\[L_t\\dot{\\mathbf{v}}=2\\mathrm{i}\\star(T_t^1(\\dot{\\eta}-\\bar{\\partial}_E\\dot{h})+T_t^2(\\dot{\\Phi}+[\\dot{h},\\Phi_t])+E_t\\dot{\\mathbf{v}}):=\\dot{\\mathbf{w}},\\]\nwhere $E_t=[F_{h_t^{\\mathrm{app}}}+[\\Phi_t,\\Phi_t^{\\ast_{h_t^{\\mathrm{app}}}}],\\cdot]$. Note that $\\nabla_{A_t}\\gamma_t, [\\Psi_t,\\gamma_t]\\in L_{-2+\\delta}^{1,2}$ and $L_{-2+\\delta}^{1,2}\\hookrightarrow L_{-1+\\delta}^p$ for any fixed $p\\in [2,\\infty)$, $L_{-1+\\delta}^p\\cdot L^2\\subset L_{-2+\\delta}^{2p\/(p+2)}$, similar to the proof of Theorem \\ref{DefSol_thm}, we have\n \\[\\lVert\\star T_t^1(\\dot{\\eta}-\\bar{\\partial}_E\\dot{h})\\rVert_{L_{-2+\\delta}^{p_1}}+\\lVert\\star T_t^2(\\dot{\\Phi}+[\\dot{h},\\Phi_t])\\rVert_{L_{-2+\\delta}^{p_1}}\\leqslant c\\mathrm{e}^{-c't^\\sigma}\\lVert [(\\dot{\\eta},\\dot{\\Phi})]\\rVert_{g_{L^2}},\\]\n where $p_1=2p\/(p+2)<2$. By the construction of $h_t^{\\mathrm{app}}$, $E_t$ is supported on a union of annuli and the $C^0$ norm is exponentially decaying, so that\n \\[\\lVert E_t\\dot{\\mathbf{v}}\\rVert_{L_{-2+\\delta}^{p_1}}\\leqslant c\\mathrm{e}^{-c't^\\sigma}\\lVert \\dot{\\mathbf{v}}\\rVert_{L_{-2+\\delta}^{p_1}}.\\] Therefore,\n\\begin{equation}\\label{p1NormEst_eq1}\n  \\lVert \\dot{\\mathbf{w}}\\rVert_{L_{-2+\\delta}^{p_1}}\\leqslant c\\mathrm{e}^{-c't^\\sigma}(\\lVert [(\\dot{\\eta},\\dot{\\Phi})]\\rVert_{g_{L^2}}+\\lVert \\dot{\\mathbf{v}}\\rVert_{L_{-2+\\delta}^{p_1}}).\n\\end{equation} By \\eqref{intbyParts_eq},\n\\begin{equation}\\label{p1NormEst_eq2}\n  \\lVert \\nabla_{A_t}\\dot{\\mathbf{v}}\\rVert_{L^2}^2+\\lVert [\\Psi_t,\\dot{\\mathbf{v}}]\\rVert_{L^2}^2=\\langle \\dot{\\mathbf{w}},\\dot{\\mathbf{v}}\\rangle_{L^2}\\leqslant \\lVert \\dot{\\mathbf{w}}\\rVert_{L_{-2+\\delta}^{p_1}}\\lVert \\dot{\\mathbf{v}}\\rVert_{L_{-\\delta}^{p_1^\\ast}},\n\\end{equation}\n where $1\/p_1+1\/p_1^\\ast=1$. Let $u$ be a function on the unit disk vanishing on the boundary, then we have the Sobolev inequality $\\lVert u\\rVert_{L^{p_1^\\ast}}\\leqslant c\\lVert \\nabla u\\rVert_{L^2}$. On the other hand, by Lemma \\ref{SobEmbed_lem}, $L_{-\\delta-1}^{1,2}=\\hat{L}_{-\\delta-1}^{1,2}\\hookrightarrow L_{-\\delta}^{p_1^\\ast}$, then \\[\\lVert u\\rVert_{L_{-\\delta}^{p_1^\\ast}}\\leqslant c(\\lVert\\nabla u\\rVert_{L_{-\\delta-1}^2}+\\lVert u\\rVert_{L_{-\\delta-1}^2})\\leqslant c(\\lVert\\nabla u\\rVert_{L_{-\\delta-1}^2}+\\lVert\\nabla u\\rVert_{L_{-\\delta-2}^2})\\leqslant c\\lVert \\nabla u\\rVert_{L^2}.\\] These estimates can be translated to $C$ as in the proof of \\eqref{PoincareIneq_eq}, and we obtain\n\\[  \\lVert \\dot{\\mathbf{v}}\\rVert_{L_{-\\delta}^{p_1^\\ast}}\\leqslant ct^M(\\lVert \\nabla_{A_t}\\dot{\\mathbf{v}}\\rVert_{L^2}+\\lVert [\\Psi_t,\\dot{\\mathbf{v}}]\\rVert_{L^2}),\\]\nfor some constant $M$. Combining with \\eqref{p1NormEst_eq1}, \\eqref{p1NormEst_eq2}, we deduce that \\begin{align*}\n  \\lVert \\nabla_{A_t}\\dot{\\mathbf{v}}\\rVert_{L^2}+\\lVert [\\Psi_t,\\dot{\\mathbf{v}}]\\rVert_{L^2}&\\leqslant ct^M \\lVert\\dot{\\mathbf{w}}\\rVert_{L_{-2+\\delta}^{p_1}}\\leqslant c\\mathrm{e}^{-c't^\\sigma}(\\lVert [(\\dot{\\eta},\\dot{\\Phi})]\\rVert_{g_{L^2}}+\\lVert \\dot{\\mathbf{v}}\\rVert_{L_{-2+\\delta}^{p_1}})\\\\\n  &\\leqslant c\\mathrm{e}^{-c't^\\sigma}(\\lVert [(\\dot{\\eta},\\dot{\\Phi})]\\rVert_{g_{L^2}}+\\lVert\\dot{\\mathbf{v}}\\rVert_{L_{-\\delta}^{p_1^\\ast}})\\\\\n  &\\leqslant c\\mathrm{e}^{-c't^\\sigma}(\\lVert [(\\dot{\\eta},\\dot{\\Phi})]\\rVert_{g_{L^2}}+\\lVert \\nabla_{A_t}\\dot{\\mathbf{v}}\\rVert_{L^2}+\\lVert [\\Psi_t,\\dot{\\mathbf{v}}]\\rVert_{L^2}),\\\\\n   \\lVert \\nabla_{A_t}\\dot{\\mathbf{v}}\\rVert_{L^2}+\\lVert [\\Psi_t,\\dot{\\mathbf{v}}]\\rVert_{L^2}&\\leqslant c\\mathrm{e}^{-c't^\\sigma}\\lVert [(\\dot{\\eta},\\dot{\\Phi})]\\rVert_{g_{L^2}},\n\\end{align*}\nfor $t$ large. By \\eqref{L2AppCompTerm1_eq}, we conclude that\n\\[\\uppercase\\expandafter{\\romannumeral1}\\leqslant c\\mathrm{e}^{-c't^\\sigma}(\\lVert \\dot{\\eta}-\\bar{\\partial}_E\\dot{h}_{\\mathrm{app}}\\rVert_{L^2}^2+\\lVert [(\\dot{\\eta},\\dot{\\Phi})]\\rVert_{g_{L^2}}^2).\\]\nSimilarly, with the roll of $\\lVert \\nabla_{A_t}\\dot{\\mathbf{v}}\\rVert_{L^2}$ replaced by $\\lVert [\\Psi_t,\\dot{\\mathbf{v}}]\\rVert_{L^2}$, we have\n\\[\\uppercase\\expandafter{\\romannumeral2}\\leqslant c\\mathrm{e}^{-c't^\\sigma}(\\lVert \\dot{\\Phi}+[\\dot{h}_{\\mathrm{app}},\\Phi_t]\\rVert_{L^2}^2+\\lVert [(\\dot{\\eta},\\dot{\\Phi})]\\rVert_{g_{L^2}}^2).\\]\nTherefore $|\\lVert [(\\dot{\\eta},\\dot{\\Phi})]\\rVert_{g_{\\mathrm{app}}}^2-\\lVert [(\\dot{\\eta},\\dot{\\Phi})]\\rVert_{g_{L^2}}^2|\\leqslant c\\mathrm{e}^{-c't^\\sigma}\\lVert [(\\dot{\\eta},\\dot{\\Phi})]\\rVert_{g_{L^2}}^2$, meaning that along the curve $[(\\bar{\\partial}_E,\\Phi_t)]$ in $\\mathcal{M}$, $g_{L^2}$ and $g_{\\mathrm{app}}$ are exponentially close in $t$.\n\n\\subsection{Comparing $g_{\\mathrm{app}}$ and $g_{\\mathrm{sf}}$}\nBy the construction, $h_t^{\\mathrm{app}}$ and $h_{t,\\infty}$ coincide on the complement of the disks $\\widetilde{B}_{x,j,t}$ ($x\\in I$) and $\\widetilde{B}_{x,t}$ ($x\\in T$), and so do the infinitesimal deformations $\\dot{h}_{\\mathrm{app}}$ and $\\dot{h}_\\infty$. Accordingly, the difference $g_{\\mathrm{app}}-g_{\\mathrm{sf}}$ localizes on these disks.\n\n  On $\\widetilde{B}_{x,j,t}$, in a local holomorphic coordinate $\\zeta_{x,j,t}$ and a holomorphic frame we have \\[\\bar{\\partial}_E=\\bar{\\partial},\\quad \\Phi_t=\\lambda_{x,j}(t)\\begin{pmatrix}\n    0&1\\\\\\zeta_{x,j,t}&0\n  \\end{pmatrix}\\,\\mathrm{d}\\zeta_{x,j,t},\\quad h_{t,\\infty}=\\begin{pmatrix}\n    |\\zeta_{x,j,t}|^{1\/2}&0\\\\0& |\\zeta_{x,j,t}|^{-1\/2}\n  \\end{pmatrix},\\]\nand $h_t^{\\mathrm{app}}$ is given by \\eqref{approxMetricZ_eq}. Let $\\hat{\\zeta}_{x,j,t}=\\kappa^{-1}t^{1\/j(x)}\\zeta_{x,j,t}$ and $\\hat{r}_{x,j,t}=|\\hat{\\zeta}_{x,j,t}|$, then the disk $\\widetilde{B}_{x,j,t}=\\{\\,\\hat{r}_{x,j,t}<1\\,\\}$. Applying the local holomorphic gauge transformation\n\\[  g=\\begin{pmatrix}\n    \\kappa^{-1\/4}t^{1\/4j(x)}&0\\\\0&\\kappa^{1\/4}t^{-1\/4j(x)}\n  \\end{pmatrix},\\text{ then }\n  g^{-1}\\Phi_tg =t_{x,j}\\begin{pmatrix}\n    0&1\\\\\\hat{\\zeta}_{x,j,t}&0\n\\end{pmatrix}\\,\\mathrm{d}\\hat{\\zeta}_{x,j,t},\\]\nwhere $t_{x,j}=\\lambda_{x,j}(t)\\kappa^{3\/2}t^{-3\/2j(x)}$.\n\\[ g^\\ast h_{t,\\infty}g=\\begin{pmatrix}\n  \\hat{r}_{x,j,t}^{1\/2}&0\\\\0&\\hat{r}_{x,j,t}^{-1\/2}\n\\end{pmatrix},\\quad g^\\ast h_t^{\\mathrm{app}}g=\\begin{pmatrix}\n  \\hat{r}_{x,j,t}^{1\/2}\\mathrm{e}^{l_{t_{x,j}(\\hat{r}_{x,j,t})}\\chi(\\hat{r}_{x,j,t})}&0\\\\0&\\hat{r}_{x,j,t}^{-1\/2}\\mathrm{e}^{-l_{t_{x,j}}(\\hat{r}_{x,j,t})\\chi(\\hat{r}_{x,j,t})}\n\\end{pmatrix},\\]\nwhere $l_{t_{x,j}}(\\hat{r}_{x,j,t})=\\psi_1(8t_{x,j}\\hat{r}_{x,j,t}^{3\/2}\/3)$. In this local holomorphic frame, the following result holds.