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+{"text":"\\section{INTRODUCTION}\n\nAn experimental study of relativistic heavy ion collisions augments\nour understanding of the QCD phase diagram~\\cite{cite:QCD_Diagram}.\nThe high energy density reached in such collisions at the Relativistic\nHeavy Ion Collider (RHIC) is believed to result in a novel state of\nhot and dense matter with properties strikingly different from that\nof a hadron gas or ordinary nuclear matter~\\cite{cite:STARWitePaper}.  \n\nThe bulk properties of particle production are studied using\nidentified particle spectra at low momentum. Model-dependent\ninterpretations of the measured data provide insight into the\ncomplex dynamics of the collision and further explore the QCD\nphases through which the collision evolves.  The dense system\nformed in the early stages of the collision continuously\nexpands and cools, until kinetic freeze-out, beyond which the\nparticles stream freely into the detector.  Through\nmeasurements of species abundances and transverse momentum\ndistributions, information about the final stages of the\ncollision evolution at  chemical and kinetic freeze-out can\nbe inferred.\n\nThe relative particle abundances and spectral shapes discussed\nhere were tested within the frameworks of statistical (chemical\nequilibrium)~\\cite{cite:StatModel} and\nBlast-wave~\\cite{cite:BlastWaveModel} models.  In the chemical\nequilibrium model, particle abundances relative to the total system\nvolume (assumed to be the same for all particle species) are\ndescribed by system temperature at freeze-out,  the baryon and\nstrangeness chemical potentials, and the strangeness\nsuppression factor. The Blast-wave model describes spectral shapes\nassuming a locally thermalized source with a common  transverse\nflow velocity field.  The results from Au+Au collisions at\n\\snn=200 and at\n62.4~GeV~\\cite{cite:200spectra,cite:62spectra,cite:LongPRC} have\nshown that the chemical freeze-out temperature, $T_{\\rm ch}$, has\nlittle dependence on centrality whereas the kinetic freeze-out\ntemperature, $T_{\\rm kin}$, decreases with increasing centrality of the\ncollision. Further, the radial flow velocity, $\\beta$, increases\nwith increasing centrality.  The observed changes in $T_{\\rm kin}$ and\n$\\beta$ with  centrality are consistent with higher energy and\npressure in the initial state for more central events.  On the\nother hand, the centrality independence of the extracted chemical\nfreeze-out temperature indicates that even for different initial\nconditions, collisions always evolve to the same chemical\nfreeze-out. Moreover, the value for the chemical freeze-out\ntemperature in Au+Au is close to the critical temperature,\npredicted by some lattice QCD calculations~\\cite{Tcrit}.  This\nsuggests that chemical freeze-out  coincides with hadronization\nand therefore  $T_{\\rm ch}$  provides a lower limit estimate\nfor the temperature of the prehadronic state~\\cite{cite:Olgaposter}.\nThe systematic behavior of the kinetic freeze-out properties with\ncharged hadron multiplicity appears to follow the same trend for\nall energies and systems at RHIC~\\cite{cite:200spectra,cite:62spectra}.\nIn this paper, the systematic studies of the QCD phase diagram\nfrom heavy ion collisions are enriched by the addition of new\nRHIC data from Cu+Cu collisions at \\snn=200 and 62.4~GeV.  \n\n\\section{THE STAR EXPERIMENT}\nThe Cu+Cu data presented here were collected by the STAR experiment\nduring the RHIC 2005 run.  Copper nuclei ($^{63}$Cu) were collided\nat \\snn=200 and 62.4~GeV.  Data were recorded with a minimum bias\ntrigger obtained from  the Beam-Beam Cherenkov counters~\\cite{cite:BBCs}\ncoupled with information from the Zero-Degree Calorimeters~\\cite{cite:ZDCs}.\nThis trigger is found to be sensitive to the top $\\sim$85\\% of\nthe inelastic cross-section.  The data studied here correspond to\nthe top 60\\% of the inelastic cross-section (minimum bias) where\nlittle or no inefficiency of the triggering or vertex reconstruction\nis found.  These 0-60\\% minimum bias events, with 24~M and 10~M\nevents recorded at 200 and 62.4~GeV, respectively,  were divided\ninto six centrality bins each corresponding to a 10\\% interval of\nthe geometric cross section.  Transverse mass distributions for\ncharged pions, charged kaons, protons and antiprotons, previously\nreported by STAR for \\snn=200 GeV $pp$ collisions and Au+Au\ncollisions  at 62.4 and\n200~GeV~\\cite{cite:200spectra,cite:62spectra,cite:LongPRC}, are used\nfor comparison.\n\n\\begin{figure*}[!ht]\n\\includegraphics[width=0.475\\textwidth]{DrawDedxvsp}\n\\includegraphics[width=0.475\\textwidth]{DrawZ_pi}\n\\caption{\\label{fig:dedx}  The left panel shows the truncated mean\nionization energy loss ($\\langle dE\/dx \\rangle$) in the TPC as a \nfunction of transverse momentum for positively charged tracks from\n200~GeV Cu+Cu collisions.  The right panel shows $Z(\\pi)$, the\nlogarithm of the measured $\\langle dE\/dx \\rangle$ divided by the\ntheoretical expectation for energy loss of charged pions, for\n$0.40 < \\pT < 0.55$~GeV\/$c$.  Also shown is an example four-Gaussian\nfit that is used to extract the raw yields for different species.}\n\\end{figure*}\n\n\\begin{figure}[!ht]\n\\includegraphics[width=0.475\\textwidth]{pBg_Central_200}\n\\vspace*{-0.3cm}\n\\caption{\\label{fig:pBack} Estimated fraction of background protons in\nthe raw proton sample as function of transverse momentum, for the most\ncentral \\snn=200~GeV Cu+Cu collisions.  No strong centrality or energy\ndependence for this correction was observed for all Cu+Cu data available.}\n\\end{figure}\n\n\\begin{figure*}[!ht]\n\\includegraphics[width=0.95\\textwidth]{MinusSpectraCuCu200_and62_SplitPanels.eps}\n\\vspace{-0.5cm}\n\\caption{\\label{fig:SpectraMinus}The top row shows negatively charged pion\n(leftmost column), kaon (center) and anti-proton (right) spectra from Cu+Cu\ncollisions at \\snn=200~GeV.  Six centrality classes are shown as dark\n(central 0-10\\%) to light (50-60\\%) shades.  \nSpectra for 62.4~GeV Cu+Cu are shown on the bottom row.  Statistical and\nsystematic errors (which do not exceed 7\\%) are smaller than the symbol size.}\n\\end{figure*}\n\n\\begin{figure*}[!ht]\n\\includegraphics[width=0.95\\textwidth]{PlusSpectraCuCu200_and62_SplitPanels.eps}\n\\vspace{-0.8cm}\n\\caption{\\label{fig:SpectraPlus}The top row shows positively charged pion\n(leftmost column), kaon (center) and proton (right) spectra from Cu+Cu\ncollisions at \\snn=200~GeV.  Six centrality classes are shown as dark\n(central 0-10\\%) to light (50-60\\%) shades.  \nSpectra for 62.4~GeV Cu+Cu are shown on the bottom row.  Statistical and\nsystematic errors (which do not exceed 7\\%) are smaller than the symbol size.}\n\\end{figure*}\n \n\\begin{figure*}[!ht]\n\\includegraphics[width=0.98\\textwidth]{DrawDiffFits2_Plus200_SplitPanels.eps}\n\\caption{\\label{fig:SpecFits}\nComparison between the 10\\% most central Cu+Cu collision data (symbols)\nat \\snn=200~GeV  and the corresponding Blast-wave model fit (dashed\nline) to $\\pi^{+}$ (left), K$^{+}$ (center) and proton (right) spectra\n-- note fit is performed simultaneously across species.  The pion data\npoints below \\pT~=~0.5~GeV\/$c$ were not included in the Blast-wave fits\nto reduce the effect of resonance decays.\nA Bose-Einstein fit  to the pion spectra over the entire\nfiducial range is also shown.  The lower panels illustrate the quality\nof the fits by showing the difference between the measured points and\nthe fit expressed as the number of standard deviations.}\n\\end{figure*} \n\nThe STAR Time Projection Chamber (TPC)~\\cite{cite:STAR_TPC} tracks\nparticle trajectories over a wide range of momentum at mid-rapidity\n($|\\eta| < 1.8$).  The particle identification at low-\\pT~uses\nmeasurements of truncated mean ionization energy loss,\n$\\langle dE\/dx \\rangle$, of the charged particles traversing the\nTPC.  Particles of different mass show distinct patterns in the\n$\\langle dE\/dx \\rangle$ dependence, as shown in Fig.~\\ref{fig:dedx},\nleft panel.  This allows statistical separation of pions and kaons\nin the momentum range $0.25 < \\pT < 0.80$~GeV\/{\\em c} at mid-rapidity\n($|y| < 0.1$), and of protons and anti-protons from other species in\nthe range $0.40 < \\pT < 1.20$~GeV\/{\\em c}.\n\nThe momentum measurement is given by the curvature of the particle\ntrajectories as they pass through the 0.5~T magnetic field of the\nSTAR detector.  \nTo ensure optimal $dE\/dx$ resolution, only primary tracks, with {\\em dca}\n(distance of closest approach between the particle trajectory and the\nevent vertex) less than 3~cm, and at least 25 out of 45 possible fit\npoints are used in this analysis.\nParticle identification  at mid-rapidity ($|y| < 0.1$)\nis achieved by fits to the $Z$-variable, defined as a logarithm of\n$\\langle dE\/dx \\rangle$ divided by the theoretically expected value\nfor each particle type, given by Bethe-Bloch formula~\\cite{cite:BB}.\nThis new variable is introduced to remove the\nstrong \\pT~dependence at low momenta.  Such a normalized distribution\nis created for a given particle and centrality and is divided into\nnarrow transverse momentum slices (width $\\Delta$\\pT~=~50~MeV\/{\\em c}).\nThese momentum projections are fit with a combined four-Gaussian\nfunction, one for each of the particle species of a given charge:\n$\\pi$, K, $p$ and $e$.  \nThe integral of each Gaussian provides the raw yield at each momentum.\nThis procedure is repeated for each particle species in order to assign\nthe correct rapidity for each track, using the mass of the particle.\nThus, fits to the auxiliary particles in each distribution (for example\nK, $p$ and $e$ for $\\pi$ analysis) are used only to estimate the\ncontamination when bands  overlap. The right panel of Fig.~\\ref{fig:dedx}\nshows an example for pion yield extraction for one momentum slice. For\nmore details see Ref.~\\cite{cite:LongPRC}.\n\nThe raw yields extracted from each of the four-Gaussian fits are then\ncorrected for detector acceptance, single-track reconstruction\nefficiencies, and other effects as discussed below. To determine the\ncorrection factors, simulated tracks were embedded into real data on\nthe raw signal level and run through the standard reconstruction chain.\nThe estimated single-track reconstruction efficiency is about $80\\%$\nfor $\\pi^{\\pm}$ in Cu+Cu collisions and exhibits a small centrality\nand $p_T$ dependence.  The \\pT-spectrum has also been corrected for the\nenergy loss\nby multiple scattering beyond that for pions which is calculated\nduring reconstruction.  This affects the reconstructed momentum at\nlow values. The maximum value of this  mass-dependent correction to\nthe measured \\pT~value for ${\\rm K}^{\\pm}$ and $p$($\\overline{p}$) was found\nto be $2\\%$ and $3\\%$, respectively, for the lowest measured \\pT~bin.  \nAn additional correction for the background contamination in the proton\nsample is made.  The background protons arise predominantly from\nsecondary interactions in the beam pipe and detector material\n(knock-out protons). It is estimated from data to be about 40\\% at\n\\pT~=~400~MeV\/{\\em c}, diminishing to near zero at \\pT~=~1~GeV\/{\\em c},\nas shown in Fig.~\\ref{fig:pBack}.\nTo estimate this correction factor we compare the distribution of proton\n{\\em dca}\nto that of the anti-protons (see\nRef.~\\cite{cite:LongPRC} for more details).  In the measured {\\em dca}\nregion, the integral difference between protons and anti-protons (after\nnormalization by the anti-proton to proton ratio) is considered to be\nthe background contribution to the proton yield.\nPion yields are additionally corrected for feed-down contributions from\nweakly decaying particles, muon contamination, and background pions\nfrom detector material. This correction is found to decrease from about\n$15\\%$ at 0.3~GeV\/$c$ to about $5\\%$ at 1~GeV\/{\\em c}.