\n\\begin{proposition}[{\\cite[Th.~3.1]{fredrickson_2019}}]\nLet $[(\\dot{\\eta},\\dot{\\Phi})]\\in T_{[(\\bar{\\partial}_E,\\Phi_t)]}\\mathcal{M}$, then on the disk $\\{\\,\\hat{r}_{x,j,t}<1\\,\\}$ there is a unique representative in the equivalence class where\n\\[\\dot{\\eta}=0,\\quad \\dot{\\Phi}=t_{x,j}\\begin{pmatrix}\n  0&0\\\\\\dot{P}&0\n\\end{pmatrix}\\,\\mathrm{d}\\hat{\\zeta}_{x,j,t},\\quad \\dot{h}_{\\infty}=\\frac{\\dot{P}}{4\\hat{\\zeta}_{x,j,t}}\\begin{pmatrix}\n  1&0\\\\0&-1\n\\end{pmatrix}\\quad\\text{for }\\bar{\\partial}\\dot{P}=0.\\]\nThese infinitesimal deformations satisfy \\eqref{DecoupInftyDef_eq}, and $\\dot{h}_{\\mathrm{app}}$ solving \\eqref{AppLinHitCol_eq} is diagonal. For any $[(\\dot{\\eta},\\dot{\\Phi})]\\in T_{[(\\bar{\\partial}_E,\\Phi_t)]}\\mathcal{M}$, there exists a constant $c>0$ such that \\[\\lVert [(\\dot{\\eta},\\dot{\\Phi})]\\rVert_{g_{\\mathrm{app}}(\\widetilde{B}_{x,j,t})}^2-\\lVert [(\\dot{\\eta},\\dot{\\Phi})]\\rVert_{g_{\\infty}(\\widetilde{B}_{x,j,t})}^2=O(\\mathrm{e}^{-ct_{x,j}}).\\]\n\\end{proposition}\nOn $\\widetilde{B}_{x,t}$ ($x\\in T$), in a local holomorphic coordinate $\\zeta_{x,t}$ and holomorphic frame we have\n\\[\\bar{\\partial}_E=\\bar{\\partial},\\quad \\Phi_t=\\lambda_{x}(t)\\begin{pmatrix}\n  0&1\\\\\\frac{1}{\\zeta_{x,t}}&0\n\\end{pmatrix}\\,\\mathrm{d}\\zeta_{x,t},\\quad h_{t,\\infty}=\\begin{pmatrix}\n  |\\zeta_{x,t}|^{1\/2}&0\\\\0& |\\zeta_{x,t}|^{3\/2}\n\\end{pmatrix},\\]\nand $h_t^{\\mathrm{app}}$ is given by \\eqref{approxMetricT_eq}. Let $\\hat{\\zeta}_{x,t}=\\kappa^{-1}\\zeta_{x,t}$ and $\\hat{r}_{x,t}=|\\hat{\\zeta}_{x,t}|$, then the disk $\\widetilde{B}_{x,t}=\\{\\,\\hat{r}_{x,t}<1\\,\\}$. Applying the local holomorphic gauge transformation\n\\[  g=\\kappa^{-1\/2}\\begin{pmatrix}\n    \\kappa^{1\/4}&0\\\\0&\\kappa^{-1\/4}\n  \\end{pmatrix},\\text{ then }\n  g^{-1}\\Phi_tg =t_x\\begin{pmatrix}\n    0&1\\\\\\hat{\\zeta}_{x,t}&0\n\\end{pmatrix}\\,\\mathrm{d}\\hat{\\zeta}_{x,t},\\]\nwhere $t_x=\\lambda_x(t)\\kappa^{1\/2}$.\n\\[ g^\\ast h_{t,\\infty}g=\\begin{pmatrix}\n  \\hat{r}_{x,t}^{1\/2}&0\\\\0&\\hat{r}_{x,t}^{3\/2}\n\\end{pmatrix},\\quad g^\\ast h_t^{\\mathrm{app}}g=\\begin{pmatrix}\n  \\hat{r}_{x,t}^{1\/2}\\mathrm{e}^{m_{t_{x}(\\hat{r}_{x,t})}\\chi(\\hat{r}_{x,t})}&0\\\\0&\\hat{r}_{x,t}^{3\/2}\\mathrm{e}^{-m_{t_{x}}(\\hat{r}_{x,t})\\chi(\\hat{r}_{x,t})}\n\\end{pmatrix},\\]\nwhere $m_{t_x}(\\hat{r}_{x,t})=\\psi_{2,x}(8t_x\\hat{r}_{x,t}^{1\/2})$. In this local holomorphic frame, the following result holds.\n\\begin{proposition}[{\\cite[Lem.~7.10,~Prop.~7.11]{fredrickson2022asymptotic}}]\nLet $[(\\dot{\\eta},\\dot{\\Phi})]\\in T_{[(\\bar{\\partial}_E,\\Phi_t)]}\\mathcal{M}$, then on the disk $\\{\\,\\hat{r}_{x,t}<1\\,\\}$ there is a unique representative in the equivalence class where\n\\[\\dot{\\eta}=0,\\quad \\dot{\\Phi}=t_x\\begin{pmatrix}\n  0&0\\\\\\frac{\\dot{P}}{\\hat{\\zeta}_{x,t}}&0\n\\end{pmatrix}\\,\\mathrm{d}\\hat{\\zeta}_{x,t},\\quad \\dot{h}_{\\infty}=\\frac{\\dot{P}}{4}\\begin{pmatrix}\n  1&0\\\\0&-1\n\\end{pmatrix}\\quad\\text{for }\\bar{\\partial}\\dot{P}=0.\\]\nThese infinitesimal deformations satisfy \\eqref{DecoupInftyDef_eq}. For any $[(\\dot{\\eta},\\dot{\\Phi})]\\in T_{[(\\bar{\\partial}_E,\\Phi_t)]}\\mathcal{M}$, there exists a constant $c>0$ such that \\[\\lVert [(\\dot{\\eta},\\dot{\\Phi})]\\rVert_{g_{\\mathrm{app}}(\\widetilde{B}_{x,t})}^2-\\lVert [(\\dot{\\eta},\\dot{\\Phi})]\\rVert_{g_{\\infty}(\\widetilde{B}_{x,t})}^2=O(\\mathrm{e}^{-ct_{x}}).\\]\n\\end{proposition}\nThe above two propositions immediately imply $g_{\\mathrm{app}}-g_{\\mathrm{sf}}=O(\\mathrm{e}^{-ct^{\\sigma}})$ along the curve $[(\\bar{\\partial}_E,\\Phi_t)]$ in $\\mathcal{M}$. Hence Theorem \\ref{Main_thm} is established.\n\\begin{remark}\n $g_{\\mathrm{L^2}}-g_{\\mathrm{app}}$ and $g_{\\mathrm{app}}-g_{\\mathrm{sf}}$ are still exponentially decaying when $I_u=\\varnothing$, only slight modifications of the previous proofs are needed.\n\\end{remark}\n\n\\subsection{Comparing $g_{\\mathrm{sf}}$ and $g_{\\mathrm{model}}$}\nIn this subsection we assume that $\\mathrm{dim}_{\\mathbb{C}}\\,\\mathcal{M}=2$, or equivalently $N=\\mathrm{deg}\\,D=4$. The Hitchin base now has complex dimension one, parametrized by $t\\in \\mathbb{C}$. The constants in this subsection will not depend on $\\mathrm{Arg}(t)$ and the choice of the Higgs bundle in the Hitchin fiber $H^{-1}(t)$.\n\\subsubsection{An untwisted order four pole}\\label{U4_subsubsec}\nThe Hitchin base is given by\n\\begin{equation}\\label{HitBaseU4_eq}\n  \\mathcal{B}=\\left\\{\\,\\left(\\sum_{k=5}^8\\frac{\\mu_k}{z^k}+\\frac{t}{z^4}\\right)\\,\\mathrm{d}z^2\\,\\middle|\\,t\\in \\mathbb{C}\\,\\right\\},\n\\end{equation}\nwhere $\\mu_5,\\ldots,\\mu_8$ are constants determined by the irregular type at $0$, and $\\mu_8\\neq 0$. We may assume that $\\mu_8=-1$, using a rescaling of $z$ if necessary.\nFix \\[\\nu=\\nu(z)\\,\\mathrm{d}z^2=\\left(\\sum_{k=5}^8 \\frac{\\mu_k}{z^k}+\\frac{t}{z^4}\\right)\\,\\mathrm{d}z^2\\in\\mathcal{B}',~\\dot{\\nu}=\\frac{\\dot{t}}{z^4}\\,\\mathrm{d}z^2\\in T_\\nu\\mathcal{B}'.\\]\nThen the special K\u00e4hler metric $g_{\\mathrm{sK}}$, which is the restriction of $g_{\\mathrm{sf}}$ on $\\mathcal{B}'$, is given by \\cite[Prop.~8.3]{fredrickson2022asymptotic} \\begin{align*}\n  g_{\\mathrm{sK}}(\\dot{\\nu},\\dot{\\nu})&=\\int_{C}\\frac{|\\dot{\\nu}|^2}{|\\nu|}\\,\\mathrm{dvol}_C=\\frac{|\\dot{t}|^2}{|t|}\\int_{\\mathbb{C}}\\frac{1}{\\left|\\prod_{k=1}^4(z-z_k(t))\\right|}~\\mathrm{i}\\,\\mathrm{d}z\\mathrm{d}\\bar{z},\n\\end{align*}\nwhere $z_k(t)=t^{-1\/4}\\mathrm{e}^{\\mathrm{i}(k-1)\\pi\/2}+t^{-1\/2}(-1)^k \\mu_7 \/ 4 + t^{-3\/4} \\mathrm{e}^{-\\mathrm{i}(k-1)\\pi\/2} (-\\mu_6\/4 - \\mu_7^2\/32)+O(|t|^{-1})$ ($k=1,\\ldots,4$) are the four roots of $z^8\\nu(z):=\\tilde{\\nu}(z)$.  Let $z=t^{-1\/4}\\xi$ and $z_k=t^{-1\/4}\\xi_k$, then the integral becomes\n\\[\n|t|^{1\/2}\\int_{\\mathbb{C}}\\frac{1}{\\left|\\prod_{k=1}^4(\\xi-\\xi_k)\\right|}~\\mathrm{i}\\,\\mathrm{d}\\xi \\mathrm{d}\\bar\\xi.\n\\]\nWe define a funtion \\[F(\\xi_1,...\\xi_4,\\bar\\xi_1,...\\bar\\xi_4) := \\int_{\\mathbb{C}}\\frac{1}{\\left|\\prod_{k=1}^4(\\xi-\\xi_k)\\right|}~\\mathrm{i}\\,\\mathrm{d} \\xi \\mathrm{d} \\bar\\xi.\\]\nThen we can prove that\n\\[\\frac{\\partial F}{\\partial \\xi_k}= \\lim_{\\epsilon \\to 0}\\int_{|\\xi-\\xi_k|>\\epsilon}\\frac{\\bar\\xi-\\bar\\xi_k}{2\\left|\\xi-\\xi_k\\right|^3} \\frac{1}{\\left|\\prod_{l\\not =k}(\\xi-\\xi_l)\\right|}~\\mathrm{i}\\,\\mathrm{d} \\xi \\mathrm{d} \\bar{\\xi},\\,\\quad\\frac{\\partial F}{\\partial \\bar\\xi_k}=\\overline{\\frac{\\partial F}{\\partial \\xi_k}}.\\]\nThen, for $l\\not= k$,\n\\begin{align*}\n\\frac{\\partial^2 F}{\\partial \\xi_k\\partial \\xi_l}= &\\lim_{\\epsilon\\to 0}\\int_{|\\xi-\\xi_k|>\\epsilon, |\\xi-\\xi_l|>\\epsilon}\\frac{\\bar\\xi-\\bar\\xi_k}{2\\left|\\xi-\\xi_k\\right|^3}\\frac{\\bar\\xi-\\bar\\xi_l}{2\\left|\\xi-\\xi_l\\right|^3} \\frac{1}{\\left|\\prod_{m\\not =k,l}(\\xi-\\xi_m)\\right|}~\\mathrm{i}\\,\\mathrm{d} \\xi \\mathrm{d} \\bar{\\xi},\\\\\n\\frac{\\partial^2 F}{\\partial \\xi_k\\partial \\bar\\xi_l}= &\\lim_{\\epsilon\\to 0}\\int_{|\\xi-\\xi_k|>\\epsilon, |\\xi-\\xi_l|>\\epsilon}\\frac{\\bar\\xi-\\bar\\xi_k}{2\\left|\\xi-\\xi_k\\right|^3}\\frac{\\xi-\\xi_l}{2\\left|\\xi-\\xi_l\\right|^3} \\frac{1}{\\left|\\prod_{m\\not =k,l}(\\xi-\\xi_m)\\right|}~\\mathrm{i}\\,\\mathrm{d} \\xi \\mathrm{d} \\bar{\\xi},\\\\\n\\frac{\\partial^2 F}{\\partial \\bar\\xi_k\\partial \\bar\\xi_l}= &\\lim_{\\epsilon\\to 0}\\int_{|\\xi-\\xi_k|>\\epsilon, |\\xi-\\xi_l|>\\epsilon}\\frac{\\xi-\\xi_k}{2\\left|\\xi-\\xi_k\\right|^3}\\frac{\\xi-\\xi_l}{2\\left|\\xi-\\xi_l\\right|^3} \\frac{1}{\\left|\\prod_{m\\not =k,l}(\\xi-\\xi_m)\\right|}~\\mathrm{i}\\,\\mathrm{d} \\xi \\mathrm{d} \\bar{\\xi}.\n\\end{align*}\nFor any $k=1,2,3,4$,\n\\[\n\\frac{\\partial^2 F}{\\partial \\xi_k\\partial \\xi_k}=-\\sum_{l\\not=k}\\frac{\\partial^2 F}{\\partial \\xi_k\\partial \\xi_l},\\,\n\\frac{\\partial^2 F}{\\partial \\xi_k\\partial \\bar\\xi_k}=-\\sum_{l\\not=k}\\frac{\\partial^2 F}{\\partial \\xi_k\\partial \\bar\\xi_l},\\,\n\\frac{\\partial^2 F}{\\partial \\bar\\xi_k\\partial \\bar\\xi_k}=-\\sum_{l\\not=k}\\frac{\\partial^2 F}{\\partial \\bar\\xi_k\\partial \\bar\\xi_l}\\]\nbecause we can shift $\\xi$ by any variation of $\\xi_k$ in $\\frac{\\partial F}{\\partial \\xi_k}$ or $\\frac{\\partial F}{\\partial \\bar{\\xi}_k}$.\n\nNow we use these to calculate\n\\begingroup\n\\allowdisplaybreaks\n\\begin{align*}\n  &g_{\\mathrm{sK}}(\\dot{\\nu},\\dot{\\nu}) =|t|^{1\/2}\\int_{\\mathbb{C}}\\frac{1}{\\left|\\prod_{k=1}^4(\\xi-\\xi_k)\\right|}~\\mathrm{i}\\,\\mathrm{d}\\xi \\mathrm{d}\\bar{\\xi} = |t|^{1\/2}F(\\xi_k, \\bar\\xi_k)\\\\\n =&|t|^{1\/2}\\left( F(\\mathrm{e}^{\\mathrm{i}(k-1)\\pi\/2}, \\mathrm{e}^{-\\mathrm{i}(k-1)\\pi\/2}) + \\sum_{k=1}^{4}2\\mathrm{Re}\\left(\\frac{\\partial F}{\\partial \\xi_k}\\frac{(-1)^k\\mu_7t^{-1\/4}}{4}\\right)\\right.\\\\\n &+\\sum_{k=1}^{4}2\\mathrm{Re}\\left(\\frac{\\partial F}{\\partial \\xi_k}(t^{-1\/2} \\mathrm{e}^{-\\mathrm{i}(k-1)\\pi\/2})(-\\mu_6\/4 - \\mu_7^2\/32)\\right)\\\\\n&+\\mathrm{Re}\\left(\\sum_{k=1}^{4}\\sum_{l=1}^{4}\\frac{\\partial^2 F}{\\partial \\xi_k\\partial \\xi_l}(-1)^{k+l}\\frac{\\mu_7^2 t^{-1\/2}}{16}\\right)\\\\& +\\left. \\mathrm{Re}\\left(\\sum_{k=1}^{4}\\sum_{l=1}^{4}\\frac{\\partial^2 F}{\\partial \\xi_k\\partial \\bar\\xi_l}\\right)(-1)^{k+l}\\frac{|\\mu_7|^2 |t|^{-1\/2}}{16}+O(|t|^{-3\/4})\\right).\n\\end{align*}\n\\endgroup\nThe terms involving $\\frac{\\partial F}{\\partial \\xi_k}$ cancel with each other. So in polar coordinates, as $|t|=r\\to\\infty$, the special K\u00e4hler metric is \\[(C_0+\\mathrm{Re} (C_1 \\mu_7^2 t^{-1\/2}) + C_2 |\\mu_7|^2 |t|^{-1\/2} + O(r^{-3\/4})))\\frac{\\mathrm{d}r^2+r^2\\,\\mathrm{d}\\theta^2}{\\sqrt{r}},\\]\nwhich is asymptotic to a conic metric with cone angle $3\\pi\/2$.\n\nNext we consider $g_{\\mathrm{sf}}$ along the fiber direction. By the discussion in Section \\ref{Prelim_sec}, the Hitchin fiber $H^{-1}(t)$ can be written as\n\\begin{align}\n  H^{-1}(t)&=\\left\\{\\,\\left[\\left(\\bar{\\partial}_{E}^0,\\Phi\\right)\\right]\\,\\middle|\\,\\Phi=\\frac{\\mathrm{d}z}{z^4}\\begin{pmatrix}\n    \\pm (-t)^{1\/2}z^2&1-\\mu_7z-\\mu_6z^2-\\mu_5z^3\\\\1&\\mp(-t)^{1\/2}z^2\n\\end{pmatrix}\\,\\right\\}\\notag\\\\\n&\\cup\\left\\{\\,\\left[\\left(\\bar{\\partial}_{E}^0,\\Phi\\right)\\right]\\,\\middle|\\,\\Phi=\\frac{\\mathrm{d}z}{z^4}\\begin{pmatrix}\n    a_0&\\begin{smallmatrix}\n      -tz^3+(-\\mu_5+tc_0)z^2+(-\\mu_6+\\mu_5c_0-tc_0^2)z - {} \\\\\n      \\mu_7+\\mu_6c_0-\\mu_5c_0^2+tc_0^3\n    \\end{smallmatrix}\\\\ c_0+z&-a_0\n  \\end{pmatrix},a_0^2=-\\tilde{\\nu}(-c_0)\\right\\},\\label{HitFibU4_eq}\n\\end{align}\nwhere $(E,\\bar{\\partial}_E^0)\\cong \\mathcal{O}\\oplus\\mathcal{O}(-1)$. Therefore $H^{-1}(t)$ is the closure of the second set, which is the elliptic curve \\[a_0^2=-\\tilde{\\nu}(-c_0)=-t\\prod_{k=1}^4(-c_0-z_k(t)),~(a_0,c_0)\\in \\mathbb{C}^2,\\]\nwith modulus $\\tau(t)=\\lambda^{-1}(l(t))$. Here $\\lambda$ is the modular lambda function, and\n\\begin{align*}\n  l(t)&=\\frac{(z_4(t)-z_1(t))(z_3(t)-z_2(t))}{(z_3(t)-z_1(t))(z_4(t)-z_2(t))}=\\frac{1}{2}+\\frac{3\\mathrm{i}\\mu_7^2}{32}t^{-1\/2}+O(|t|^{-3\/4}), \\\\\n  \\tau(t)&=\\mathrm{i}+\\frac{3\\pi \\mu_7^2}{32K(\\sqrt{1\/2})^2}t^{-1\/2}+O(|t|^{-3\/4}),\n\\end{align*}\nwhere $K(x)=\\int_0^{\\pi\/2} (1-x^2\\sin^2\\theta)^{-1\/2}\\,\\mathrm{d}\\theta$ is the complete elliptic integral of the first kind. The semiflat metric induces a constant multiple of the Euclidean metric on $H^{-1}(t)$:\n\\begin{proposition}[{\\cite[Proposition 8.4]{fredrickson2022asymptotic}}]\n  The area of $H^{-1}(t)$ in the semiflat metric is $4\\pi^2$, so $H^{-1}(t)\\cong \\mathbb{C}\\,\/\\,c_t(\\mathbb{Z}\\oplus\\tau(t)\\mathbb{Z})$ with Euclidean metric $\\mathrm{d}x^2+\\mathrm{d}y^2$ and $c_t=2\\pi\/\\sqrt{\\mathrm{Im}\\,\\tau(t)}$.\n\\end{proposition}\nLet $T_{\\hat{\\tau}}^2=\\mathbb{C}\\,\/\\,\\hat{c}(\\mathbb{Z}\\oplus \\hat{\\tau}\\mathbb{Z})$, where $\\hat{\\tau}=i, \\hat{c}=2\\pi\/\\sqrt{\\mathrm{Im}\\,\\hat{\\tau}}$. Define $\\mu_t: T_{\\hat{\\tau}}^2\\to H^{-1}(t)$ as \\[x+\\mathrm{i}y\\mapsto \\frac{c_t}{\\hat{c}}\\left(x+\\frac{\\mathrm{Re}\\,(\\tau(t)-\\hat{\\tau})}{\\mathrm{Im}\\,\\hat{\\tau}}y+\\mathrm{i}\\frac{\\mathrm{Im}\\,\\tau(t)}{\\mathrm{Im}\\,\\hat{\\tau}}y\\right)=\\frac{c_t}{\\hat{c}}(x+y\\mathrm{Re}\\,\\tau(t)+\\mathrm{i}y \\mathrm{Im}\\,\\tau(t)).\\]\nThen we have\n\\begin{align*}\n\\mu_t^\\ast (g_{\\mathrm{sf}}|_{H^{-1}(t)})&=\\frac{c_t^2}{\\hat{c}^2}\\left((\\mathrm{d}x+\\mathrm{Re}\\,\\tau(t)\\,\\mathrm{d}y)^2+(\\mathrm{Im}\\,\\tau(t))^2\\,\\mathrm{d}y^2\\right)\n\\\\&=\\mathrm{d}x^2+\\mathrm{d}y^2+\\mathrm{Im}\\left(\\frac{3\\pi \\mu_7^2}{32K(\\sqrt{1\/2})^2}t^{-1\/2}\\right)(-\\mathrm{d}x^2+\\mathrm{d}y^2)\\\\\n&+2\\mathrm{Re}\\left(\\frac{3\\pi \\mu_7^2}{32K(\\sqrt{1\/2})^2}t^{-1\/2}\\right)\\,\\mathrm{d}x\\otimes \\mathrm{d}y+O(|t|^{-3\/4}).\n\\end{align*}\n\n\\subsubsection{A twisted order four pole}\nIn this case,\n\\[\\mathcal{B}=\\left\\{\\,\\left(\\sum_{k=5}^7\\frac{\\mu_k}{z^k}+\\frac{t}{z^4}\\right)\\,\\mathrm{d}z^2\\,\\middle|\\,t\\in\\mathbb{C}\\,\\right\\},\\]\nwhere $\\mu_5,\\mu_6,\\mu_7$ are fixed and $\\mu_7\\neq 0$, we may assume that $\\mu_7=-1$. Fix \\[\\nu=\\left(\\sum_{k=5}^7 \\frac{\\mu_k}{z^k}+\\frac{t}{z^4}\\right)\\,\\mathrm{d}z^2\\in\\mathcal{B}',~\\dot{\\nu}=\\frac{\\dot{t}}{z^4}\\,\\mathrm{d}z^2\\in T_\\nu\\mathcal{B}'.