\nThe (anti)protons presented in this paper are inclusive measurements\n(not corrected for weak decays). It has been found in previous studies\nthat the analysis cuts used for the low-\\pT~identified proton studies\n($dca < 3$~cm) reject only a negligible fraction of daughter protons from\nthe hyperon decays~\\cite{cite:200spectra}. Therefore, our  sample \nreflects the total baryon production in the collision. Earlier Au+Au\nstudies~\\cite{cite:200spectra} and preliminary Lambda-hyperon spectra\nfrom Cu+Cu collisions~\\cite{cite:AntStrange} indicate that the\nfreeze-out spectral shapes are similar for $\\Lambda$s and protons,\nresulting in similar spectra shapes for primary and feed-down protons.\nThe fraction of the weak-decay feed-down protons is estimated to be\nabout $30\\%$~\\cite{cite:JuneWWND08}.\n\nThis analysis technique is used to obtain the low-\\pT~particle spectra\nfor all centrality bins at both 200 and 62.4~GeV center-of-mass\nenergies and for the Cu+Cu and Au+Au colliding systems.  Additional\ntechnical details on  the analysis  and applied corrections can be\nfound in Refs.~\\cite{cite:200spectra,cite:62spectra} with a thorough\noverview in Ref.~\\cite{cite:LongPRC}.\n\n\n\\section{RESULTS}\n\nThe transverse momentum spectra are shown in\nFigs.~\\ref{fig:SpectraMinus}~and~\\ref{fig:SpectraPlus} for $\\pi^{\\pm}$\n(leftmost column), K$^{\\pm}$ (center) and (anti)protons (right) in\nCu+Cu collisions. The top row presents the data for \\snn=200~GeV, whilst \ndata for 62.4~GeV are shown in the bottom row.  The symbol shades represent\ndifferent centrality bins.  The particle and anti-particle spectral\nshapes are similar for all species in each centrality bin. At both\ncollision energies a mass dependence is observed in the slope of the\nparticle spectra.  Due to the large number of events recorded and good\ntracking efficiency, the statistical errors are less than 1\\%.  The\nsystematic uncertainties are similar to those determined in prior\nanalyses of low-\\pT~spectra in Au+Au collisions~\\cite{cite:LongPRC}.\nSystematic errors are divided into two classes: point-to-point and\nscale uncertainties.  The overall scale uncertainty, mostly due to the\nembedding procedure for the single-track reconstruction efficiency, is\nestimated to be 5\\%  for all particle species.  Point-to-point\nuncertainties are determined for each \\pT~bin and particle species.\nFor pions and kaons, this error is evaluated to be less than 7\\% and\n13\\%, respectively.  These maximal errors represent \\pT~bins where a\nsignificant $\\langle dE\/dx \\rangle$ overlap occurs between $\\pi^{\\pm}$,\nK$^{\\pm}$ or $e^{\\pm}$.  For protons and anti-protons, the maximum\nerror is 5\\%.  At low-\\pT, the proton uncertainty is greater than that\nfor anti-protons (4.0\\% versus 1.3\\%, respectively, at \n\\pT~=~400-450~MeV\/$c$) owing to the additional uncertainty from the\nproton's background.  The uncertainty due to the background decreases\nrapidly from 3.7\\% at \\pT~=~400-450~MeV\/$c$ to 1.5\\% for $\\pT > 1$~GeV\/$c$.\n\nFor the anti-particle to particle yield ratios, systematic errors are\nmuch reduced due to a cancellation of the efficiency uncertainties\nand a partial cancellation of extrapolation uncertainties, as described\nabove.  A  systematic uncertainty of 2\\%, 3\\%, and 5\\% is assigned to\n$\\pi^{-}\/\\pi^{+}$, K$^{-}$\/K$^{+}$, $\\overline{p}\/p$, respectively.\n\nWe further fit the obtained \\pT~distributions  to extract system\nproperties at different stages of the collision evolution.  The first\nfit to the data probes collision properties at kinetic freeze-out.\nHere, a Blast-wave model~\\cite{cite:BlastWaveModel} is used to\nsimultaneously fit the $\\pi^{\\pm}$, K$^{\\pm}$ and (anti)proton spectra\nat a given centrality.  This fit provides a good description of the\nspectra shapes, as illustrated in Fig.~\\ref{fig:SpecFits} with results\nfrom most central 200~GeV Cu+Cu data.  The $\\pi^{\\pm}$ data points for\n$\\pT < 0.5$~GeV\/$c$ are excluded from the Blast-wave fits to reduce the\neffects of resonance decay contributions as done in previous\nworks~\\cite{cite:200spectra,cite:62spectra,cite:LongPRC}.  Including\nthis  low-\\pT~region in the fit leads to a poorer description of proton\nand kaon shapes, however the resultant modification of the extracted\nparameters remains well within  their systematic uncertainty.  The\nfreeze-out parameters obtained from this  model are discussed later.\nAlso shown in Fig.~\\ref{fig:SpecFits} are Bose-Einstein\n($\\propto 1\/(\\exp{\\frac{\\mT}{T}}-1)$) fits to the $\\pi^{\\pm}$, which\nprovide a slightly better description of these data. For evaluation\nof the systematic uncertainties from extrapolation, $m_{T}$-exponential\n($\\propto 1\/\\exp{\\frac{\\mT}{T}}$) and Boltzmann\n($\\propto \\mT\/\\exp{\\frac{\\mT}{T}}$) fits are also used in the\nanalysis (for more details see Ref.~\\cite{cite:LongPRC}).\n\nThe particle mean-\\pT~and total particle yields at mid-rapidity\n($|y| < 0.1$) are shown in Fig.~\\ref{fig:MeanPt} and Fig.~\\ref{fig:dN\/dy},\nrespectively. The values presented for kaons and (anti)protons are\ndetermined from the measured spectra points, extrapolated outside the\nfiducial range using Blast-wave fits discussed above. Similarly, a\ncombination of the measured data-points and extrapolation from\nBose-Einstein fits is used for the pions.  The measured fraction of\nthe total yield is found to be 62\\% for $\\pi^{\\pm}$, 58\\% for K$^{\\pm}$\nand 65\\% for (anti)protons  for the most central 200 GeV data; these\nfractions are slightly higher in other centrality bins and at lower\nenergy~\\cite{cite:STAR9GeV}.  The systematic uncertainty on $dN\/dy$ and\nmean-\\pT, shown in\nthe figures, includes the extrapolation uncertainty evaluated by means\nof various model fits  mentioned earlier. Overall, these are estimated\nto be near 15\\% of the yields outside the fiducial range for pions and\nkaons and 15\\%-25\\% of the extrapolated yields for protons and\nanti-protons, depending on centrality.\n\nWe also determine the total charged hadron production per unit of\npseudo-rapidity, \\nch, at mid-rapidity.  The total particle yield at\nmid-rapidity for each species, obtained by extrapolating the fits to\nthe measured spectrum in the momentum range outside our fiducial\ncoverage,  was corrected for the Jacobian transformation\n\\nchy$\\rightarrow$\\nch. The sum of the total charged pion, kaon and\n(anti)proton yields was then corrected for  the feed-down of weakly\ndecaying neutral strange particles, providing the estimate of\nprimordial charged hadron yield at mid-rapidity.  A complementary\nmethod was also used,  integrating over the charged hadron spectra\ncorrected for efficiency, feed-down and the Jacobian transformation,\nand yielded consistent results.\n\nThe mean-\\pT~of each particle species ($\\pi$, K, $p$) increases with\nthe number of charged hadrons at mid-rapidity \\nch,  as shown in\nFig.~\\ref{fig:MeanPt}.  Moreover, the mean-\\pT~for each particle\nspecies appears to scale with \\nch~at mid-rapidity, and to be\nindependent of the colliding system and the center-of-mass energy.  \nThe particle yields show the same systematic scaling features with\n\\nch~as mean-\\pT~across system and collision energy, see\nFig.~\\ref{fig:dN\/dy}.  In this logarithmic representation the particle\nyields for each species appear to increase linearly with multiplicity,\nwith Cu+Cu matching the Au+Au data at similar values of \\nch. When\nshown on a linear scale, the integrated yields exhibit a near-linear\ndependence with \\nch.  The logarithmic scale for both axes used here\npreserves the apparent linear dependencies whilst better illustrating\nthe lower multiplicity Cu+Cu data.  For the detailed features we\ninvestigate the relative particle production  in the following.\n\n\\begin{figure}[!t]\n\\includegraphics[width=0.475\\textwidth]{DrawMeanPt_OnePanel.eps}\n\\caption{\\label{fig:MeanPt} Mean transverse momentum as a function of\ncharged hadron multiplicity at mid-rapidity for pions, kaons and\nanti-protons.  For comparison, the mean-\\pT~values for Au+Au data are\nshown as bands.  Open (closed) symbols\/bands depict data at\n\\snn=200~GeV (62.4~GeV).  Error-bars represent statistical and systematic\nuncertainties added in quadrature. }\n\\end{figure}\n\n\\begin{figure}[!t]\n\\includegraphics[width=0.475\\textwidth]{DrawYield_ThreePanels_Log.eps}\n\\caption{\\label{fig:dN\/dy} Integrated yields at mid-rapidity for pions, kaons and\nanti-protons  as a function of the charged particle density (\\nch),\nwhich is used as a measure of the centrality.  For comparison Au+Au\ndata are shown as bands.  Filled (open) points\/bands depict data at\n\\snn=200~GeV (62.4~GeV).  Error-bars represent statistical and systematic\nuncertainties added in quadrature.}  \n\\end{figure}\n\n\\begin{figure}[!t]\n\\includegraphics[width=0.475\\textwidth]{DrawKtoPiandPtoPiMultiPanel2.eps}\n\\caption{\\label{fig:Ratios} Integrated particle yield ratios at\n\\snn=200~GeV (closed symbols) and 62.4~GeV (open) for Cu+Cu (black)\nand Au+Au collisions (grey bands) versus \\nch~at mid-rapidity. Error-bars\nrepresent statistical and systematic uncertainties added in quadrature.}\n\\end{figure}\n\nThe relative abundances of particles provide an important insight\ninto the chemical properties of the system.  The relative kaon yield\nreflects the strangeness production  in the collision, whereas proton\nwith respect to pion production is dependent on the baryon production\nand transport.  Figure~\\ref{fig:Ratios}($a$) shows the ratios for the\nnegatively charged particles, $\\overline{p}\/\\pi^{-}$ and ${\\rm K}^{-}\/\\pi^{-}$,\nas a function of \\nch, which exhibit similar \\nch-scaling behavior at\neach collision energy.  \nThe slight decrease of the values for both ratios seen at the\nlower collision energy of\n 62.4~GeV is insignificant within\nexperimental uncertainties.  Figure~\\ref{fig:Ratios}($b$) shows the\nratios for positively charged particles, $p\/\\pi^{+}$ and ${\\rm K}^{+}\/\\pi^{+}$,\nwhich also exhibit a \\nch-scaling behavior within the same collision\nenergy. The beam-energy effect is reversed here as compared to the ratio\nof negatively charged particles. Summing over the two charges\n(Fig.~\\ref{fig:Ratios}($c$)), the corresponding ratios exhibit a common\nscaling behavior with \\nch, independent of colliding system and collision\nenergy. The energy dependence of the positive and negative particle\nratios considered separately, points to the effects of baryon transport\nto mid-rapidity, which decreases with increasing energy (see also~\\cite{cite:STAR9GeV}).\n\nWe further explore the kaon production in Cu+Cu collisions to gain\nbetter insight into production of strange quarks.  The ${\\rm K}^{-}\/\\pi^{-}$\nratio scales with \\nch~at both energies and there is no hint of an\nadditional strangeness enhancement of charged kaons in the smaller\nCu+Cu system compared to the larger Au+Au system. Early works from\nSPS energies reported such additional relative strangeness\nenhancement in the ${\\rm K}\/\\pi$ ratio for smaller systems, although no\nfinal confirmation of this observation is\navailable~\\cite{cite:SPS_NA49QM02,cite:SPS_NA49PRL}.  The pion and\nkaon enhancement factors are compared in Fig.~\\ref{fig:Enhancement}.\nThis factor is defined as the yield per mean number of participating\nnucleons (estimated using a Glauber model), \\npart,  in heavy-ion\ncollisions divided by the respective value in \\pp~collisions. A\nprogressive enhancement of kaon production with respect to pions as\na function of collision centrality is evident, as shown earlier by\nthe K\/$\\pi$ ratios (Fig.~\\ref{fig:Ratios}($a$) and\nFig.~\\ref{fig:Ratios}($b$)). A comparison of these enhancement\nfactors between Cu+Cu and Au+Au data is also shown. The enhancement\nfactors for kaons do not show universal scaling features with\nrespect to \\npart, and are indeed found to be higher  in Cu+Cu\ncollisions compared to the Au+Au system. However, these features do\nnot appear to be unique to kaons. A similar trend is observed in\nFig.~\\ref{fig:Enhancement} in the\npion enhancement factors for the two systems. This suggests that the\nadditional enhancement, seen in the charged kaon  yields, is not\nrelated to strangeness production, but other physics mechanisms, for\nexample, additional entropy production. It should be noted that while\ncomparing more spherical central Cu+Cu collisions with semi-peripheral\nAu+Au collisions, the initial conditions may not be reflected by\n\\npart~alone.  \n\n\\begin{figure*}[t]\n\\includegraphics[width=0.85\\textwidth]{DrawYield_per_pp.eps}\n\\caption{\\label{fig:Enhancement} Enhancement factors for negatively\ncharged pions (left), kaons (center), and anti-protons (right) as\nfunction of \\npart~in \\snn=200  Cu+Cu and Au+Au collisions.\nError-bars represent statistical and systematic uncertainties on the\nA+A measurements added in quadrature. The shaded bands depict model\nuncertainties on number of participants calculation. The bands on the\nleft show uncertainties from the $pp$ measurements that are correlated\nfor all data points. }\n\\end{figure*}\n\n\nIn contrast to pions and kaons, protons show minimal evolution with\ncentrality and no difference between Cu+Cu and Au+Au systems is\nobserved. Figure~\\ref{fig:Ratios}($d$) illustrates the difference in\nanti-proton and proton production across energies.  As observed in\nother collision systems, the ratio is found to increase and becomes\ncloser to unity for higher energy\ncollisions~\\cite{cite:PHOBOS_ppPartRatios}.  The anti-proton to\nproton ratio gives information on the amount of baryon transport.\nIn line with the earlier STAR results, our measurements indicate\nthat while a finite excess of baryons over anti-baryons is still\npresent at RHIC energies, $p-\\bar{p}$ pair production becomes an\nimportant factor.  Little or no change due to an increase in the\nsystem size (centrality) is apparent in the Cu+Cu data at 200~GeV,\nwhile 62.4~GeV data show a decreasing trend with increasing centrality\nfor this ratio  for both Cu+Cu and Au+Au data.\n\n\\section{FREEZE-OUT PROPERTIES}\n\nThe particle yields and their ratios provide further information on the\nthermal properties of the system at kinetic and chemical freeze-out.\n\n\\subsection{Kinetic Properties}\nThe completion of all elastic scattering marks the final stage of\ncollision evolution and could be interpreted as a kinetic freeze-out,\nwhere the particle momentum spectra are fixed.  To quantify this\nstage, fits are made simultaneously to the spectra of all particle\nspecies, but independently for each centrality class (see\nFig.~\\ref{fig:SpecFits} for example).  The fits used here are based\non the previously discussed Blast-wave model~\\cite{cite:BlastWaveModel},\nwhich assumes a radially boosted thermal source.  These\nhydrodynamically-motivated fits describe the mass dependence of\nparticle spectral shapes in terms of the radial flow velocity ($\\beta$),\nthe kinetic freeze-out temperature ($T_{\\rm kin}$) and the flow velocity\nprofile exponent ($n$) at the final freeze-out.  The extracted value for\n$n$ is not used to derive any physics interpretation.  The effects from\nresonance contributions to the pion spectral shape are reduced by\nexcluding the  low-\\pT~data points (below 0.5~GeV\/{\\em c}).  To enable a\ncomparison with earlier results on \\pp~and Au+Au\ncollisions~\\cite{cite:200spectra,cite:LongPRC}, the same model and the\nsame procedures for the fits are adopted, thereby avoiding any possible\nsystematic bias.\n\n\\begin{figure*}[t]\n\\includegraphics[width=0.95\\textwidth]{DrawKineticFreezeOut_3Panel.eps}\n\\caption{\\label{fig:KineticFO} Comparison of kinetic freeze-out\nproperties obtained from fits to Cu+Cu (symbols) and Au+Au (bands)\ncollision data at \\snn=200 (closed symbols\/bands) and 62.4~GeV (open).\nThe kinetic freeze-out temperature, $T_{\\rm kin}$, is shown versus flow\nvelocity, $\\beta$, and multiplicity in panels (a) and (b) respectively\n(more central collisions are to the right side of each plot).\nPanel (b) also shows the multiplicity dependence of the chemical\nfreeze-out temperature, $T_{\\rm ch}$, (square symbols).  Panel (c) shows\nthe multiplicity dependence of the average radial flow velocity.}\n\\end{figure*}\n\n\nThe Blast-wave fit results for the temperature of freeze-out are shown in\nFig.~\\ref{fig:KineticFO}.  $T_{\\rm kin}$ and $\\beta$ show similar\ndependences as a function of \\nch~in both Cu+Cu and Au+Au collisions,\nevolving smoothly from the lowest to the highest multiplicity, from \\pp~to\ncentral Au+Au.  $T_{\\rm kin}$ decreases smoothly with centrality implying\nthat freeze-out occurs at a lower temperature in more central collisions.  \nThe similarity of kinetic freeze-out parameters in the events with similar\nmultiplicity from different colliding species is confirmed by the data alone.\nAs noted earlier, the particle mean-\\pT~increases with increasing \\nch,\nwhich is consistent with an increase of radial flow with centrality. \nWe note, however, that other physics mechanisms, for example, hard and\nsemi-hard scatterings, can contribute to higher mean-\\pT~values observed\nfor kaon and proton spectra~\\cite{cite:Trainor}.  Direct spectral shape\ncomparisons of Cu+Cu and Au+Au events from similar multiplicity bins, shown\nin Fig.~\\ref{fig:CompareSpectra}, show the same \\pT-dependencies between\npion spectra from the two systems. The same is seen to hold for the\nrespective kaon and proton spectra. The middle panel of Fig.~\\ref{fig:KineticFO}\nshows in addition the chemical\nfreeze-out temperatures for different\ncolliding systems at different energies. Both the chemical freeze-out\nand the kinetic freeze-out temperature show similar scaling features,\nreflecting the common trends in mean-\\pT~and the ratios of $p\/\\pi$ and\n${\\rm K}\/\\pi$, discussed earlier. Similarly, on the left panel of\nFig.~\\ref{fig:KineticFO} we observe a\ncommon \\nch-dependence for the average radial flow velocity at kinetic\nfreeze-out. \n\nA more important observation is that the obtained kinetic freeze-out\nparameters for pions, kaons and (anti)protons follow the same trends\nwith \\nch, independent of collision energy, even though the production\ncross-sections of the underlying spectra are different.  This observation\nsuggests that the kinetic freeze-out properties are determined by the\ninitial state.  Furthermore, a model-dependent connection between the\nnumbers of produced charged particles and the initial gluon density of\nthe colliding system~\\cite{cite:CGC} can be used to deduce that the\nfreeze-out properties are most probably determined at the initial\nstages of the collision and are driven by the initial energy density.\n\n\\begin{figure*}[t]\n\\includegraphics[width=0.95\\textwidth]{MinusSpectraComp_SplitPanelsLin.eps}\n\\caption{\\label{fig:CompareSpectra}\nComparison of the spectral shape between Cu+Cu and Au+Au data at\n\\snn=200~GeV.  Centrality classes are chosen with a similar average\ncharged hadron multiplicity at mid-rapidity.  Pion (left), kaon\n(center) and anti-proton (right) spectra are shown for 10-20\\% central\n(40-50\\%) Cu+Cu (symbols) compared to 40-50\\% mid-peripheral (60-70\\%)\nAu+Au (lines).}\n\\end{figure*}\n\n\\subsection{Chemical Properties}\n\n\\begin{figure}[!ht]\n\\includegraphics[width=0.475\\textwidth]{ratiosChem2}\n\\caption{\\label{fig:ChemicalFit200}\nThe upper panel shows statistical model fit predictions (grey lines)\nfor the measured particle ratios (circles) from central 200~GeV Cu+Cu\ncollisions.  The lower panel illustrates the fit quality by showing\nthe difference between the measured data and the model prediction in\nterms of the number of standard deviations ($N_{\\sigma}$) determined\nby systematic (data) uncertainty.}\n\\end{figure}\n\n\\begin{figure}[ht]\n\\vspace{0.5cm}\n\\includegraphics[width=0.475\\textwidth]{DrawChemicalFreezeOutMuBNch.eps}\n\\caption{\\label{fig:ChemicalFO2} Baryon and strangeness chemical potentials, $\\mu_{\\rm B}$ and $\\mu_{\\rm S}$,\n as a function of \\nch~for 200\nand 62.4~GeV in Cu+Cu (symbols) and Au+Au collisions (bands). }\n\\end{figure}\n\n\\begin{figure*}[t]\n\\includegraphics[width=0.95\\textwidth]{DrawChemicalFreezeOut.eps}\n\\caption{\\label{fig:ChemicalFO} Left: chemical freeze-out temperature,\n$T_{\\rm ch}$, as function of the baryon-chemical potential, $\\mu_{\\rm B}$,\nderived for central Au+Au (0-5\\% for 200 and 62.4 GeV~\\cite{cite:LongPRC} \nand 0-10\\% for 9.2 GeV~\\cite{cite:STAR9GeV}) and Cu+Cu (0-10\\%) collisions.\nFor comparison, results for minimum bias \\pp~collisions at 200~GeV are also\nshown along with additional heavy-ion data points compiled for lower\ncollision energies~\\cite{cite:Claymans}.  The dashed line represents\na common fit to all\navailable heavy-ion data described in the text. Right: strangeness\nsuppression factor, $\\gamma_{\\rm S}$, as a function of \\nch~for 200\nand 62.4~GeV in Cu+Cu (symbols) and Au+Au collisions (bands). }\n\\end{figure*}\n\n\n\nChemical freeze-out occurs at the stage of the collision when all\ninelastic interactions cease and the produced particle composition in\nterms of yields is fixed.  Valuable information for this collision\nstage can be obtained directly from the experimental results by\nforming particle ratios and comparing them across different collision\nsystems and energies.\n\nThe ratios of the different particle yields in Cu+Cu collisions\nare further analyzed within the framework of the statistical\nmodel~\\cite{cite:StatModel}.  This model describes the chemical\nfreeze-out of the colliding system by  several fit\nparameters: the {\\em temperature} at which freeze-out occurs\n($T_{\\rm ch}$), the {\\em cost} of producing matter in terms of\nbaryon and strangeness chemical potentials ($\\mu_{\\rm B}$,\n$\\mu_{\\rm S}$), and an additional {\\it ad-hoc} parameter, known as\nthe strangeness suppression  factor, ($\\gamma_{\\rm s}$),  to reconcile the\nlower yield of strange hadrons in collisions\ninvolving smaller species (for example \\pp~and d+Au).\n\nThese statistical fits are performed on the relative particle\nabundances from $\\pi^{\\pm}$, K$^{\\pm}$ and $p$($\\overline{p}$) alone.\nFigure~\\ref{fig:ChemicalFit200} shows an example of the resultant fit\nto the identified hadron  ratios from  central \\snn=200~GeV Cu+Cu\ncollisions.  The lower panel of this figure illustrates the fit\nquality. We note, that the successful description of the ratios by the\nmodel could not prove the attainment of chemical equilibrium, but\nsuggests the statistical nature of particle  production in these\ncollisions~\\cite{cite:prc126}.  The results obtained for the\nfreeze-out parameters are shown in\nFigs.~\\ref{fig:KineticFO}, \\ref{fig:ChemicalFO2}~and~\\ref{fig:ChemicalFO}.  \n\n\nStatistical model fits to a wider variety of hadron yields were also\nattempted using preliminary results for the  $\\Lambda$, $\\Xi$ and\n$\\phi$ particles and anti-particles from 200~GeV Cu+Cu data\nfrom~\\cite{cite:AntStrange}.  Including more particles into the model\nfits reduces the systematic uncertainty on the extracted parameters\nand resulted in parameter values consistent with those  obtained from\nfits using $\\pi^{\\pm}$, K$^{\\pm}$ and $p$($\\overline{p}$) alone\nreported here.  In general, the observed systematic trends in the\nfreeze-out parameters as a function of the collision centrality are\npreserved ~\\cite{cite:LongPRC, cite:STARWitePaper}.\n\nFigure~\\ref{fig:ChemicalFO} (left panel) shows the evolution of the chemical\nfreeze-out temperature versus baryon chemical potential in central\nheavy-ion collisions from the very low energy SIS data through AGS\nand SPS to RHIC (STAR data points only).  The overall evolution of\n$T_{\\rm ch}$ can be reproduced by the phenomenological model\nfit~\\cite{cite:Claymans} applied here to all the data points shown\n(dashed line).  As the collision energy increases,\n the temperature at freeze-out is found to increase up to\n SPS energies.  This\nis followed by a plateau at RHIC energies at a value close to that\nof the hadronization temperature expected from lattice QCD\ncalculations. At RHIC, for all systems and center-of-mass energies,\n$T_{\\rm ch}$ appears to be universal, as shown in\nFig.~\\ref{fig:KineticFO} (middle panel).\n\nThe value of the baryon-chemical potential at a given center-of-mass\nenergy is found to be slightly higher for the larger system, with Au+Au\nand Cu+Cu measurements showing common trends with charged hadron\nmultiplicity (Fig.