\\]\nThen $g_{\\mathrm{sK}}$ on $\\mathcal{B}'$ is \\begin{align*}\n  g_{\\mathrm{sK}}(\\dot{\\nu},\\dot{\\nu})&=\\int_{C}\\frac{|\\dot{\\nu}|^2}{|\\nu|}\\,\\mathrm{dvol}_C=\\frac{|\\dot{t}|^2}{|t|}\\int_{\\mathbb{C}}\\frac{1}{\\left|\\prod_{k=1}^3 z(z-z_k(t))\\right|}~\\mathrm{i}\\,\\mathrm{d}z\\mathrm{d}\\bar{z},\n\\end{align*}\nwhere $z_k(t)=t^{-1\/3}\\mathrm{e}^{2(k-1)\\pi \\mathrm{i}\/3}-\\mu_6\\mathrm{e}^{-2(k-1)\\pi \\mathrm{i}\/3}t^{-2\/3}\/3-\\mu_5t^{-1}\/3+O(|t|^{-4\/3})$ ($k=1,2,3$) are the three roots of $z^7\\nu(z):=\\tilde{\\nu}(z)$. Let $z=t^{-1\/3}\\xi$, the integral becomes\n\\[\n  |t|^{2\/3}\\int_{\\mathbb{C}}\\frac{1}{|\\xi\\prod_{k=1}^3 (\\xi-t^{1\/3}z_k(t))|}~\\mathrm{i}\\,\\mathrm{d} \\xi \\mathrm{d}\\bar\\xi.\n\\]\nAs before, the linear terms cancel with each other.\nTherefore in polar coordinates, as $|t|=r\\to\\infty$, the special K\u00e4hler metric is \\[(C_0+\\mathrm{Re}(C_1 \\mu_5 t^{-2\/3})+\\mathrm{Re}(C_2 \\mu_6^2t^{-2\/3})+C_3|\\mu_6|^2|t|^{-2\/3}+O(r^{-1}))\\frac{\\mathrm{d}r^2+r^2\\,\\mathrm{d}\\theta^2}{r^{1\/3}},\\]\nwhich is asymptotic to a conic metric with cone angle $5\\pi\/3$.\n\nThe fiber $H^{-1}(t)$ can be written as\n\\begin{align*}\n  H^{-1}(t)&=\\left\\{\\,\\left[\\left(\\bar{\\partial}_{E}^0,\\Phi\\right)\\right]\\,\\middle|\\,\\Phi=\\frac{\\mathrm{d}z}{z^4}\\begin{pmatrix}\n    \\pm (-t)^{1\/2}z^2&z-\\mu_6z^2-\\mu_5z^3\\\\1&\\mp(-t)^{1\/2}z^2\n\\end{pmatrix}\\,\\right\\}\\\\\n&\\cup\\left\\{\\,\\left[\\left(\\bar{\\partial}_{E}^0,\\Phi\\right)\\right]\\,\\middle|\\,\\Phi=\\frac{\\mathrm{d}z}{z^4}\\begin{pmatrix}\n    a_0&\\begin{smallmatrix}\n      -tz^3+(-\\mu_5+tc_0)z^2+(-\\mu_6+\\mu_5c_0-tc_0^2)z + {} \\\\\n      1+\\mu_6c_0-\\mu_5c_0^2+tc_0^3\n    \\end{smallmatrix}\\\\ c_0+z&-a_0\n  \\end{pmatrix},a_0^2=c_0\\tilde{\\nu}(-c_0)\\right\\},\n\\end{align*}\nwhere $(E,\\bar{\\partial}_E^0)\\cong \\mathcal{O}\\oplus\\mathcal{O}(-1)$. Therefore $H^{-1}(t)$ is the closure of the second set, which is the elliptic curve \\[a_0^2=c_0\\tilde{\\nu}(-c_0)=tc_0\\prod_{k=1}^3(-c_0-z_k(t)),~(a_0,c_0)\\in \\mathbb{C}^2,\\]\nwith modulus $\\tau(t)=\\lambda^{-1}(l(t))$, where\n\\begin{align*}\n  l(t)&=\\frac{z_2(t)(z_3(t)-z_1(t))}{z_3(t)(z_2(t)-z_1(t))}=\\mathrm{e}^{-\\pi \\mathrm{i}\/3}+\\frac{\\mathrm{e}^{\\pi \\mathrm{i}\/6}}{3\\sqrt{3}}(3\\mu_5+\\mu_6^2)t^{-2\/3}+O(|t|^{-4\/3}), \\\\\n  \\tau(t)&=\\mathrm{e}^{2\\pi \\mathrm{i}\/3}+\\frac{\\mathrm{e}^{-\\pi \\mathrm{i}\/3}\\pi (3\\mu_5+\\mu_6^2)}{12\\sqrt{3}K(\\mathrm{e}^{-\\pi \\mathrm{i}\/6})^2}t^{-2\/3}+O(|t|^{-4\/3}).\n\\end{align*}\nNow $H^{-1}(t)\\cong \\mathbb{C}\\,\/\\,c_t(\\mathbb{Z}\\oplus\\tau(t)\\mathbb{Z})$. Let $T_{\\hat{\\tau}}^2=\\mathbb{C}\\,\/\\,\\hat{c}(\\mathbb{Z}\\oplus \\hat{\\tau}\\mathbb{Z})$, where $\\hat{\\tau}=\\mathrm{e}^{2\\pi \\mathrm{i}\/3}, \\hat{c}=2\\pi\/\\sqrt{\\mathrm{Im}\\,\\hat{\\tau}}$. Define $\\mu_t: T_{\\hat{\\tau}}^2\\to H^{-1}(t)$ as \\begin{align*}\nx+\\mathrm{i}y&\\mapsto \\frac{c_t}{\\hat{c}}\\left(x+\\frac{\\mathrm{Re}\\,(\\tau(t)-\\hat{\\tau})}{\\mathrm{Im}\\,\\hat{\\tau}}y+\\mathrm{i}\\frac{\\mathrm{Im}\\,\\tau(t)}{\\mathrm{Im}\\,\\hat{\\tau}}y\\right)\\\\&=\\frac{c_t}{\\hat{c}}(x+2y(\\mathrm{Re}\\,\\tau(t)+1\/2)\/\\sqrt{3}+2\\mathrm{i}y \\mathrm{Im}\\,\\tau(t)\/\\sqrt{3}).\n\\end{align*}\nThen we have\n\\begin{align*}\n\\mu_t^\\ast (g_{\\mathrm{sf}}|_{H^{-1}(t)})&=\\frac{c_t^2}{\\hat{c}^2}\\left((\\mathrm{d}x+2(\\mathrm{Re}\\,\\tau(t)+1\/2)\/\\sqrt{3}\\,\\mathrm{d}y)^2+(2 \\mathrm{Im}\\,\\tau(t)\/\\sqrt{3})^2\\,\\mathrm{d}y^2\\right)\n\\\\&=\\mathrm{d}x^2+\\mathrm{d}y^2+\\frac{2}{\\sqrt{3}}\\mathrm{Im}\\left(\\frac{\\mathrm{e}^{-\\pi \\mathrm{i}\/3}\\pi (3\\mu_5+\\mu_6^2)}{12\\sqrt{3}K(\\mathrm{e}^{-\\pi \\mathrm{i}\/6})^2}t^{-2\/3}\\right)(-\\mathrm{d}x^2+\\mathrm{d}y^2)\\\\\n&+\\frac{4}{\\sqrt{3}}\\mathrm{Re}\\left(\\frac{\\mathrm{e}^{-\\pi \\mathrm{i}\/3}\\pi (3\\mu_5+\\mu_6^2)}{12\\sqrt{3}K(\\mathrm{e}^{-\\pi \\mathrm{i}\/6})^2}t^{-2\/3}\\right)\\,\\mathrm{d}x\\otimes \\mathrm{d}y+O(|t|^{-4\/3}).\n\\end{align*}\n\n\n\\subsubsection{An untwisted order three pole and a simple pole} Suppose these poles are located at $0$ and $\\infty$ respectively, then\n\\begin{align*}\n  \\mathcal{B}&=\\left\\{\\,\\left(\\sum_{k=4}^6\\frac{\\mu_k}{z^k}+\\frac{t}{z^3}+\\frac{\\mu_2}{z^2}\\right)\\,\\mathrm{d}z^2\\,\\middle|\\,t\\in\\mathbb{C}\\,\\right\\}\\\\\n  &=\\left\\{\\,\\frac{1}{w^2}\\left(\\mu_6w^4+\\mu_5w^3+\\mu_4w^2+tw+\\mu_2\\right)\\,\\mathrm{d}w^2\\,\\middle|\\,t\\in\\mathbb{C}\\,\\right\\}\n\\end{align*}\nwhere $w=1\/z$, $\\mu_2,\\mu_4,\\mu_5,\\mu_6$ are fixed and $\\mu_6\\neq 0$, we may assume that $\\mu_6=-1$. Here we do not require that the simple pole is twisted, if it is, then $\\mu_2=0$. Fix \\[\\nu=\\frac{1}{w^2}\\left(-w^4+\\mu_5w^3+\\mu_4w^2+tw+\\mu_2\\right)\\,\\mathrm{d}w^2\\in\\mathcal{B}',~\\dot{\\nu}=\\frac{\\dot{t}}{w}\\,\\mathrm{d}w^2\\in T_\\nu\\mathcal{B}'.\\]\nThen $g_{\\mathrm{sK}}$ on $\\mathcal{B}'$ is \\begin{align*}\n  g_{\\mathrm{sK}}(\\dot{\\nu},\\dot{\\nu})&=\\int_{C}\\frac{|\\dot{\\nu}|^2}{|\\nu|}\\,\\mathrm{dvol}_C=|\\dot{t}|^2\\int_{\\mathbb{C}}\\frac{1}{\\left|\\prod_{k=1}^4 (w-w_k(t))\\right|}~\\mathrm{i}\\,\\mathrm{d}w\\mathrm{d}\\bar{w},\n\\end{align*}\nwhere $w_k(t)=t^{1\/3}\\mathrm{e}^{2(k-1)\\pi \\mathrm{i}\/3}+1+O(|t|^{-1\/3})$ ($k=1,2,3$), $w_4(t)=\\mu_2(-t^{-1}-\\mu_2\\mu_4t^{-3}+O(|t|^{-5}))$ are the four roots of $w^2\\nu(w):=\\tilde{\\nu}(w)$. Let $w=t^{1\/3}\\eta $, the integral becomes\n\\begin{align*}\n  &|t|^{-2\/3}\\int_{\\mathbb{C}}\\frac{1}{\\left|\\prod_{k=1}^4(\\eta -t^{-1\/3}w_k(t))\\right|}~\\mathrm{i}\\,\\mathrm{d}\\eta \\mathrm{d}\\bar{\\eta }\\\\\n  =&|t|^{-2\/3}\\left(\\int_{\\mathbb{C}}\\frac{1}{|\\eta |\\left|\\prod_{k=1}^3(\\eta -\\mathrm{e}^{\\mathrm{i}(k-1)\\pi\/2})\\right|}~\\mathrm{i}\\,\\mathrm{d}\\eta \\mathrm{d}\\bar{\\eta }+O(|t|^{-1\/3})\\right)\\\\\n  =&|t|^{-2\/3}(C_0+\\mathrm{Re}(C_1t^{-1\/3})+O(|t|^{-2\/3}))\n\\end{align*}\nTherefore in polar coordinates, as $|t|=r\\to\\infty$, the special K\u00e4hler metric is \\[(C_0+\\mathrm{Re}(C_1t^{-1\/3})+O(r^{-2\/3}))\\frac{\\mathrm{d}r^2+r^2\\,\\mathrm{d}\\theta^2}{r^{2\/3}},\\]\nwhich is asymptotic to a conic metric with cone angle $4\\pi\/3$.