~\\ref{fig:ChemicalFO2}).  We note that, presented in\nthe same figure, values of strangeness chemical potential are close to\nzero with no obvious systematic trends for all energies and colliding\nsystems studied at RHIC.\nWithin a given system, $\\mu_{\\rm B}$ reflects the decrease in net-baryon\ndensity with increasing collision energy from \\snn=62.4 to 200~GeV.\nThis behavior can be observed directly from the particle ratios,\nwhere $\\overline{p}\/p$ increases as a function of energy\n(Fig.~\\ref{fig:Ratios}).  For the most central Cu+Cu events we measure\n$\\overline{p}\/p = 0.80\\pm0.04$ at 200~GeV, and $0.55\\pm0.03$ at 62.4~GeV.\nThe lack of centrality dependence in the baryon to meson ratios in\nCu+Cu and Au+Au data, points to similar freeze-out temperatures for\nthe studied systems.  The constant values of $T_{\\rm ch}$ at RHIC\nenergies for collisions with different initial conditions, energy and\nnet-baryon density, points to a common hadronization temperature of\nthe systems. \n\nAnother parameter extracted from the fit, which is related to\nstrangeness production, is the strangeness suppression factor,\n$\\gamma_{\\rm s}$, shown versus \\nch~ in Fig.~\\ref{fig:ChemicalFO}.\nThe suppression of strange hadron yields is observed\nin smaller systems, such as \\pp~and peripheral collisions. Within\nstatistical models this can be explained by a reduced production\nvolume~\\cite{cite:StrangenessSupp}.  At low beam energies, where\nequilibration of $s$ quarks with respect to $u$ and $d$ is not\nexpected, the suppression is also seen.  We find that within the\nsystematic errors on the fit parameters the strangeness suppression\nfactor in Cu+Cu is consistent with that for  Au+Au  for the same number\nof charged particles, \\nch.  As only charged kaon yields were included\nin the fit, this observation is directly related to an absence of any\nadditional enhancement in ${\\rm K}\/\\pi$ at the same \\nch~in the smaller Cu+Cu \nsystem with respect to the larger Au+Au system as discussed previously.\n \nThe $\\gamma_{\\rm s}$ parameter shows a similar increase with centrality\nfor both systems and energies.  The value of $\\gamma_{\\rm s}$ approaching\nunity for the central Au+Au collisions in the context of thermal model\nwould imply that the produced strangeness is close to equilibrium.  \n \n\n\n\\section{Summary}\nWe have presented measurements of identified charged hadron spectra\nin Cu+Cu collisions for two center-of-mass energies, 200 and 62.4~GeV.\nThese new results of $\\pi^{\\pm}$, K$^{\\pm}$ and $p$($\\overline{p}$)\nhave further enriched the variety of low-\\pT~spectra at RHIC.  The\ndata have been studied within the statistical hadronization and\nBlast-wave model frameworks in order to characterize the properties\nof the final hadronic state of the colliding system as a function of\nsystem size, collision energy and centrality.\n\nThese multidimensional systematic studies reveal remarkable\nsimilarities between the different colliding systems.  No additional\nenhancement of kaon yields with respect to pions is observed for the\nsmaller Cu+Cu system compared to Au+Au.  The obtained particle ratios,\nmean-\\pT~and the freeze-out parameters, including the strangeness\nsuppression factor, $\\gamma_{\\rm s}$, are found to exhibit a smooth\nevolution with \\nch, and similar properties at the same number of\nproduced charged hadrons are observed for all collision systems and\ncenter-of-mass energies.  The bulk properties studied \nhave a strong correspondence with the total particle yield. Within\nthermal models this reflects a relation between the energy per\nparticle at freeze-out and the entropy derived from particle yields,\nwhich reflects the initial state properties for adiabatic expansion.\nThe baryon chemical potential could in addition be influenced by\nthe initial valence quark distribution and by baryon transport\nduring expansion, leading to a more complicated dependence.  The\nscaling features of freeze-out properties are not presented at the\nsame \\npart~for lighter and heavier ions as scaling is badly\nbroken when data measured at different energies are compared. This\nsuggests that \\npart~does not reflect the initial state of the system\naccurately.\n\nWe thank the RHIC Operations Group and RCF at BNL, the NERSC Center at LBNL and the Open Science Grid consortium for providing resources and support. This work was supported in part by the Offices of NP and HEP within the U.S. DOE Office of Science, the U.S. NSF, the Sloan Foundation, the DFG cluster of excellence `Origin and Structure of the Universe' of Germany, CNRS\/IN2P3, STFC and EPSRC of the United Kingdom, FAPESP CNPq of Brazil, Ministry of Ed. and Sci. of the Russian Federation, NNSFC, CAS, MoST, and MoE of China, GA and MSMT of the Czech Republic, FOM and NWO of the Netherlands, DAE, DST, and CSIR of India, Polish Ministry of Sci. and Higher Ed., Korea Research Foundation, Ministry of Sci., Ed. and Sports of the Rep. Of Croatia, Russian Ministry of Sci. and Tech, and RosAtom of Russia.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Setup}\n\n\\subsection*{Fundamentals of viscous streaming}\nViscous streaming \\cite{holtsmark1954boundary,bertelsen1973nonlinear}, an inertial phenomenon, arises when a small feature (typically a solid inclusion or a microbubble) of characteristic length $a$, is immersed in a liquid of kinematic viscosity $\\nu$, that oscillates with angular frequency $\\omega$ and amplitude $A\\ll a$. Inertial rectification, facilitated by stress concentration at the boundaries of the immersed body, results in steady flows that can be effectively leveraged to manipulate fluid, suspended particles, and chemicals \\cite{lutz2003microfluidics, lutz2006hydrodynamic, lutz2005microscopic, parthasarathy2019streaming, bhosale2021multi}. For simple geometries of uniform curvature (cylinders, spheres), viscous streaming flows have been theoretically, computationally, and experimentally well characterized as function of the Stokes layer thickness $\\delta_\\text{AC}\\sim\\mathcal{O}(\\sqrt{\\nu\/\\omega})$ and length-scale $a$ \\cite{holtsmark1954boundary,riley2001steady,lutz2005microscopic,lane1955acoustical,bhosale_parthasarathy_gazzola_2020}. More recently, the use of multi-curvature streaming bodies has revealed rich flow repertoires, furthering opportunities in transport, separation, and assembly \\cite{parthasarathy2019streaming,bhosale_parthasarathy_gazzola_2020,chan2021three,bhosale2021multi,bhosale2022soft}.\n\n\\subsection*{Biostreamer fabrication and flow imaging integration}\n\nWe aim to build a millimeter-scale, 3D muscle tissue to harvest contractile motions for the generation of oscillatory flows. All the while, the geometry of the tissue itself is selected and leveraged to modulate identifiable streaming fields, facilitate biofabrication, $\\mu$PIV imaging, and flow analysis.\n\nHere, skeletal muscle myoblasts are chosen for their ability to form 3D tissue and fuse into muscle fibers (myotubes) that align spontaneously along stress fields, thus providing directed contractions \\cite{neal2014formation}. This can be achieved by allowing the tissue to anchor on a pair of posts \\cite{aydin2019neuromuscular,aydin2020development}, providing axial tension thus longitudinal alignment and contraction, or by letting the tissue wrap around a single cylindrical pillar \\cite{uzel2014microfabrication}, generating tangential stresses and tangentially aligned fibers. These, upon contraction, cause the circumference of the tissue to shorten, resulting in overall radial actuation \\cite{li2019biohybrid,li2022adaptive}.\n\nThe latter approach is particularly suited to our investigation (\\cref{fig:1}a). Indeed, the cylindrical pillar minimizes uneven stress distributions, favoring regular fiber organization, tissue shape, and uniform radial contractions. This, in turn, is conducive of reproducible flow responses. Further, geometric axisymmetry implies the existence of a midplane where the flow becomes two-dimensional, simplifying flow imaging and analysis. Finally, a ring shape can be conveniently suspended in bulk liquid by simply sliding it onto a thin submerged rod (\\cref{fig:1}b).\n\nTo fabricate our muscle ring (\\cref{fig:1}a), a polydimethylsiloxane (PDMS) mold is first created by pouring liquid precursor into a machined negative mold and allowing it to polymerize. A mixture of C2C12 mouse skeletal muscle myoblasts, ECM, and type-I collagen is then cast into the PDMS mold, where the myoblast-laden gel compacts into rings \\cite{bell1979production}. They are subsequently transferred onto a hydrogel tube submerged in culture medium, to induce differentiation, during which contractile myotubes form and align. Because of tube compliance and muscle internal tension, the diameter of the torus shrinks of typically 1--2~mm, before stabilizing.\n\n\n\\begin{figure*}[ht!]\n\\centering\n\\includegraphics[width=\\textwidth]{images\/fig2.png}\n\\caption{Computational design of the biostreamer and autonomous streaming. \n(a) Modeling of the muscle ring and its radial contractions. The computational muscle ring and surrounding liquid environment are characterized, consistent with experiments, by the non-dimensional Stokes layer thickness $\\delta_{AC} \/ a = 0.11$, where the length scale $a$ is the depth of the muscle ring.\n(b-c) Numerical simulation of the streaming flow around the muscle ring. The visualization of the flow field is presented using two different methods: (b) Stokes stream function  and (c) dynamical system representation.\n(d-f) Observed critical points from simulation and illustrations of their corresponding local flow patterns.\n(g) Expected flow structure at the midplane of the muscle ring from simulation. Critical points are marked by circles with colors corresponding to (d-f). Colour contours on the presented plane indicate regions of clockwise (blue) and counter-clockwise (orange) recirculating fluid. Regions near the critical points where potential particle manipulations can be achieved are encircled in bold lines. \n(h) Confocal fluorescent $z$-stack image of a section of a muscle ring showing the circumferentially aligned myotubes (green) and cell nuclei (blue). \n(i) Bright-field image showing the muscle ring hanging on the thin rod. The ring has measured inner diameter of 3.5~mm, outer diameter 4.7~mm, and depth 1.4~mm. Orange curves illustrate the radial muscle contractions.\n(j) Kymograph of the top and bottom edge of the muscle ring showing its periodic contractions over 2 seconds.\n(k) Plots of the top and bottom edge displacement due to muscle actuation over 4 seconds.\n(l) Time-averaged streamlines of the upper half of the muscle ring from experiment juxtaposed with streamlines of the lower half of the muscle ring from numerical simulation, showing the matching centers and recirculating regions. \n(m) Dominant first POD mode extracted using $x$ velocity components ($u$) from experiments (upper half) juxtaposed with those from simulations (lower half), demonstrating coherent regions from comparatively high correlation zones (flow locations of the same color indicate that local fluid particles move coherently in a correlated fashion; orange\/blue colors indicate positive and negative values; the change in sign indicates an opposite sense of motion). \nThe location used for obtaining the experimental flow spectrum is marked. \nSimilar results are observed when analyzing at other flow locations (Fig. S3b) and when considering the $y$ velocity component $v$ (Fig. S3d). \n(n) Frequency spectrum of the experimental flow ($u$), simulated flow, and muscle actuation. The normalized amplitude is presented on a logarithmic scale. \n}\n\\label{fig:2}\n\\end{figure*}\n\n\nFor flow measurement and analysis, a $\\mu$PIV system is realized (\\cref{fig:1}b). A rectangular aluminum enclosure is machine fabricated and bonded on a glass slide to hold culture medium laden with tracer particles of diameter 6--8~$\\mu$m. Besides serving as a container, the aluminum box also shields the tracers from external electric fields, minimizing drift caused by their slight negative charge. Two of the box opposing walls present cutouts at their centers to hold a thin aluminum rod onto which the muscle ring is slid. The box has inner dimensions of 1~cm$\\times$1~cm$\\times$1~cm. This choice is dictated by the need to minimize convection, which is observed to be significant (compared to expected streaming velocities) for larger volumes. A direct consequence of using a relatively small box is that flow boundary effects come into play. Nonetheless, we note that streaming is robust to moderate confinement, as demonstrated in periodic lattices both experimentally and numerically \\cite{bhosale2021multi}. We thus expect that an eventual streaming field generated by the muscle ring will be topologically unaffected by the presence of the walls, although we do anticipate a geometric compression of the main flow structures towards the center of the enclosure. This, in turn, may facilitate flow visualization. Indeed, by bringing characteristic flow features closer together, the area to be imaged is reduced, allowing improved resolution and $\\mu$PIV reconstruction accuracy.