\n\nThe fiber $H^{-1}(t)$ can be written as\n\\begin{align*}\n  H^{-1}(t)&=\\left\\{\\,\\left[\\left(\\bar{\\partial}_{E}^1,\\Phi\\right)\\right]\\,\\middle|\\,\\Phi=\\frac{\\mathrm{d}w}{w}\\begin{pmatrix}\n    0&w^4-\\mu_5w^3-\\mu_4w^2-tw-\\mu_2\\\\1&0\n\\end{pmatrix}\\,\\right\\}\\\\\n  &\\cup\\left\\{\\,\\left[\\left(\\bar{\\partial}_{E}^0,\\Phi\\right)\\right]\\,\\middle|\\,\\Phi=\\frac{\\mathrm{d}w}{w}\\begin{pmatrix}\n    -\\mu_5w\/2+w^2&-(\\mu_5^2\/4+\\mu_4)w^2-tw-\\mu_2\\\\1&\\mu_5w\/2-w^2\n\\end{pmatrix}\\,\\right\\}\\\\\n  &\\cup\\left\\{\\,\\left[\\left(\\bar{\\partial}_{E}^0,\\Phi\\right)\\right]\\,\\middle|\\,\\Phi=\\frac{\\mathrm{d}w}{w}\\begin{pmatrix}\n  a_0+w^2&\\begin{smallmatrix}\n    -\\mu_5w^2+(c_0\\mu_5-\\mu_4-2a_0)w-{}\\\\c_0^2\\mu_5+c_0\\mu_4+2a_0c_0-t\n  \\end{smallmatrix}\\\\c_0+w & -a_0-w^2\n\\end{pmatrix},(a_0+c_0^2)^2=-\\tilde{\\nu}(-c_0)\\right\\},\n\\end{align*}\nwhere $(E,\\bar{\\partial}_E^0)\\cong \\mathcal{O}(-1)\\oplus \\mathcal{O}(-1)$, $(E,\\bar{\\partial}_E^1)\\cong \\mathcal{O}\\oplus \\mathcal{O}(-2)$. $H^{-1}(t)$ is the closure of the last set, which is the elliptic curve\n\\[(a_0+c_0^2)^2=-\\tilde{\\nu}(-c_0)=\\prod_{k=1}^4(-c_0-w_k(t)),~(a_0,c_0)\\in \\mathbb{C}^2,\\]\nwith modulus $\\tau(t)=\\lambda^{-1}(l(t))$, where\n\\begin{align*}\n  l(t)&=\\frac{(w_1(t)-w_2(t))(w_3(t)-w_4(t))}{(w_3(t)-w_2(t))(w_1(t)-w_4(t))}=\\mathrm{e}^{-\\pi \\mathrm{i}\/3}+\\sqrt{3}\\mathrm{i} t^{-1\/3} +O(|t|^{-2\/3}), \\\\\n  \\tau(t)&=\\mathrm{e}^{2\\pi \\mathrm{i}\/3}+\\frac{\\sqrt{3}\\pi}{4K(\\mathrm{e}^{-\\pi\/6})^2}t^{-1\/3}+O(|t|^{-2\/3}).\n\\end{align*}\nDefine $\\mu_t: T_{\\hat{\\tau}}^2\\to H^{-1}(t)$ as above, then we have\n\\begin{align*}\n\\mu_t^\\ast (g_{\\mathrm{sf}}|_{H^{-1}(t)})&=\\mathrm{d}x^2+\\mathrm{d}y^2+\\frac{2}{\\sqrt{3}}\\mathrm{Im}\\left(\\frac{\\sqrt{3}\\pi}{4K(\\mathrm{e}^{-\\pi\/6})^2}t^{-1\/3}\\right)(-\\mathrm{d}x^2+\\mathrm{d}y^2)\\\\\n&+\\frac{4}{\\sqrt{3}}\\mathrm{Re}\\left(\\frac{\\sqrt{3}\\pi}{4K(\\mathrm{e}^{-\\pi\/6})^2}t^{-1\/3}\\right)\\,\\mathrm{d}x\\otimes \\mathrm{d}y+O(|t|^{-2\/3}).\n\\end{align*}\n\n\\subsubsection{A twisted order three pole and a simple pole} Suppose these poles are located at $0$ and $\\infty$ respectively, then\n\\begin{align*}\n  \\mathcal{B}&=\\left\\{\\,\\left(\\frac{\\mu_5}{z^5}+\\frac{\\mu_4}{z^4}+\\frac{t}{z^3}+\\frac{\\mu_2}{z^2}\\right)\\,\\mathrm{d}z^2\\,\\middle|\\,t\\in\\mathbb{C}\\,\\right\\}\\\\\n  &=\\left\\{\\,\\frac{1}{w^2}\\left(\\mu_5w^3+\\mu_4w^2+tw+\\mu_2\\right)\\,\\mathrm{d}w^2\\,\\middle|\\,t\\in\\mathbb{C}\\,\\right\\}\n\\end{align*}\nwhere $w=1\/z$, $\\mu_2,\\mu_4,\\mu_5$ are fixed and $\\mu_5\\neq 0$, we may assume that $\\mu_5=-1$. Fix \\[\\nu=\\frac{1}{w^2}\\left(-w^3+\\mu_4w^2+tw+\\mu_2\\right)\\,\\mathrm{d}w^2\\in\\mathcal{B}',~\\dot{\\nu}=\\frac{\\dot{t}}{w}\\,\\mathrm{d}w^2\\in T_\\nu\\mathcal{B}'.\\]\nThen $g_{\\mathrm{sK}}$ on $\\mathcal{B}'$ is \\begin{align*}\n  g_{\\mathrm{sK}}(\\dot{\\nu},\\dot{\\nu})&=\\int_{C}\\frac{|\\dot{\\nu}|^2}{|\\nu|}\\,\\mathrm{dvol}_C=|\\dot{t}|^2\\int_{\\mathbb{C}}\\frac{1}{\\left|\\prod_{k=1}^3 (w-w_k(t))\\right|}~\\mathrm{i}\\,\\mathrm{d}w\\mathrm{d}\\bar{w},\n\\end{align*}\nwhere $w_k(t)=(-1)^{k-1}(t^{1\/2}+\\mu_4^2t^{-1\/2}\/8)+\\mu_4\/2+O(|t|^{-1})$ ($k=1,2$), $w_3(t)=\\mu_2(-t^{-1}-\\mu_2\\mu_4t^{-3}+O(|t|^{-4}))$ are the three roots of $w^2\\nu(w):=\\tilde{\\nu}(w)$. Let $w=t^{1\/2}\\eta $, the integral becomes\n\\begin{align*}\n  &|t|^{-1\/2}\\int_{\\mathbb{C}}\\frac{1}{\\left|\\prod_{k=1}^3(\\eta -t^{-1\/2}w_k(t))\\right|}~\\mathrm{i}\\,\\mathrm{d}\\eta \\mathrm{d}\\bar{\\eta }\\\\\n  =&|t|^{-1\/2}\\left(\\int_{\\mathbb{C}}\\frac{1}{|\\eta (\\eta -1)(\\eta +1)|}~\\mathrm{i}\\,\\mathrm{d}\\eta \\mathrm{d}\\bar{\\eta }+O(|t|^{-1\/2})\\right)\\\\\n  =&|t|^{-1\/2}(C_0+\\mathrm{Re}(C_1\\mu_4t^{-1\/2})+O(|t|^{-1}))\n\\end{align*}\nTherefore in polar coordinates, as $|t|=r\\to\\infty$, the special K\u00e4hler metric is \\[(C_0+\\mathrm{Re}(C_1\\mu_4t^{-1\/2})+O(r^{-1}))\\frac{\\mathrm{d}r^2+r^2\\,\\mathrm{d}\\theta^2}{r^{1\/2}},\\]\nwhich is asymptotic to a conic metric with cone angle $3\\pi\/2$.\n\nThe fiber $H^{-1}(t)$ can be written as\n\\begin{align*}\n  H^{-1}(t)&=\\left\\{\\,\\left[\\left(\\bar{\\partial}_{E}^1,\\Phi\\right)\\right]\\,\\middle|\\,\\Phi=\\frac{\\mathrm{d}w}{w}\\begin{pmatrix}\n    0&w^3-\\mu_4w^2-tw-\\mu_2\\\\1&0\n\\end{pmatrix}\\,\\right\\}\\\\\n  &\\cup\\left\\{\\,\\left[\\left(\\bar{\\partial}_{E}^0,\\Phi\\right)\\right]\\,\\middle|\\,\\Phi=\\frac{\\mathrm{d}w}{w}\\begin{pmatrix}\n  a_0&\\begin{smallmatrix}\n    w^2-(\\mu_4+c_0)w+{}\\\\\\mu_4c_0+c_0^2-t\n  \\end{smallmatrix}\\\\c_0+w & -a_0\n\\end{pmatrix},a_0^2=-\\tilde{\\nu}(-c_0)\\right\\},\n\\end{align*}\nwhere $(E,\\bar{\\partial}_E^0)\\cong \\mathcal{O}(-1)\\oplus \\mathcal{O}(-1)$, $(E,\\bar{\\partial}_E^1)\\cong \\mathcal{O}\\oplus \\mathcal{O}(-2)$. $H^{-1}(t)$ is the closure of the last set, which is the elliptic curve\n\\[a_0^2=-\\tilde{\\nu}(-c_0)=\\prod_{k=1}^3(-c_0-w_k(t)),~(a_0,c_0)\\in \\mathbb{C}^2,\\]\nwith modulus $\\tau(t)=\\lambda^{-1}(l(t))$, where\n\\begin{align*}\n  l(t)&=\\frac{w_3(t)-w_2(t)}{w_1(t)-w_2(t)}=\\frac{1}{2}-\\mu_4t^{-1\/2}\/4 +O(|t|^{-1}), \\\\\n  \\tau(t)&=\\mathrm{i}+\\frac{\\mu_4\\pi \\mathrm{i} }{4K(\\sqrt{1\/2})^2}t^{-1\/2}+O(|t|^{-1}).\n\\end{align*}\nDefine $\\mu_t: T_{\\hat{\\tau}}^2\\to H^{-1}(t)$ as above, then we have\n\\begin{align*}\n\\mu_t^\\ast (g_{\\mathrm{sf}}|_{H^{-1}(t)})&=\\mathrm{d}x^2+\\mathrm{d}y^2+\\mathrm{Im}\\left(\\frac{\\mu_4\\pi \\mathrm{i} }{4K(\\sqrt{1\/2})^2}t^{-1\/2}\\right)(-\\mathrm{d}x^2+\\mathrm{d}y^2)\\\\\n&+2\\mathrm{Re}\\left(\\frac{\\mu_4\\pi \\mathrm{i} }{4K(\\sqrt{1\/2})^2}t^{-1\/2}\\right)\\,\\mathrm{d}x\\otimes \\mathrm{d}y+O(|t|^{-1}).\n\\end{align*}\n\n\\subsubsection{Two untwisted order two poles} Suppose these poles are located at $0$ and $\\infty$ respectively, then \\[\\mathcal{B}=\\left\\{\\,\\left(\\frac{\\mu_4}{z^4}+\\frac{\\mu_3}{z^3}+\\frac{t}{z^2}+\\frac{\\mu_1}{z}+\\mu_0\\right)\\,\\mathrm{d}z^2\\,\\middle|\\,t\\in\\mathbb{C}\\,\\right\\},\\]\nwhere $\\mu_0,\\mu_1,\\mu_3,\\mu_4$ are fixed and $\\mu_0,\\mu_4\\neq 0$, we may assume that $\\mu_4=-1$. Fix \\[\\nu=\\left(\\frac{-1}{z^4}+\\frac{\\mu_3}{z^3}+\\frac{t}{z^2}+\\frac{\\mu_1}{z}+\\mu_0\\right)\\,\\mathrm{d}z^2\\in\\mathcal{B}',~\\dot{\\nu}=\\frac{\\dot{t}}{z^2}\\,\\mathrm{d}z^2\\in T_\\nu\\mathcal{B}'.\\]\nThen $g_{\\mathrm{sK}}$ on $\\mathcal{B}'$ is\n\\begin{align*}\n  g_{\\mathrm{sK}}(\\dot{\\nu},\\dot{\\nu})&=\\int_{C}\\frac{|\\dot{\\nu}|^2}{|\\nu|}\\,\\mathrm{dvol}_C=|\\dot{t}|^2\\int_{\\mathbb{C}}\\frac{1}{\\left|\\mu_0\\prod_{k=1}^4 (z-z_k(t))\\right|}~\\mathrm{i}\\,\\mathrm{d}z\\mathrm{d}\\bar{z}\\\\\n  &=\\frac{|\\dot{t}|^2}{|\\mu_0|}\\int_{|z|\\leqslant 1}\\frac{1}{\\left|\\prod_{k=1}^4(z-z_k(t))\\right|}\\,\\mathrm{i}\\,\\mathrm{d}z\\mathrm{d}\\bar{z}+|\\dot{t}|^2\\int_{|w|\\leqslant 1}\\frac{1}{\\left|\\prod_{k=1}^4(w-z_k(t)^{-1})\\right|}~\\mathrm{i}\\,\\mathrm{d}w\\mathrm{d}\\bar{w},\n\\end{align*}\nwhere $z_k(t)=(-1)^{k-1}t^{-1\/2}-\\mu_3t^{-1}\/2+O(|t|^{-3\/2})$ ($k=1,2$), $z_k(t)=(-1)^{k-1}(-\\mu_0)^{-1\/2}t^{1\/2}-\\mu_1\/2\\mu_0+O(|t|^{-1\/2})$ ($k=3,4$) are the four roots of $z^4\\nu(z)=:\\tilde{\\nu}(z)$. In $\\{|z|\\leqslant 1\\}$, $|z-z_k(t)|=|\\mu_0^{-1\/2}t^{1\/2}|(1+O(|t|^{-1\/2}))$ for $k=3,4$, then the first integral is\n\\begin{align*}\n  &\\int_{|z|\\leqslant 1}\\frac{|\\mu_0t^{-1}|}{|z-z_1(t)||z-z_2(t)|}(1+O(|t|^{-1\/2}))~\\mathrm{i}\\,\\mathrm{d}z\\mathrm{d}\\bar{z}\\quad (\\text{let }\\xi=t^{1\/2}z)\\\\=&\\int_{|\\xi|\\leqslant |t^{1\/2}|}\\frac{1}{|\\xi-t^{1\/2}z_1(t)||\\xi-t^{1\/2}z_2(t)|}~\\mathrm{i}\\,\\mathrm{d}\\xi \\mathrm{d}\\bar{\\xi}\\cdot|\\mu_0t^{-1}|(1+O(|t|^{-1\/2}))\\\\\n  =&\\int_{|\\xi|\\leqslant |t^{1\/2}|}\\frac{1}{|\\xi-1||\\xi+1|}~\\mathrm{i}\\,\\mathrm{d}\\xi \\mathrm{d}\\bar{\\xi}\\cdot|\\mu_0t^{-1}|(1+O(|t|^{-1\/2}))\\\\\n  =&2\\pi |\\mu_0||t|^{-1}\\log|t|+O(|t|^{-1}).\n\\end{align*}\nSimilarly, the second integral is\n\\[2\\pi |t|^{-1}\\log|t|+O(|t|^{-1}).\\]\nTherefore in polar coordinates, as $|a|=r\\to\\infty$, the special K\u00e4hler metric is \\[(4\\pi\\log\\,r+O(1))\\frac{\\mathrm{d}r^2+r^2\\,\\mathrm{d}\\theta^2}{r}.\\]\n\nThe fiber $H^{-1}(t)$ can be written as\n\\begin{align*}\n  H^{-1}(t)&=\\left\\{\\,\\left[\\left(\\bar{\\partial}_{E}^1,\\Phi\\right)\\right]\\,\\middle|\\,\\Phi=\\frac{\\mathrm{d}z}{z^2}\\begin{pmatrix}\n    0&1-\\mu_3z-tz^2-\\mu_1z^3-\\mu_0z^4\\\\1&0\n\\end{pmatrix}\\,\\right\\}\\\\\n  &\\cup\\left\\{\\,\\left[\\left(\\bar{\\partial}_{E}^0,\\Phi\\right)\\right]\\,\\middle|\\,\\Phi=\\frac{\\mathrm{d}z}{z^2}\\begin{pmatrix}\n  -\\frac{\\mu_1z}{2(-\\mu_0)^{1\/2}}+(-\\mu_0)^{1\/2}z^2&-(t+\\mu_1^2\/4\\mu_0)z^2-\\mu_3z+1\\\\1&  \\frac{\\mu_1z}{2(-\\mu_0)^{1\/2}}-(-\\mu_0)^{1\/2}z^2\n\\end{pmatrix}\\,\\right\\}\\\\\n  &\\cup\\left\\{\\,\\left[\\left(\\bar{\\partial}_{E}^0,\\Phi\\right)\\right]\\,\\middle|\\,\\Phi=\\frac{\\mathrm{d}z}{z^2}\\begin{pmatrix}\n   a_0+(-\\mu_0)^{1\/2}z^2&\\begin{smallmatrix}\n     -\\mu_1z^2+(\\mu_1c_0-2a_0(-\\mu_0)^{1\/2}-t)z-{}\\\\\n     \\mu_1c_0^2+2a_0c_0(-\\mu_0)^{1\/2}+tc_0-\\mu_3\n   \\end{smallmatrix}\\\\c_0+z&-a_0-(-\\mu_0)^{1\/2}z^2\n \\end{pmatrix},\\right.\\notag\\\\&\\left.\\hspace{8cm}\\vphantom{\\begin{pmatrix}\n     1+a_1z&\\begin{smallmatrix}\n       \\mu_1c_0-2a_0(-\\mu_0)^{1\/2}\\\\a_0c_0(-\\mu_0)^{1\/2}\n     \\end{smallmatrix}\\\\1+c_1z&-(-\\mu_0)^{1\/2}z^2\n  \\end{pmatrix}}  (a_0+(-\\mu_0)^{1\/2}c_0^2)^2=-\\tilde{\\nu}(-c_0)\\,\\right\\},\n\\end{align*}\nwhere $(E,\\bar{\\partial}_E^0)\\cong \\mathcal{O}(-1)\\oplus \\mathcal{O}(-1)$, $(E,\\bar{\\partial}_E^1)\\cong \\mathcal{O}\\oplus \\mathcal{O}(-2)$. $H^{-1}(t)$ is the closure of the last set, which is the elliptic curve\n\\[(a_0+(-\\mu_0)^{1\/2}c_0^2)^2=-\\tilde{\\nu}(-c_0)=-\\mu_0\\prod_{k=1}^4(-c_0-z_k(t)),~(a_0,c_0)\\in \\mathbb{C}^2,\\]\nwith modulus $\\tau(t)=\\lambda^{-1}(l(t))$, where\n\\begin{align*}\n  l(t)&=\\frac{(z_1(t)-z_2(t))(z_3(t)-z_4(t))}{(z_3(t)-z_2(t))(z_1(t)-z_4(t))}\\\\&=4(-\\mu_0)^{1\/2}t^{-1}+(8\\mu_0+(-\\mu_0)^{-1\/2}(\\mu_1^2-\\mu_0\\mu_3^2)\/2)t^{-2}+O(|t|^{-3}),\\\\\n  \\tau(t)&=\\frac{\\mathrm{i}}{\\pi}\\log(4(-\\mu_0)^{-1\/2}t)+\\frac{\\mathrm{i}}{\\pi}(\\mu_0^{-1}(\\mu_1^2-\\mu_0\\mu_3^2)\/8)t^{-1}+O(|t|^{-2}).\n\\end{align*}\n $H^{-1}(t)\\cong \\mathbb{C}\\,\/\\,c_t(\\mathbb{Z}\\oplus\\tau(t)\\mathbb{Z})$ with Euclidean metric $\\mathrm{d}x^2+\\mathrm{d}y^2$ and $c_t=2\\pi\/\\sqrt{\\mathrm{Im}\\,\\tau(t)}$.\n\n\\subsubsection{Two order two poles with one twisted} Suppose the pole at $0$ is untwisted and the pole at $\\infty$ is twisted, then\n\\[\\mathcal{B}=\\left\\{\\left(\\frac{\\mu_4}{z^4}+\\frac{\\mu_3}{z^3}+\\frac{t}{z^2}+\\frac{\\mu_1}{z}\\right)\\,\\mathrm{d}z^2\\middle|\\,t\\in\\mathbb{C}\\,\\right\\},\\]\nwhere $\\mu_1,\\mu_3,\\mu_4$ are fixed and $\\mu_1,\\mu_4\\neq 0$, we may assume that $\\mu_4=-1$. Fix \\[\\nu=\\left(\\frac{\\mu_4}{z^4}+\\frac{\\mu_3}{z^3}+\\frac{t}{z^2}+\\frac{\\mu_1}{z}\\right)\\,\\mathrm{d}z^2\\in\\mathcal{B}',~\\dot{\\nu}=\\frac{\\dot{t}}{z^2}\\,\\mathrm{d}z^2\\in T_\\nu\\mathcal{B}'.\\]\nThen $g_{\\mathrm{sK}}$ on $\\mathcal{B}'$ is \\begin{align*}\n  g_{\\mathrm{sK}}(\\dot{\\nu},\\dot{\\nu})&=\\int_{C}\\frac{|\\dot{\\nu}|^2}{|\\nu|}\\,\\mathrm{dvol}_C=\\frac{|\\dot{t}|^2}{|\\mu_1|}\\int_{\\mathbb{C}}\\frac{1}{\\left|\\prod_{k=1}^3 (z-z_k(t))\\right|}~\\mathrm{i}\\,\\mathrm{d}z\\mathrm{d}\\bar{z}\\\\\n  &=\\frac{|\\dot{t}|^2}{|\\mu_1|}\\int_{|z|\\leqslant 1}\\frac{1}{|\\prod_{k=1}^3(z-z_k(t))|}~\\mathrm{i}\\,\\mathrm{d}z\\mathrm{d}\\bar{z}+|\\dot{t}|^2\\int_{|w|\\leqslant 1}\\frac{1}{|w||\\prod_{k=1}^3(w-z_k(t)^{-1})|}~\\mathrm{i}\\,\\mathrm{d}w\\mathrm{d}\\bar{w},\n\\end{align*}\nwhere $z_k(t)=(-1)^{k-1}t^{-1\/2}-\\mu_3t^{-1}\/2+O(|t|^{-3\/2})$ ($k=1,2$), $z_3(t)=-t\/\\mu_1+\\mu_3t^{-1}+O(|t|^{-2})$ are the three roots of $z^4\\nu(z)=:\\tilde{\\nu}(z)$. In $\\{|z|\\leqslant 1\\}$, $|z-z_3(t)|=|t\/\\mu_1|(1+O(|t|^{-1}))$, then the first integral is\n\\begin{align*}\n &\\int_{|z|\\leqslant 1}\\frac{|\\mu_1t^{-1}|}{|z-z_1(t)||z-z_2(t)|}(1+O(|t|^{-1}))~\\mathrm{i}\\,\\mathrm{d}z\\mathrm{d}\\bar{z}\\\\=&2\\pi|\\mu_1||t|^{-1}\\log\\,|t|+O(|t|^{-1}),\n\\end{align*}\nas before. Similarly, the second integral is $4\\pi |t|^{-1}\\log\\,|t|+O(|t|^{-1})$. Therefore in polar coordinates, as $|t|=r\\to\\infty$, the special K\u00e4hler metric is \\[(6\\pi\\log\\,r+O(1))\\frac{\\mathrm{d}r^2+r^2\\,\\mathrm{d}\\theta^2}{r}.\\]\n\nThe fiber $H^{-1}(t)$ can be written as\n\\begin{align*}\n  H^{-1}(t)&=\\left\\{\\,\\left[\\left(\\bar{\\partial}_{E}^1,\\Phi\\right)\\right]\\,\\middle|\\,\\Phi=\\frac{\\mathrm{d}z}{z^2}\\begin{pmatrix}\n    0&1-\\mu_3z-tz^2-\\mu_1z^3\\\\1&0\n\\end{pmatrix}\\,\\right\\}\\notag\\\\\n  &\\cup\\left\\{\\,\\left[\\left(\\bar{\\partial}_{E}^0,\\Phi\\right)\\right]\\,\\middle|\\,\\Phi=\\frac{\\mathrm{d}z}{z^2}\\begin{pmatrix}\n   a_0&\\begin{smallmatrix}\n     -\\mu_1z^2+(\\mu_1c_0-t)z-{}\\\\\n     \\mu_3+tc_0-\\mu_1c_0^2\n   \\end{smallmatrix}\\\\c_0+z&-a_0\n \\end{pmatrix},a_0^2=-\\tilde{\\nu}(-c_0)\\,\\right\\},\n\\end{align*}\nwhere $\\bar{\\partial}_E^0,\\bar{\\partial}_E^1$ are holomorphic structures as before. $H^{-1}(t)$ is the closure of the last set, which is the elliptic curve\n\\[a_0^2=-\\tilde{\\nu}(-c_0)=-\\mu_1\\prod_{k=1}^3(-c_0-z_k(t)),~(a_0,c_0)\\in \\mathbb{C}^2,\\]\nwith modulus $\\tau(t)=\\lambda^{-1}(l(t))$, where\n\\begin{align*}\n  l(t)&=\\frac{z_1(t)-z_2(t)}{z_1(t)-z_3(t)}=2\\mu_1t^{-3\/2}+\\mu_1\\mu_3^2t^{-5\/2}\/4+O(|t|^{-3}),\\\\\n  \\tau(t)&=\\frac{\\mathrm{i}}{\\pi}\\left(\\log(8\\mu_1^{-1}t^{3\/2})-\\mu_3^2t^{-1}\/8-\\mu_1t^{-3\/2}\\right)+O(|t|^{-2}).\n\\end{align*}\n\n\\subsubsection{Two twisted order two poles}\nSuppose the poles are $0$ and $\\infty$, then\n\\[\\mathcal{B}=\\left\\{\\left(\\frac{\\mu_3}{z^3}+\\frac{t}{z^2}+\\frac{\\mu_1}{z}\\right)\\,\\mathrm{d}z^2\\middle|\\,t\\in\\mathbb{C}\\,\\right\\},\\]\nwhere $\\mu_1,\\mu_3\\neq 0$ are fixed, we may assume that $\\mu_3=-1$. Fix\n\\[\\nu=\\left(\\frac{\\mu_3}{z^3}+\\frac{t}{z^2}+\\frac{\\mu_1}{z}\\right)\\,\\mathrm{d}z^2\\in\\mathcal{B}',~\\dot{\\nu}=\\frac{\\dot{t}}{z^2}\\,\\mathrm{d}z^2\\in T_\\nu\\mathcal{B}'.