\n\n\nThe box and glass slide are mounted on the $x$-$y$ translation stage of an Olympus IX81 inverted microscope equipped with a $2\\times$ objective. The differentiated biostreamer is then transferred into the $\\mu$PIV system, and we adjust the stage such that the muscle ring is at the center of the field of view. We focus the microscope to the midplane of the toroidal tissue. A Hamamatsu digital camera connected to the microscope is utilized to capture images at 50 Hz, which are paired with a 5-frame step size and integrated with spatial cross correlation to produce velocity field data \\cite{hart1998elimination,keane1992theory}.\nThe biostreamer is thus allowed to contract spontaneously, or upon stimulation by an external optical fiber setup, generating an oscillatory flow in its surrounding. In the case of optical stimulation, tissue biofabrication entails the use of C2C12 cells with a mutated variant of the blue light-sensitive ion channel, Channelrhodopsin-2 (ChR2). Myotubes differentiated from these myoblasts selectively respond to precise light wavelengths, in our case 470~nm \\cite{raman2016optogenetic}. To reach the illumination intensity required for muscle contraction \\cite{sakar2012formation,bryson2014optical,klapoetke2014independent}, we connect the fiber to a laser diode source producing light pulses of intensity 1.6~mW\/mm$^2$. The specific combination of light wavelength and intensity prevents the $\\mu$PIV illumination system from interfering with the biostreamer activity.\n\n\n\n\n\n\n\n\\section*{Results}\n\n\\subsection*{Computational design of the biostreamer} Streaming literature has almost exclusively examined the use of bodies without holes (genus-0), unlike our muscle ring (genus-1). While in one recent instance \\cite{chan2021three} a rigid torus was computationally investigated for externally driven flow oscillations (perpendicular to the body axis of symmetry), radial self-contractions have not been considered so far. Further complicating the design specifics of our system, 3D skeletal muscle constructs have been demonstrated in a range of dimensions and actuation conditions. Viable muscle rings have been indeed reported for inner diameters ranging from 0.5~mm to 14~mm \\cite{okano1997tissue,gwyther2011engineered,li2019biohybrid,li2022adaptive,pagan2018simulation}, with wall thicknesses between 0.5~mm and 4~mm \\cite{gwyther2011engineered,pagan2018simulation}. Spontaneous contractions have been found to occur within the frequency range of 1~Hz to 3~Hz, and oscillation amplitudes observed to vary between 20~$\\mu$m and 500~$\\mu$m. \\cite{li2019biohybrid,li2022adaptive,pagan2018simulation}. Additionally, optogenetic C2C12 muscle tissues have been shown to produce consistent force responses to optical stimulation across the 1~Hz to 4~Hz range \\cite{raman2016optogenetic}.\n\n\n\nTo provide insight into the behavior of our coupled flow-muscle system, and in search of design solutions able to robustly generate streaming, we simulate different biostreamer instances within the above outlined, realizable parameter space. We model the muscle ring as a solid undergoing prescribed radial oscillations (while conserving volume, \\cref{fig:2}a), immersed in an incompressible viscous fluid of unbounded domain. The coupled system is numerically discretized and simulated using a remeshed vortex method coupled with Brinkmann penalization (Methods), an approach \\cite{gazzola2011simulations} demonstrated in a variety of settings \\cite{bhosale2021remeshed,gazzola2011simulations,gazzola2012c,gazzola2012flow,gazzola2014reinforcement,gazzola2016learning}, including streaming \\cite{parthasarathy2019streaming,bhosale_parthasarathy_gazzola_2020,chan2021three}.\n\n\nIn simulations, the investigated biostreamers are observed to produce qualitatively similar streaming flows, although quantitative details vary depending on muscle geometry and actuation properties. A representative instance is reported in \\cref{fig:2}b-g. On the left (\\cref{fig:2}b), isosurfaces of the time-averaged Stokes stream function (blue and orange representing clockwise and counter-clockwise flow rotations) highlight the presence of four recirculating annular regions (two on each side of the muscle ring). Two large regions are found to envelope the biostreamer, while a pair of smaller, nested ones approximately lie within the muscle itself. On the right (\\cref{fig:2}c) is the corresponding dynamical representation, whereby critical points (characterized by zero velocity) are extracted from the flow and classified based on their stability properties (through the Jacobian of the local velocity field).\n\n\n\nThe flow is found to be organized around four types of critical points: centers (pink -- \\cref{fig:2}d), saddles (yellow -- \\cref{fig:2}e) and attractive (blue) \/ repulsive (red) node-saddle-saddle (NSS -- \\cref{fig:2}f). In our axisymmetric case, both centers and saddles are 2D degenerate \\cite{strogatz2018nonlinear}, since the surrounding local flow has no out-of-plane (i.e. azimuthal) component. These degenerate points, when mapped to 3D space, form four continuous center-rings (pink) and four continuous saddle-rings (yellow), as illustrated in \\cref{fig:2}c. They collectively define the four annular recirculating regions described above.\n\nThe extracted flow skeleton underscores the structures that are most likely to be captured experimentally. A streamfunction slice through the midplane (\\cref{fig:2}g) mimics experimental visualization conditions and provides intuition. The locations at which the rings cross the midplane are identified by dots of corresponding colors, elucidating the role of centers and saddles in shaping the flow while highlighting their practical utility. Specifically, centers initiate recirculation and can be employed to attract\/retain particles \\cite{chong2013inertial,chong2016transport} and mix fluids, while saddles (and connecting streamlines) partition the flow, enabling particle separation and sorting \\cite{lutz2003microfluidics,thameem2016particle} or spatially controlled chemistry \\cite{lutz2003microfluidics,lutz2006characterizing}. Based on \\cref{fig:2}g, we anticipate that inner rings, being tucked within the muscle, will be difficult to image experimentally. Further, repulsive (red) and attractive (blue) saddles, that lie along the axis of symmetry in unperturbed simulations, may not be found on the experimentally imaged midplane, as a consequence of implementation imperfections and perturbations induced by the aluminum rod. Thus, the two external center-rings are likely the most robustly detectable structures. We then expect to observe, projected on the $\\mu$PIV imaging plane, four centers, located around the muscle ring and associated with recirculatory flows.\n\nOverall, we numerically find that muscle rings of approximately $\\sim$4~mm inner diameter and $\\sim$5~mm outer diameter may be ideal candidates for experimental realization. Indeed, at typical oscillation amplitudes ($\\sim 100$~$\\mu$m) and frequencies (1~Hz to 4~Hz), such rings are observed to generate streaming velocities ($\\sim 10~\\mu\\text{m\/s}$) significantly larger than background disturbances ($\\sim 1~\\mu\\text{m\/s}$). Further, key flow structures are predicted to closely surround the muscle, thus falling within a compact field of view, while remaining sufficiently separated for experimental detection. \n\n\n\n\n\n\n\n\n\\subsection*{Autonomous streaming}\nWe grow the biostreamer targeting the computationally identified specifications. A PDMS toroidal mold of 5~mm inner diameter, 12~mm outer diameter, and 3~mm depth is fabricated and the mixture of muscle cells and ECM is seeded (\\cref{fig:1}a). Mold dimensions are empirically determined to account for compaction, differentiation, and associated shrinking, leading to muscles of the approximately desired geometry ($\\sim$4~mm inner, $\\sim$5~mm outer diameters). Confocal fluorescent imaging also confirms myotubes circumferential alignment, for radial contraction (\\cref{fig:2}h).\n\nUpon transfer to the $\\mu$PIV system, the rings' specific dimensions and spontaneous contraction amplitudes\/frequencies are determined by tracking and averaging top and bottom edge displacements across 1,000 camera images acquired at 50~Hz. For the representative muscle of \\cref{fig:2}h-n, we measure a 3.5~mm inner diameter, 4.7~mm outer diameter, 1.4~mm tissue depth, 3.1~Hz dominant contraction frequency, and 72~$\\mu$m\/117~$\\mu$m average peak-to-peak top\/bottom amplitudes. Obtained amplitudes and frequencies are reflected in the kymographs of \\cref{fig:2}j, over 2 seconds, and in the time-varying edge displacements of \\cref{fig:2}k, over 4 seconds.\n\nTo facilitate comparison, we input these muscle-specific parameters into our simulations and analyze both experimentally and computationally obtained flows. We first consider experimental steady streamlines, depicted in \\cref{fig:2}l, top half. These are obtained by averaging recorded instantaneous velocities in the midplane, over 8 contraction cycles. This procedure removes (approximately) the flow's oscillatory components, revealing the underlying rectified, steady streaming field. As anticipated, we observe two distinct centers and corresponding flow recirculation regions. These features are also visible in the mirrored streaming field obtained from simulations, revealing close agreement (\\cref{fig:2}l, bottom half). We note the slight asymmetry that characterizes experiments, on account of ring geometry and actuation imperfections as well as due to the presence of the aluminum rod.  These perturbations are also likely responsible for the missing NSS along the ring axis of symmetry, features that we previously identified as non-robustly observable. We further highlight how in experiments, centers are found closer to the biostreamer than in simulations.  This is consistent with the intuition that wall effects (enclosure) would result in flow geometric compression. Finally, we note that the overall streaming response is found to be repeatable, with associated features seen across cyclic muscle contractions and samples (Fig. S1, S3).\n\nWe proceed with identifying additional hallmarks of streaming, through a combination of Proper Orthogonal Decomposition (POD) \\cite{sirovich1991analysis} and flow spectral analyses. POD considers space-time correlations in the velocity field and extracts energetic coherent structures lingering in the flow. We can employ this information to determine flow locations highly representative, in terms of their spectral fingerprint, of the underlying flow. We thus consider experimental and simulated midplane velocities over 20 seconds, extract the corresponding dominant POD mode (\\cref{fig:2}m, 90\\% of total energy, orange\/blue intensity correlates to local flow energy), and sample it at different coordinates (\\cref{fig:2}m and S3) at which spectral analysis of raw velocity data is performed.