\\]\nBy a similar computation as before, $g_{\\mathrm{sK}}$ on $\\mathcal{B}'$ is \\begin{align*}\n  g_{\\mathrm{sK}}(\\dot{\\nu},\\dot{\\nu})&=\\int_{C}\\frac{|\\dot{\\nu}|^2}{|\\nu|}\\,\\mathrm{dvol}_C=\\frac{|\\dot{t}|^2}{|\\mu_1|}\\int_{\\mathbb{C}}\\frac{1}{\\left|z(z-z_1(t))(z-z_2(t))\\right|}~\\mathrm{i}\\,\\mathrm{d}z\\mathrm{d}\\bar{z}\\\\\n  &=\\frac{|\\dot{t}|^2}{|\\mu_1|}\\int_{|z|\\leqslant 1}\\frac{1}{|z||z-z_1(t)||z-z_2(t)|}~\\mathrm{i}\\,\\mathrm{d}z\\mathrm{d}\\bar{z}\\\\&\\quad+|\\dot{t}|^2\\int_{|w|\\leqslant 1}\\frac{1}{|w||w-z_1(t)^{-1}||w-z_2(t)^{-1}|}~\\mathrm{i}\\,\\mathrm{d}w\\mathrm{d}\\bar{w}\\\\&=|\\dot{t}|^2( 8\\pi |t|^{-1}\\log\\,|t|+O(|t|^{-1})).\n\\end{align*}\nHere $z_1(t)=t^{-1}-\\mu_1t^{-3}+O(|t|^{-5})$, $z_2(t)=-t\/\\mu_1-t^{-1}+O(|t|^{-3})$ are the two roots of $z^3\\nu(z):=\\tilde{\\nu}(z)$. Therefore in polar coordinates, as $|t|=r\\to\\infty$, the special K\u00e4hler metric is \\[(8\\pi\\log\\,r+O(1))\\frac{\\mathrm{d}r^2+r^2\\,\\mathrm{d}\\theta^2}{r}.\\]\n\nThe fiber $H^{-1}(t)$ can be written as\n\\begin{align*}\n  H^{-1}(t)&=\\left\\{\\,\\left[\\left(\\bar{\\partial}_{E}^1,\\Phi\\right)\\right]\\,\\middle|\\,\\Phi=\\frac{\\mathrm{d}z}{z^2}\\begin{pmatrix}\n    0&z(1-tz+\\mu_1z^2)\\\\1&0\n\\end{pmatrix}\\,\\right\\}\\\\\n  &\\cup\\left\\{\\,\\left[\\left(\\bar{\\partial}_{E}^0,\\Phi\\right)\\right]\\,\\middle|\\,\\Phi=\\frac{\\mathrm{d}z}{z^2}\\begin{pmatrix}\n   a_0&\\begin{smallmatrix}\n     -\\mu_1z^2+(\\mu_1c_0-a)z+{}\\\\1+tc_0-\\mu_1c_0^2\n   \\end{smallmatrix} \\\\c_0+z&-a_0\n \\end{pmatrix},a_0^2=c_0\\tilde{\\nu}(-c_0)\\,\\right\\},\n\\end{align*}\nwhere $\\bar{\\partial}_E^0,\\bar{\\partial}_E^1$ are as above. $H^{-1}(t)$ is the closure of the last set, which is the elliptic curve\n\\[a_0^2=c_0\\tilde{\\nu}(-c_0)=\\mu_1c_0(c_0+z_1(t))(c_0+z_2(t)),~(a_0,c_0)\\in \\mathbb{C}^2,\\]\nwith modulus $\\tau(t)=\\lambda^{-1}(l(t))$, where\n\\begin{align*}\n  l(t)&=\\frac{z_1(t)-0}{z_1(t)-z_2(t)}=\\mu_1t^{-2}-3\\mu_1^2t^{-4}+O(|t|^{-6}),\\\\\n  \\tau(t)&=\\frac{\\mathrm{i}}{\\pi}(\\log(16\\mu_1^{-1}t^2)+5\\mu_1t^{-2}\/2)+O(|t|^{-4}).\n\\end{align*}\n\n\\subsubsection{An untwisted order two pole and two simple poles}\nSuppose the two simple poles are $0$ and $1$, the order two pole is $\\infty$, then\n\\[\\mathcal{B}=\\left\\{\\,\\left(\\frac{\\mu_0}{z^2}+\\frac{\\mu_1}{(z-1)^2}+\\frac{\\mu_3}{z}+\\mu_4+\\frac{t}{z(z-1)}\\right)\\,\\mathrm{d}z^2\\,\\middle|\\,t\\in\\mathbb{C}\\,\\right\\},\\]\nwhere $\\mu_0,\\mu_1,\\mu_3,\\mu_4$ are fixed, $\\mu_4\\neq 0$. Fix\n\\[\\nu=\\left(\\frac{\\mu_0}{z^2}+\\frac{\\mu_1}{(z-1)^2}+\\frac{\\mu_3}{z}+\\mu_4+\\frac{t}{z(z-1)}\\right)\\,\\mathrm{d}z^2\\in\\mathcal{B}',~\\dot{\\nu}=\\frac{\\dot{t}}{z(z-1)}\\,\\mathrm{d}z^2\\in T_\\nu\\mathcal{B}'.\\]\nThen $g_{\\mathrm{sK}}$ on $\\mathcal{B}'$ is\n\\begin{align*}\n  g_{\\mathrm{sK}}(\\dot{\\nu},\\dot{\\nu})&=\\int_{C}\\frac{|\\dot{\\nu}|^2}{|\\nu|}\\,\\mathrm{dvol}_C=|\\dot{t}|^2\\int_{\\mathbb{C}}\\frac{1}{\\left|\\mu_4\\prod_{k=1}^4 (z-z_k(t))\\right|}~\\mathrm{i}\\,\\mathrm{d}z\\mathrm{d}\\bar{z}\\\\\n  &=\\frac{|\\dot{t}|^2}{|\\mu_4|}\\int_{|z|\\leqslant 2}+\\int_{2\\leqslant |z|\\leqslant |\\mu_4|^{-1\/2}|t|^{1\/2}\/2}\\frac{1}{\\left|\\prod_{k=1}^4(z-z_k(t))\\right|}\\,\\mathrm{i}\\,\\mathrm{d}z\\mathrm{d}\\bar{z}\\\\&+\\frac{|\\dot{t}|^2}{|\\mu_0|}\\int_{|w|\\leqslant 2|\\mu_4|^{1\/2}|a|^{-1\/2}}\\frac{1}{\\left|\\prod_{k=1}^4(w-z_k(t)^{-1})\\right|}~\\mathrm{i}\\,\\mathrm{d}w\\mathrm{d}\\bar{w}\\\\\n  &=|\\dot{t}|^2(O(1\/|t|)+2\\pi |t|^{-1}\\log\\,|t|+O(1\/|t|))=2\\pi |\\dot{t}|^2|t|^{-1}(\\log\\,|t|+O(1)).\n\\end{align*}\nHere $z_1(t)=\\mu_0t^{-1}+\\mu_0(\\mu_3-\\mu_0)t^{-2}+O(|t|^{-3})$, $z_2(t)=1-\\mu_1t^{-1}+\\mu_1^2t^{-2}+O(|t|^{-3})$, $z_k(t)=(-1)^{k-1}(-\\mu_4)^{-1\/2}t^{1\/2}+(1-\\mu_3\/\\mu_4)\/2+O(|t|^{-1\/2})$ ($k=3,4$) are the four roots of $z^2(z-1)^2\\nu(z):=\\tilde{\\nu}(z)$. Therefore in polar coordinates, as $|t|=r\\to\\infty$, the special K\u00e4hler metric is \\[(2\\pi\\log\\,r+O(1))\\frac{\\mathrm{d}r^2+r^2\\,\\mathrm{d}\\theta^2}{r}.\\]\n\nThe fiber $H^{-1}(t)$ can be written as\n\\begin{align*}\n  H^{-1}(t)&=\\left\\{\\,\\left[\\left(\\bar{\\partial}_{E}^0,\\Phi\\right)\\right]\\,\\middle|\\,\\Phi=\\frac{\\mathrm{d}z}{z(z-1)}\\begin{pmatrix}\n    \\pm (-\\mu_4)^{1\/2}z^2&\\mu_4z^4-\\tilde{\\nu}(z)\\\\1&\\mp(-\\mu_4)^{1\/2}z^2\n\\end{pmatrix}\\,\\right\\}\\\\\n&\\hspace{-2em}\\cup\\left\\{\\,\\left[\\left(\\bar{\\partial}_{E}^0,\\Phi\\right)\\right]\\,\\middle|\\,\\Phi=\\frac{\\mathrm{d}z}{z(z-1)}\\begin{pmatrix}\n a_0&\\begin{smallmatrix}\n   -\\mu_4z^3+(\\mu_4c_0+2\\mu_4-\\mu_3)z^2+{}\\\\\n   (2\\mu_3-\\mu_4-\\mu_1-t-\\mu_4c_0^2-2\\mu_4c_0+\\mu_3c_0)z+{}\\\\2\\mu_0+t-\n  \\mu_3+(\\mu_4-2\\mu_3+\\mu_1+t)c_0+{}\\\\(2\\mu_4-\\mu_3)c_0^2+\\mu_4c_0^3\n \\end{smallmatrix}\\\\c_0+z&-a_0\n\\end{pmatrix}, a_0^2=-\\tilde{\\nu}(-c_0)\\right\\},\n\\end{align*}\nwhere $(E,\\bar{\\partial}_E^0)\\cong \\mathcal{O}(-1)\\oplus \\mathcal{O}(-2)$. $H^{-1}(t)$ is the closure of the last set, which is the elliptic curve\n\\[a_0^2=-\\tilde{\\nu}(-c_0)=-\\mu_4\\prod_{k=1}^4(-c_0-z_k(t)),~(a_0,c_0)\\in \\mathbb{C}^2,\\]\nwith modulus $\\tau(t)=\\lambda^{-1}(l(t))$, where\n\\begin{align*}\nl(t)&=\\frac{(z_1(t)-z_2(t))(z_3(t)-z_4(t))}{(z_3(t)-z_2(t))(z_1(t)-z_4(t))}=-2(-\\mu_4)^{1\/2}t^{-1\/2}+2\\mu_4t^{-1}+O(|t|^{-3\/2}),\\\\\n\\tau(t)&=\\frac{\\mathrm{i}}{\\pi}\\log(-8(-\\mu_4)^{-1\/2}t^{1\/2})+O(|t|^{-1}).\n\\end{align*}\n\n\\subsubsection{A twisted order two pole and two simple poles} Let $0,1$ be the two simple poles and $\\infty$ be the order two pole, then\n\\[\\mathcal{B}=\\left\\{\\,\\left(\\frac{\\mu_0}{z^2}+\\frac{\\mu_1}{(z-1)^2}+\\frac{\\mu_3}{z}+\\frac{t}{z(z-1)}\\right)\\,\\mathrm{d}z^2\\,\\middle|\\,t\\in\\mathbb{C}\\,\\right\\},\\]\nwhere $\\mu_0,\\mu_1,\\mu_3$ are fixed, $\\mu_3\\neq 0$. Fix\n\\[\\nu=\\left(\\frac{\\mu_0}{z^2}+\\frac{\\mu_1}{(z-1)^2}+\\frac{\\mu_3}{z}+\\frac{t}{z(z-1)}\\right)\\,\\mathrm{d}z^2\\in\\mathcal{B}',~\\dot{\\nu}=\\frac{\\dot{t}}{z(z-1)}\\,\\mathrm{d}z^2\\in T_\\nu\\mathcal{B}'.\\]\nThen $g_{\\mathrm{sK}}$ on $\\mathcal{B}'$ is\n\\begin{align*}\n  g_{\\mathrm{sK}}(\\dot{\\nu},\\dot{\\nu})&=\\int_{C}\\frac{|\\dot{\\nu}|^2}{|\\nu|}\\,\\mathrm{dvol}_C=|\\dot{t}|^2\\int_{\\mathbb{C}}\\frac{1}{\\left|\\mu_3\\prod_{k=1}^3 (z-z_k(t))\\right|}~\\mathrm{i}\\,\\mathrm{d}z\\mathrm{d}\\bar{z}\\\\\n  &=\\frac{|\\dot{t}|^2}{|\\mu_3|}\\int_{|z|\\leqslant 2}+\\int_{2\\leqslant |z|\\leqslant |t|\/(2|\\mu_3|)}\\frac{1}{\\left|\\prod_{k=1}^3(z-z_k(a))\\right|}\\,\\mathrm{i}\\,\\mathrm{d}z\\mathrm{d}\\bar{z}\\\\&+\\frac{|\\dot{t}|^2}{|\\mu_0|}\\int_{|w|\\leqslant 2|\\mu_3|\/|t|}\\frac{1}{|w|\\left|\\prod_{k=1}^3(w-z_k(t)^{-1})\\right|}~\\mathrm{i}\\,\\mathrm{d}w\\mathrm{d}\\bar{w}\\\\\n  &=|\\dot{t}|^2(O(1\/|t|)+4\\pi |t|^{-1}\\log\\,|t|+O(1\/|t|))=4\\pi |\\dot{t}|^2 |t|^{-1}(\\log\\,|t|+O(1)).