\n\nIn frequency domain, streaming flows present a well-known signature: a driving-frequency main peak capturing first order $\\mathcal{O}(\\epsilon)$ oscillatory effects, a zero-frequency peak (about ten times smaller) capturing second order $\\mathcal{O}(\\epsilon^2)$ streaming rectification, and a series of peaks of exponentially decaying magnitude at multiples of the driving frequency, corresponding to increasingly higher-order effects \\cite{holtsmark1954boundary}. \nOur data are found to quantitatively recapitulate this structure, as illustrated in \\cref{fig:2}n where we report the spectra of measured (black) and simulated (blue) velocities at the representative location marked as \\textcircled{$\\times$} in \\cref{fig:2}m. \nAdditionally, for reference, we plot the spectrum of the muscle top edge displacement (orange) of \\cref{fig:2}k. As can be noticed, the measured\/simulated flows indeed respond to the input muscle frequency (peak at 3.1~Hz) and produce rectification (peak at 0~Hz). The relative magnitude of these two peaks is also in line with theory, with rectification about one order of magnitude weaker, on account of being a second order effect.\n\nWe also note the presence of the characteristic peaks at multiples of the driving frequency. In simulations (blue bars), as expected from theory, the peak corresponding to the first frequency doubling ($\\sim$6~Hz) presents a magnitude comparable to the rectification peak, consistent with the fact that this is a second order effect too. Remaining frequency-multiples peaks are also detected from simulated flows (Fig. S2), although they are not visible in \\cref{fig:2}n given their negligible normalized amplitudes (that fall below the plotted lower bound), on account of their exponential decay \\cite{holtsmark1954boundary}.\n\nThe experimental flow response (black bars) does present a similar structure, whereby driving, rectification and cascading peaks are all found at the theoretically and computationally expected frequencies. However, the relative magnitude of the peaks at frequencies multiples of the driving one, is significantly stronger than predicted. This is not necessarily surprising for the following complementary reasons. First, the muscle is soft and can deform in response to the flow, unlike in theory where the body is considered to be rigid or in simulations where the ring's deformations are imposed. Disregarding body compliance leads to underestimating the strength of the streaming response, as recently demonstrated in \\cite{bhosale2022soft} whereby flow-induced elastic deformations are shown to be an additional source of streaming. Second, frequency multiples might originate from the tissue itself on account of its non-linear mechanical response to a driving oscillatory forcing (in this case cyclic contractions), through a mechanism mathematically similar to fluids \\cite{holtsmark1954boundary}. This effect, pertaining to the solid phase, might also couple with the flow, strengthening both responses. That a combination of these factors may be at play is suggested by the muscle spectrum (orange bars) of \\cref{fig:2}n, whereby deformations are found to precisely sync with the flow, presenting a set of highly structured harmonics that are unlikely to be the result of spontaneous, higher-frequency tissue contractions. While an explanation for the observed peak enhancement is provided, dissecting the exact mechanisms at play is beyond the scope of this work and not strictly necessary to confirm the presence of streaming or demonstrate the utility of our bio-hybrid platform.\n\nOverall, given the observed qualitative and quantitative agreement between experiments, simulations and theory, we conclude that our tissue-engineered biostreamer is indeed capable of generating streaming autonomously.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\begin{figure*}[t!]\n\\centering\n\\includegraphics[width=\\textwidth]{images\/fig3.png}\n\\caption{Optically controlled streaming. \n(a) Confocal fluorescent $z$-stack images of a section of a muscle ring showing the myotubes (MF 20, green) and the ChR2 marker (tdTomato, red) at the same locations. \n(b) Muscle edge displacement of the top half of the muscle ring over 3 seconds (pink) and corresponding light stimulation pattern made of pulses of 10~ms duration (blue) at different frequencies. (top) 2~Hz stimulation results in an average peak-to-peak amplitude of 81~$\\mu$m, (middle) 3~Hz stimulation and 59~$\\mu$m peak-to-peak amplitude, and (bottom) 4~Hz stimulation and 61~$\\mu$m peak-to-peak amplitude. (c) Frequency spectrum of the experimental flow ($u$), simulated flow, and muscle actuation at optical driving frequencies of (top) 2~Hz, (middle) 3~Hz, and (bottom) 4~Hz. The normalized amplitude is presented on a logarithmic scale. \n}\n\\label{fig:3}\n\\end{figure*}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection*{Optically controlled streaming}\nNow, we demonstrate controllability of the native function of the muscle ring by exploiting muscle contractions in response to an external light stimulus. \nTo control the actuation frequency using optical stimulation, we fabricate our biostreamer following the same process of the autonomous streamer, however, we now utilize optogenetic C2C12 myoblasts.\nAfter fabrication, we use confocal fluorescence microscopy to confirm the expression of ChR2 after differentiation.\nIndeed, we observe the persistent expression of ChR2 within the tissue (\\cref{fig:3}a) and, therefore, expect the muscle ring to respond to an external light stimulation of wavelength 470~nm. \nAn optical fiber is fixed on the microscope stage and is used to uniformly illuminate the muscle ring within the enclosure. \nThe fiber delivers 10~ms pulses from the connected laser diode source at driving frequencies of 2, 3, and 4~Hz. \nThese driving frequencies are less than, equal to, and greater than the autonomous biostreamer actuation frequency ($\\sim 3.1$ Hz).\n\n\n\n\nUpon light stimulation, the representative muscle ring of \\cref{fig:3} is observed to contract robustly for all three driving frequencies, as quantified by tracking its edge displacement \\cref{fig:3}b.\nAt the lowest driving frequency (2~Hz), the muscle actuation response is characterized by rapid contractions ($\\sim 0.1$~s) followed by slower relaxations ($\\sim 0.4$~s) to its original configuration (\\cref{fig:3}b, top).\nAt higher driving frequencies (3 and 4~Hz), the muscle responds to the stimulus consistently, with contractions over $\\sim 0.1$ s (\\cref{fig:3}b, middle\/bottom).\nHowever, the relaxation time ($\\leq 0.2$~s) is no longer sufficient for returning to the original configuration and smaller edge displacement amplitudes are realized. \nFor the driving frequencies of 2, 3, and 4 Hz, the average peak-to-peak edge displacements are indeed measured to be $81 \\ \\mu$m, $59 \\ \\mu$m, and $61 \\ \\mu$m, respectively. \n\n\n\n\n\nFinally, using the same spectral analysis approach of \\cref{fig:2}, we characterize the oscillatory and rectified flow fields generated by the optically driven biostreamer, both in simulations and experiments.\nFor all driving frequencies, we find that the experimental flow (black) responds with hallmarks similar to the spontaneous case, as the dominant mode shifts to align with the optical driving frequency (\\cref{fig:3}c), underscoring the system controllability.\nFurther, and importantly, the zero-frequency peak, which corresponds to the rectified streaming flow, is observed for all three driving frequencies. \nIn good agreement with theory and simulations (blue), the magnitude of the zero-frequency peak is an order of magnitude smaller than the dominant mode. \nMoreover, the presence of the cascading peaks at multiples of the driving frequency are evident for all three cases. \nSpecifically, for a driving frequency of 2 Hz, peaks of decaying magnitude are observed at 4, 6, 8, and 10 Hz. \nSimilarly, for driving frequencies of 3 and 4 Hz, peaks of decaying magnitudes are observed at 6 and 9 Hz, and at 8 and 12 Hz, respectively. \nAs for the autonomous biostreamer, the magnitude of the cascading peaks is larger in experiments than simulations, for the same reasons previously described. \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section*{Conclusion}\nIn this work, we present a bio-hybrid setup that allows us to investigate viscous streaming in biologically-powered, untethered, three-dimensional and millimeter-scale systems, approximating conditions experienced by a host of small aquatic creatures, long speculated (but not rigorously, experimentally confirmed) to autonomously rectify surrounding flows.\n\nBy combining tissue engineering, micro-particle image velocimetry, numerical simulations and flow analysis, we design and demonstrate a $\\sim$3.5 millimeter muscle ring that, once freely suspended in bulk liquid, generates viscous streaming, autonomously or in a remotely controlled fashion, via light stimulation. This experimentally verifies the speculated ability of millimeter-scale organism to produce and sustain streaming, while paving the way for potential applications in microfluidics and robotics. Further, this work sets the stage for exploring novel streaming flows and dynamics through design and actuation strategies afforded by tissue-engineering approaches \\cite{filippi2022will}, broadening the scope of bio-hybrid technology to both fundamental and applied fluid mechanics.\n\n\n\n\\matmethods{\n\n\\subsection*{PDMS Mold}\nPDMS base, Sylgard 184, and curing agent (Dow Corning) are mixed with 10:1 ratio by weight and degassed in a vacuum desiccator. \nThe mixture is then poured into the machine fabricated aluminum negative mold and cured at 60 $^{\\circ}$C overnight. \nThe PDMS molds are peeled off from the negative structures the next day and sterilized by autoclave. The PDMS mold has an inner diameter of 5~mm, an outer diameter of 12~mm, and an depth of 3~mm.\nPrior to seeding the cell-gel solution, PDMS molds are treated overnight with 1\\% (w\/v) pluronic F-127 (Sigma) dissolved in phosphate-buffered saline to reduce adhesion of cells and proteins.\n\n\\subsection*{Cell culture and muscle ring formation}\nC2C12 murine myoblasts and those that are transfected with pLenti2-EF1$\\alpha$-ChR2[H134R]-tdTomato-WPRE plasmid to express Channelrhodopsin-2 (ChR2) are seeded in cell culture flasks with growth medium consisted of 10\\% fetal bovine serum (Gibco), 1\\% L-glutamine (Gibco) and 1\\% penicillin-streptomycin (Lonza) in Dulbecco's modified eagle medium (Gibco) and grown to $70-80\\%$ confluency. \nFor muscle tissue formation, cells are collected and mixed with Matrigel basement membrane matrix and type-I collagen diluted by growth medium. \nThe cell-gel mixture is then injected into the PDMS mold and allowed to polymerize for 1 hour at 37$^{\\circ}$C, then the sample is inundated in growth medium and incubated for 2 days. \nAfter 2 days, they are transferred onto hydrogel tubes immersed in differentiation medium, which replaces the fetal bovine serum with horse serum (Gibco) in the growth medium, to induce the formation of myotubes.\nThe muscle rings are kept on the tubes for 10 days, with the differentiation medium refreshed every 2 days, before being used for the experiment. \n\n\n\\subsection*{Experiment setup}\nThe aluminum enclosure for the experiment is machined and glued to a microscope glass slide using Kwik-Sil silicone adhesive (World Precision Instrument). \nThe box also serves to shield electric field, minimizing tracking particles motion caused by their slight negative charge. \nBefore experiment, one muscle ring is transferred from the hydrogel tube to a thin aluminum rod of 1~mm diameter, which is then transferred to the middle of the aluminum enclosure filled with culture media mixed with tracking particles. \nThe aluminum enclosure apparatus is then secured on the microscope stage. \nA cover slip is added on top of the box to reduce convection, and the particles are allowed to settle for five minutes before recording. \nRecording is done using a digital complementary metal\u2013oxide semiconductor camera (Hamamatsu) with $2\\times$ objective focused to center plane of muscle based on sharpness of the muscle ring edge. \nEach recording is 20 seconds long with 50~Hz capturing frequency, 20~ms exposure time and $2 \\times 2$ binning. \nA control experiment, which only have culture media, is conducted with same temperature and carbon-dioxide control as streaming experiment to verify the particles motion without muscle ring is insignificant. \nOptical stimulation is carried out by an optical fiber of diameter 2.6~mm connected to a laser diode (Doric), fixed on the microscope stage next to the aluminum enclosure, producing light pulses at 2, 3 and 4~Hz. \nEach light pulse is 10~ms long with intensity 1.6~mW\/mm$^2$. \n\n\\subsection*{Simulation method and numerical implementation}\nWe briefly recap the governing equations and numerical solution technique. \nWe consider incompressible viscous flows in an unbounded domain $\\Sigma$. \nIn this fluid domain, immersed solid bodies perform simple harmonic oscillations. \nThe bodies are density matched and have support $\\Omega$ and boundary $\\partial\\Omega$, respectively. \nThe flow can then be described using the incompressible Navier-Stokes equations \\cref{eq:1}.\n\n\\begin{equation}\\label{eq:1}\n\\nabla \\cdot \\boldsymbol{u}=0; \\quad\\frac{\\partial u}{\\partial t}+(\\boldsymbol{u} \\cdot \\boldsymbol{\\nabla}) \\boldsymbol{u}=-\\frac{\\boldsymbol{\\nabla} P}{\\rho}+\\nu \\nabla^{2} u, \\quad x \\in \\Sigma \\backslash \\Omega\n\\end{equation}\n\nwhere $\\rho$, $P$, $\\boldsymbol{u}$ and $\\nu$ are the fluid density, pressure, velocity and kinematic viscosity, respectively. \nThe dynamics of the fluid-solid system is coupled via the no-slip boundary condition $\\boldsymbol{u}=\\boldsymbol{u}_{s}$, where $\\boldsymbol{u}_{s}$ is the solid body velocity.\nThe muscle ring contraction is prescribed via a time-varying core radius of a torus $R(t) = R_0 + A\\sin(\\omega t)$, where $R_0$ is the core radius in rest configuration, and $A$ and $\\omega$ are the contraction amplitude and frequency, respectively.\nIn order to conserve volume, we vary the tube radius of the torus accordingly through $r(t) = r_0\\sqrt{R_0 \/ R(t)}$, where $r_0$ is the tube radius in rest configuration.\nHence, the overall muscle contraction in our simulation is described as a torus with body velocity $dR \/ dt = A \\omega \\cos(\\omega t)$ accounting for time-varying core radius due to muscle contraction, and body deformation velocity $dr \/ dt = -0.5 A \\omega r_0 R(t)^{-1.5} \\sqrt{R_0} \\cos(\\omega t)$ accounting for tube radius adjustment to conserve volume.\nThe 3D system of equations is then solved using a velocity-vorticity formulation with a combination of remeshed vortex methods and Brinkmann penalization \\cite{gazzola2011simulations}, in the axisymmetric coordinate system. This method has been validated across a range of flow\u2013structure interaction problems, from flow past bluff bodies to biological swimming \\cite{gazzola2011simulations,gazzola2012flow,gazzola2012c,gazzola2014reinforcement,gazzola2016learning,bhosale2021remeshed,bhosale2022soft}. Recently, the accuracy of this method has been demonstrated in rectified flow contexts as well, capturing steady streaming responses from arbitrary rigid shapes in 2D and 3D \\cite{parthasarathy2019streaming,bhosale_parthasarathy_gazzola_2020,chan2021three,bhosale2021multi}. \n\n\\subsection*{Particle image velocimetry}\nVelocity field measurement of streaming is carried out by micro-particle image velocimetry ($\\mu$PIV). \nThe culture medium is seeded with polystyrene spheroids of diameter 6-8~$\\mu$m and specific gravity 1.05. \nThe aluminum enclosure minimizes particles motion caused by their slight negative charge. \nA control experiment, in which only the muscle ring is absent while all other conditions are maintained, is first conducted to verify the particle motion without muscle ring is insignificant. \nAfter the muscle ring is transferred into the system, the microscope is focused to center plane of muscle ring based on sharpness of the its edges. \n1000 images is then collected at a frequency of 50~Hz at 1~MP ($1024 \\times 1024$ pixels) resolution. \nThe images are then paired with 5 frames step size and then integrated with spatial cross correlation method \\cite{hart1998elimination,keane1992theory}. \nThe final integration window resulted in $10 \\times 10$ pixels with 50\\% overlap and a final grid spacing of $\\Delta X = \\Delta Y = 32.5 ~\\mu$m.\n\n\\subsection*{Proper orthogonal decomposition}\nDominant coherent structures are obtained with snapshot proper orthogonal decomposition (POD) following Sirovich \\cite{sirovich1987turbulence}. \nVelocity fluctuations $\\boldsymbol{u}^{\\prime}(\\boldsymbol{x}, t)$ are decomposed into a deterministic spatially-correlated part $\\phi^n(\\boldsymbol{x})$ and time-dependent coefficients $a^n(t)$ according to \\cref{eq:2}\n\\begin{equation}\\label{eq:2}\n\\boldsymbol{u}^{\\prime}(\\boldsymbol{x}, t)=\\sum_{n=1}^{N} a^{n}(t) \\boldsymbol{\\phi}^{n}(\\boldsymbol{x})\n\\end{equation}\nwhere $N\\approx 1000$ is the number of snapshot. \nAccording to corresponding ratio of the eigenvalues to summation of $N$ eigenvalues, i.e. $E_{n}=\\lambda_{n} \/ \\sum_{m=1}^{N} \\lambda_{m}$, each mode is sorted by its contribution to the turbulence kinetic energy. \n\n\\subsection*{Spectral analysis}\nFor the muscle ring, the displacement of the top and bottom edges is tracked using the image analysis software Tracker(physlets.org\/tracker). An FFT is then performed on the displacement data to produce the spectrum of muscle actuation. For the experimental flow, one point within the recirculation region is selected and FFT is performed on its time-dependent POD coefficients. \n\n\\subsection*{Muscle ring staining}\nThe muscle ring is rinsed with PBS and fixed in 4\\%v\/v of paraformaldehyde for 30 minutes. It is then washed three times for 5 minutes to permeabilize the tissue sample and incubated with 0.2\\%v\/v Triton X-100 (Sigma) diluted in PBS for 15 min. The muscle ring is then blocked and stored in Image-iT FX Signal Enhancer blocking solution (Invitrogen) at 4$^\\circ$C overnight.\nThe primary antibody, mouse antimyosin heavy chain (MF-20) is used to stain for myosin heavy chain with a 1:400 dilution ratio. The sample is incubated overnight at 4$^\\circ$C and then washed three times for 5 minutes before staining with secondary antibodies on the next day. The secondary antibody, AlexaFluor-488 anti-mouse (Invitrogen) is used to stain the MF-20 antibody overnight. The sample is incubated with DAPI (Invitrogen) for 15 minutes to stain nuclei at 4$^\\circ$C. After washing with PBS three times, the LSM 700 is used for the confocal fluorescent imaging.\n\n}\n\n\\showmatmethods{}\n\n\n\\acknow{The authors thank S. Hilgenfeldt for helpful discussions over the course of this work. The authors acknowledge support by the National Science Foundation under NSF CAREER Award \\#CBET\u20131846752 (MG) and the NSF Expedition 'Mind in Vitro' Award \\#IIS\u20132123781 (MG, TS).}\n\n\\showacknow{}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{section:introduction}\n\n\n\n\\begin{figure}\n\\vskip -0.01in\n\\centering\n\\includegraphics[width=7.5cm]{noise_v8.pdf}\n\\caption{Illustration of how interleaved connections contaminate the choice of neural network connections. The red, purple, and green links denote three candidate connections, among which the red one is the best choice (according to the individual evaluation). However, when interleaved connections (the dashed links) are added for the search sampling, the judgment of the candidates gradually becomes ridiculous. The bottom figures indicate the weight change of the three candidates throughout the search process (warm-up for 20 epochs), while $(a)$ is under the interleaving-free setting and $1$--$3$ interleaved connections are added for $(b)$--$(d)$. Specifically, the interleaved connection $(3,7)$ is added for $(b)$, the connections $(1,4)$, $(3,7)$ added for $(c)$, and $(1,4)$, $(2,6)$, $(3,7)$ added for $(d)$.}\n\\label{fig:introduction}\n\\end{figure}\n\nNeural architecture search (NAS) is a research field that aims to automatically design deep neural networks~\\cite{NASNet,AmoebaNet,metaQNN}. There are two important factors that define a NAS algorithm, namely, the search space that determines what kinds of architectures can appear, and the search strategy that explores the search space efficiently. Despite the rapid development of search algorithms which have become faster and more effective, the search space design is still in a preliminary status. In particular, for the most popular search spaces used in the community, either MobileNet-v3~\\cite{mobilenetv3} or DARTS~\\cite{liu2018darts}, the macro structure (\\emph{i.e.}, how the network blocks are connected) is not allowed to change. Such a conservative strategy is good for search stability (\\textit{e.g.}, one can guarantee to achieve good performance even with methods that are slightly above random search), but it reduces the flexibility of NAS, impeding the exploration of more complicated (and possibly more effective) neural architectures.\n\nThe goal of this paper is to break through the limitation of existing search spaces. For this purpose, we first note that the MobileNet-v3 and DARTS allow a cell to be connected to 1 and 2 precursors, respectively, resulting in relatively simple macro structures. In opposite, we propose a variant that each cell is connected to $L$ precursors ($L$ is 4, 6, or 8), and each connection can be either present or absent. We evaluate three differentiable NAS algorithms, namely DARTS~\\cite{liu2018darts}, PC-DARTS~\\cite{xu2020pcdarts}, and GOLD-NAS~\\cite{GoldNAS} in the designed $L$-chain search space, and all of them run into degraded results. We perform diagnosis in the failure cases and the devil turns out to be the so-called \\textbf{interleaved connections}, which refers to a pair of connections $(a,b)$ and $(c,d)$ that satisfies $a<c<b<d$. Figure~\\ref{fig:introduction} shows an example that how interleaved connections affects the search results. With the increasing extent of interleaving, the search results gradually deteriorate, reflecting in the reduced accuracy and the weak operator gaining heavier weights. More examples are provided in the experiment.\n\n\\iffalse\nIn this paper, we explore the challenge under a more general search space (in the form of a super-network) in which a node can receive input from one or many of its $L$ prior nodes. In this enlarged search space (called $L$-chain space in this paper), we find that existing one-shot search methods, especially the differentiable methods, can fall into sub-optimal and even dramatic solutions, which we call it \\textbf{degradation}. From the failure cases, we observe an interesting phenomenon that the algorithm tends to build shallow networks which are not a good choice empirically. We further diagnose the search procedure and owe the degradation to the \\textbf{interleaved connections}. For all nodes between any two activated ones, of which the input connections exceeding 1 or the output connections exceeding 2 are defined as interleaved connections.\nInterleaved connections appear when nodes between candidate nodes participate in more than 1 input or 2 outputs.\nFigure~\\ref{fig:introduction} shows a representative case that, with the increasing extent of interleaving, the search results gradually deteriorate, reflecting in the reduced accuracy and the weak operator gaining heavier weights. \\textcolor{green}{More examples are provided in the appendix.}\n\\fi\n\n\n\n\n\nThe above observation motivates us to maximally eliminate the emerge of interleaved connections during the search procedure. This is easily done by performing interleaving-free sampling, where we group all candidate connections into $L$ groups and there exist no interleaved connections in every single group, based on which we periodically choose one group and perform regular NAS algorithms. This schedule can optimize the weight of every connection without suffering the issue of interleaved connections. Discretization and pruning are performed afterwards to derive the final architecture. The entire algorithm is named interleaving-free NAS, or \\textbf{IF-NAS} for short.