\n\\end{align*}\nHere $z_1(t)=\\mu_0t^{-1}+\\mu_0(\\mu_3-\\mu_0)t^{-2}+O(|t|^{-3})$, $z_2(t)=1-\\mu_1t^{-1}+\\mu_1^2t^{-2}+O(|t|^{-3})$, $z_3(t)=-t\/\\mu_3+(1-(\\mu_0+\\mu_1)\/\\mu_3)+O(|t|^{-1})$ are the three roots of $z^2(z-1)^2\\nu(z):=\\tilde{\\nu}(z)$. Therefore in polar coordinates, as $|t|=r\\to\\infty$, the special K\u00e4hler metric is \\[(4\\pi\\log\\,r+O(1))\\frac{\\mathrm{d}r^2+r^2\\,\\mathrm{d}\\theta^2}{r}.\\]\nThe fiber $H^{-1}(t)$ can be written as\n\\begin{align*}\n  H^{-1}(t)&=\\left\\{\\,\\left[\\left(\\bar{\\partial}_{E}^0,\\Phi\\right)\\right]\\,\\middle|\\,\\Phi=\\frac{\\mathrm{d}z}{z(z-1)}\\begin{pmatrix}\n    0&-\\tilde{\\nu}(z)\\\\1&0\n\\end{pmatrix}\\,\\right\\}\\\\\n&\\cup\\left\\{\\,\\left[\\left(\\bar{\\partial}_{E}^0,\\Phi\\right)\\right]\\,\\middle|\\,\\Phi=\\frac{\\mathrm{d}z}{z(z-1)}\\begin{pmatrix}\n a_0&\\begin{smallmatrix}\n  -\\mu_3z^2+(\\mu_3c_0+2\\mu_3-\\mu_0-\\mu_1-t)z+{}\\\\\n  t+2\\mu_0-\\mu_3+{}\\\\(t+\\mu_0+\\mu_1-2\\mu_3)c_0-\\mu_3c_0^2\n \\end{smallmatrix}\\\\c_0+z&-a_0\n\\end{pmatrix}, a_0^2=-\\tilde{\\nu}(-c_0)\\right\\},\n\\end{align*}\nwhere $(E,\\bar{\\partial}_E^0)\\cong \\mathcal{O}(-1)\\oplus \\mathcal{O}(-2)$. $H^{-1}(t)$ is the closure of the last set, which is the elliptic curve\n\\[a_0^2=-\\tilde{\\nu}(-c_0)=-\\mu_3\\prod_{k=1}^3(-c_0-z_k(t)),~(a_0,c_0)\\in \\mathbb{C}^2,\\]\nwith modulus $\\tau(t)=\\lambda^{-1}(l(t))$, where\n\\begin{align*}\nl(t)&=\\frac{(z_1(t)-z_2(t))}{(z_1(t)-z_3(t))}=-\\mu_3t^{-1}-\\mu_3t^{-2}+O(|t|^{-3}),\\\\\n\\tau(t)&=\\frac{\\mathrm{i}}{\\pi}(\\log(-16\\mu_3^{-1}t)+(\\mu_3\/2-1)t^{-1})+O(|t|^{-2}).\n\\end{align*}\n\n\\subsection{The Uniformness}\nIn Theorem \\ref{Main_thm}, the constants in the decay rate of $g_{L^2}-g_{\\mathrm{sf}}$ depend on the choice of the curve. Now we aim to find constants uniform in these choices for a four-dimensional moduli space $\\mathcal{M}$. For simplicity, here we only consider the moduli space in Section \\ref{U4_subsubsec}.\n\nAs in \\eqref{HitBaseU4_eq}, $t\\in \\mathbb{C}$ parametrizes the quadratic differential $\\nu_t$ in the Hitchin base, and is no longer a positive real number as in the previous sections. Let $(\\bar{\\partial}_E,\\Phi_t)$ be any Higgs bundle in the Hitchin fiber $H^{-1}(t)$. By Lemma \\ref{HiggsDetRoot_lem}, the four roots of $\\tilde{\\nu}_t(z)$ have asymptotics\n\\[z_k(t)=t^{-1\/4}\\mathrm{e}^{\\mathrm{i}(k-1)\\pi\/2}+O(|t|^{-1\/2}),\\]\nas $|t|\\to\\infty$. The local mass at $z_k(t)$ $(k=1,2,3,4)$ is\n\\[\\lambda_k(t)=\\Big|t\\prod_{j\\neq k}(z_k(t)-z_j(t))z_k(t)^{-8}\\Big|^{1\/2}=4t^{9\/4}(1+O(|t|^{-1\/4})).\\]\nHere and below, the constants are independent of $\\mathrm{Arg}(t)$ and the choice of $(\\bar{\\partial}_E,\\Phi_t)$ in $H^{-1}(t)$. Proposition \\ref{NormalForm_prop} still hold on disks $\\widetilde{B}_{k,t}:=\\{\\,|\\zeta_{k,t}|<\\kappa |t|^{-1\/4}\\,\\}$, where $\\zeta_{k,t}$ is the local holomorphic coordinate centered at $z_k(t)$ such that $q_t=-\\lambda_k(t)^2\\zeta_{k,t}\\,\\mathrm{d}\\zeta_{k,t}^2$, and $\\kappa$ is independent of the above choices. One only need to check that the gauge transformations in the proof is well-defined on the whole disk $\\widetilde{B}_{k,t}$ for suitable $\\kappa$. This is clear for $\\Phi_t$ in the first subset of \\eqref{HitFibU4_eq}. For $\\Phi_t$ in the second subset, now $a_0,c_0$ may depend on $t$. Suppose $|c_0+z_k(t)|\\geqslant |t|^{-1\/4}\/2$, the first gauge transformation is $\\left(\\begin{smallmatrix}\n  a_0&1\\\\ z_k(t)+c_0&0\n\\end{smallmatrix}\\right)$, which is constant and invertible over $\\widetilde{B}_{k,t}$. The second gauge transformation is well-defined since \\[\\left|\\frac{z+c_0}{z_k(t)+c_0}-1\\right|= \\left|\\frac{z-z_k(t)}{z_k(t)}\\right|\\left|\\frac{z_k(t)}{c_0+z_k(t)}\\right|\\leqslant c\\kappa\\leqslant 1\/2,\\]\nfor $\\kappa$ small. The rest of the proof remains the same. If $|c_0+z_k(t)|\\leqslant |t|^{-1\/4}\/2$ for some $k$, then $|c_0+z_j(t)|\\geqslant |t|^{-1\/4}\/2$ for $j\\neq k$ and $|t|$ large, and the previous discussion applies for the disks $\\widetilde{B}_{j,t}$. On $\\widetilde{B}_{k,t}$,\n\\[\\Phi_t=\\frac{\\mathrm{d}z}{z^4}\\begin{pmatrix}\n    a_0&\\begin{smallmatrix}\n      -tz^3+(-\\mu_5+tc_0)z^2+(-\\mu_6+\\mu_5c_0-tc_0^2)z - {} \\\\\n      \\mu_7+\\mu_6c_0-\\mu_5c_0^2+tc_0^3\n    \\end{smallmatrix}\\\\ c_0+z&-a_0\n  \\end{pmatrix}:=\\begin{pmatrix}\n    a_t(\\zeta_{k,t})&b_t(\\zeta_{k,t})\\\\c_t((\\zeta_{k,t}))&-a_t(\\zeta_{k,t})\n  \\end{pmatrix}\\,\\mathrm{d}\\zeta_{k,t},\\]\n  where \\begin{align*}\n    b_t(\\zeta_{k,t})&=\\frac{\\mathrm{d}z}{\\mathrm{d}\\zeta_{k,t}}(\\zeta_{k,t})\\frac{-t}{z^4}(z^3-c_0z^2+c_0^2z-c_0^3)(1+O(|t|^{-1\/4}))\\\\\n    &=\\frac{\\mathrm{d}z}{\\mathrm{d}\\zeta_{k,t}}(\\zeta_{k,t})\\frac{-t}{z^4}(z-c_0)(z^2+c_0^2)(1+O(|t|^{-1\/4})).\n\\end{align*}\nThen $\\sqrt{b_t(\\zeta_{k,t})}$ is well-defined on $\\widetilde{B}_{k,t}$ by the assumption that $|c_0+z_k(t)|\\leqslant |t|^{-1\/4}\/2$, and we have \\[g^{-1}\\Phi_t g=\\begin{pmatrix}\n  0&1\\\\\\lambda_k(t)^2\\zeta_{k,t}&0\n\\end{pmatrix}\\,\\mathrm{d}\\zeta_{k,t}, \\text{ for }g=\\frac{1}{\\sqrt{b_t(\\zeta_{k,t})}}\\begin{pmatrix}\n  b_t(\\zeta_{k,t})&0\\\\-a_t(\\zeta_{k,t})&1\n\\end{pmatrix}.\\]\nThe remaining proof is the same. The approximate metric $h_t^{\\mathrm{app}}$ is constructed as in Definition \\ref{ApproxMet_def}, and the proof of the error estimate in \\eqref{ErrorEst_eq} works uniformly. Then there are uniform constants $t_0,c,c'$, such that for any $|t|\\geqslant t_0$,\n\\[\n\\left\\lVert F_{h_t^\\mathrm{app}}+\\left[\\Phi_t,\\Phi_t^{\\ast_{h_t^\\mathrm{app}}}\\right] \\right\\rVert_{L^2}\\leqslant c\\mathrm{e}^{-c' |t|^{3\/4}}.\n\\]\nThe computations in Sections 4, 5, 6 only use the expression of $q_t=\\det\\,\\Phi_t$ and the local forms of $\\Phi_t$ provided by Proposition \\ref{NormalForm_prop}, so the estimates remain valid uniformly for the choice of $\\Phi_t$ in $H^{-1}(t)$. Moreover, the constants can also be chosen to be independent of $\\mathrm{Arg}(t)$. Therefore, we find uniform constants $t_0,c,c'$ so that for $[(\\dot{\\eta},\\dot{\\Phi})]\\in T_{[(\\bar{\\partial}_E,\\Phi_t)]}\\mathcal{M}$ and $|t|\\geqslant t_0$,\n  \\[|\\lVert [(\\dot{\\eta},\\dot{\\Phi})]\\rVert_{g_{L^2}}\/\\lVert [(\\dot{\\eta},\\dot{\\Phi})]\\rVert_{g_{\\mathrm{sf}}}-1|\\leqslant c\\mathrm{e}^{-c'|t|^{3\/4}}.\\]\n\n\\bibliographystyle{amsalpha}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}