\n\n\n\\iffalse\nMotivated by the above, we propose a simple yet effective solution named interleaving-free sampling (IF-NAS). The core idea of IF-NAS is to avoid the active connections (\\emph{i.e.}, the connections on which weights are being updated in the current iteration) to interleave during the search procedure. \nThis is done by sampling sub-networks, each of which avoids the interleaved connections.\nNote that during the optimization, the super-network is not complete in each iteration, but the expectation that each connection is sampled remains fair and thus the super-network after optimization can be directly used for determining the final architecture. We will show in the experiments that IF-NAS largely alleviates the degradation and derives stronger architectures.\n\\fi\n\nWe conduct experiments on ImageNet, a popular benchmark of NAS. In the newly proposed $L$-chain space, IF-NAS significantly outperforms three differentiable search baselines, DARTS, PC-DARTS and GOLD-NAS, and the advantage becomes more evident as the number of possible input blocks grows larger, \\emph{i.e.}, heavier interleaving presents.\nMoreover, we evaluate IF-NAS in the existing search spaces of DARTS and GOLD-NAS, and show that it generalizes well to these easier search spaces.\n\nIn summary, the contributions of this paper are two-fold. \\textbf{First}, we advocate for investigating the macro structure and put forward a novel search space for this purpose. The existing NAS methods cannot guarantee satisfying performance in this space. \\textbf{Second}, we show that the major difficulty lies in dealing with the interleaved connections and hence propose an effective solution named IF-NAS. We hope that our efforts can inspire the NAS community to study the challenging new problem.\n\n\\section{Related Work}\n\n \n\\paragraph{Search Strategy}\n \nEarly NAS methods generally rely on individually evaluating the sampled sub-architectures under heuristic strategies, including reinforcement learning~\\cite{NASNet} and the evolutionary algorithm~\\cite{AmoebaNet}, which are computationally expensive. To accelerate the search, one-shot methods~\\cite{Understanding, brock2017smash, SPOS} propose to represent the search space with a super-network, where the weights of all the candidate architectures are shared. Individual architecture training from scratch is avoided and the search cost is reduced by large magnitudes. Recently differentiable NAS (DNAS) has aroused great popularity in this field, which maps the discrete search space into a parameterized super network so that the search process can be executed by gradient descent. DARTS~\\cite{liu2018darts}, as a pioneer differentiable framework, relaxes the search space by introducing updatable architecture parameters. The bi-level optimization is performed to update super-network weights and architectural parameters alternately. The target architecture is derived according to the distribution of architectural parameters. Due to its high efficiency, many works extend DNAS to more applications, including semantic segmentation~\\cite{AutoDeeplab,zhang2019customizable}, object detection~\\cite{fang2020fna++,guo2020hit}, \\textit{etc}. Some DNAS works~\\cite{cai2018proxylessnas,wu2019fbnet,DenseNAS} propose to integrate co-optimization with both accuracy and hardware properties into the search phase. In this paper, we target at promoting DNAS in terms of both flexibility and robustness. With the two factors improved, the final performance of DNAS can reach a higher level.\n\n\\paragraph{Eliminating Collapse in DNAS}\nThough DNAS has achieved great success due to its high efficiency, many inherent problems exist with it and may cause collapse during search. A series of methods propose to improve DNAS from various perspectives. P-DARTS~\\cite{PDARTS} bridges the gap between the super and searched network by gradually increasing the depths the networks. FairDARTS~\\cite{chu2019fairdarts} improves the sampling strategy and breaks the two  indispensable factors of unfair advantages and exclusive competition. \\cite{DAAS, GoldNAS} alleviate the discretization error by pushing the weights to sharp distributions. \\cite{DARTS+, StacNAS, RobustDARTS} robustify the search process by introducing regularization techniques, including early stopping and observing eigenvalues of the validation loss Hessian, \\textit{etc}. We reveal the degradation phenomenon in complicated search spaces when using the conventional DNAS method, and propose to suppress the interleaved connections during search. The proposed IF-NAS eliminates collapse most DNAS methods may encounter and shows superior performance.\n\n\\paragraph{Search Space}\n\nMost existing NAS methods only explore in the micro search space, while the limited flexibility hinders further development of NAS. The cell-based space is firstly proposed in NASNet~\\cite{NASNet}, which is widely adopted by the following works~\\cite{ENAS, AmoebaNet, BayesNAS, liu2018darts}. This type of search space takes several nodes into one cell structure, which though eases the search procedure, suppresses many possible architectures with stronger feature extraction ability. \\cite{liu2017hierarchical} proposes to search architectures under a hierarchical space which introduces flexible topology structures. Auto-DeepLab~\\cite{AutoDeeplab} searches in a space with multiple paths and allows feature extraction in diverse resolutions. DenseNAS~\\cite{DenseNAS} introduces densely connections in the search space and improves search freedom in terms of operators, depths and widths. GOLD-NAS~\\cite{GoldNAS} liberates the restriction of the cell-based design and performs search in a global range. We further extends the search space complexity to explore architectures with more possibilities and potential. Though intractable is the search in such a complicated space, our proposed IF-NAS still shows evident effectiveness and advantages over other compared DNAS methods.\n \n \n \n\n\n\n\\section{Our Approach}\n\n\\subsection{Preliminaries: NAS in a Super-Network}\n\nIn neural architecture search (NAS), a deep neural network can be formulated into a mathematical function that receives an image $\\mathbf{x}$ as the input and produces the desired information (\\textit{e.g.}, a class label $y$) as the output. We denote the function to be $y=f(\\mathbf{x};\\boldsymbol{\\alpha},\\boldsymbol{\\omega})$ where the form of $f(\\cdot)$ is determined by a set of architectural parameters, $\\boldsymbol{\\alpha}$, and the learnable weights (\\textit{e.g.}, the convolutional weights) are denoted by $\\boldsymbol{\\omega}$. The goal of NAS is to find the optimal architecture, $\\boldsymbol{\\alpha}^\\star$, that leads to the best performance, \\textit{i.e.},\n\\begin{eqnarray}\n\\label{eqn:goal}\n{} & {\\boldsymbol{\\alpha}^\\star}={\\arg\\min_{\\boldsymbol{\\alpha}}\\mathbb{E}_{\\left(\\mathbf{x},y^\\star\\right)\\in\\mathcal{D}_\\mathrm{val}}\\!\\left|y^\\star-f(\\mathbf{x})\\right|}\\\\\n\\nonumber\n\\mathrm{s.t.} & {\\boldsymbol{\\omega}^\\star(\\boldsymbol{\\alpha})}={\\arg\\min_{\\boldsymbol{\\omega}}\\mathbb{E}_{\\left(\\mathbf{x},y^\\star\\right)\\in\\mathcal{D}_\\mathrm{train}}\\!\\left|y^\\star-f(\\mathbf{x})\\right|},\n\\end{eqnarray}\nwhere $y^\\star$ denotes the ground truth label.\nMost often, $\\boldsymbol{\\alpha}$ takes discrete values, implying that solving Eqn~\\eqref{eqn:goal} requires enumerating a large number of sampled architectures and performing individual evaluation. To accelerate, researchers propose to slack $\\boldsymbol{\\alpha}$ into a continuous form so that solving Eqn~\\eqref{eqn:goal} involves optimizing a super-network, after which $\\boldsymbol{\\alpha}^\\star$ is discretized into the optimal architecture for other applications.\n\nIn particular, this paper is built upon the differentiable search algorithms, in which the super-network is solved by computing the gradient with respect to $\\boldsymbol{\\alpha}$. We will introduce the details of optimization in the experimental part.\n\n\\subsection{Exploring a Complicated Search Space}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=.99\\linewidth]{space_v10.pdf}\n\\caption{The studied search space in this paper. Each layer (also called node) can be connected to $L$ precursors. 3 stages are to be searched with N1, N2 and N3 nodes respectively. For better visualization, we show the example of $L=4$, yet we also study $L=6$ and $L=8$ which are even more complicated.}\n\\label{fig:space}\n\\end{figure}\n\nWe design a search space shown in Figure~\\ref{fig:space}. We define a fixed integer, $L$, indicating that each layer can be connected to $L$ precursors. For convenience, we name this space \\textbf{$L$-chain space}. \nWhen $L=1$, it degrades to the chain-styled network (the backbone of MobileNet-v3). \nOn the contrary, we study the cases of $L=4$, $L=6$, and even $L=8$, making the topology of the space much more complicated. We follow R-DARTS~\\cite{RobustDARTS} to allow two candidate operations, \\emph{i.e.} the $3\\times3$ separable convolution and skip-connection. Throughout the remaining part of this paper, we use $\\mathbb{S}^{(L)}$ to indicate the proposed space with $L$ precursor connections. For convenience, the connection between the $n$-th and $(n-l)$-th layers (nodes) is named the pre-$l$ connection of the $n$-th layer.\n\nBefore entering the algorithm part, we emphasize that we do not use additional rules to assist the architecture search, \\textit{e.g.}, forcing all nodes to survive by preserving at least one connection. This raises new challenges to the search algorithm, because the depth of the searched architecture becomes quite indeterminate. Although some prior work has explored the option of optimizing the macro and micro architectures jointly~\\cite{AutoDeeplab} or adding mid-range connections to the backbone~\\cite{DenseNAS}, they still rely on the cell\/block unit to perform partial search. Instead, we break the limitations of the unit, and allow search in a wider and macro space.\n\n\\begin{table}\n\\centering\n\\caption{Comparisons of the search space complexity.}\n\\label{tab.space_complexity}\n\\resizebox{0.47\\textwidth}{!}{\n\\begin{tabular}{@{}cccccc@{}}\n    \\toprule\n        \\textbf{DARTS}\n        & \\textbf{GOLD-NAS} \n        & \\textbf{$4$-chain}\n        & \\textbf{$6$-chain}\n        & \\textbf{$8$-chain}\n      \\\\\n    \\midrule\n      $1.1\\times10^{18}$\n      & $3.1\\times10^{117}$\n      & $1.8\\times10^{116}$\n      & $7.5\\times10^{163}$\n      & $1.6\\times10^{204}$\n      \\\\\n    \\bottomrule\t\n\\end{tabular}\n}\n\\end{table}\n\nIn each edge, there are two candidate operators in our $L$-chain space. Each node receives at most $2L$ input feature maps. For the purpose of validity, each node has at least one input to be preserved. However, not every node may be used as an input node, and these nodes will be deleted, which means that the architectures are allowed to be very shallow. For a node $x_k (k>L)$, there are $2^{2L}-1$ input possibilities because each of the $2L$ operators can be on or off. As a result, there are $(2^2-1)\\times(2^4-1)\\times\\cdot\\cdot\\cdot\\times(2^{2(L-1)}-1)\\times(2^{2L}-1)^{N-L}$ combinations if there are $N$ nodes in one stage. There are 3 stages to be searched in our space, Fig.~\\ref{fig:space}, with the node number of 18, 20 and 18 respectively. Therefore, if $L$ is 4, 6 and 8,  there are about $1.8\\times10^{116}$, $7.5\\times10^{163}$ and $1.6\\times10^{204}$ possible architectures respectively. The complexity comparison of popular spaces are shown in Tab.~\\ref{tab.space_complexity}. \n\n\n\n\n\n\n\\begin{figure*}[t]\n\\begin{center}\n\\includegraphics[width=.90\\linewidth]{pipeline_V6.pdf}\n\\end{center}\n\\caption{The flowchart of IF-NAS. The key is to avoid interleaved connections in \\textit{any} time. For this purpose, we repeat a loop with a length of $L$, and each time part of the connections remain active and all others are removed. The layers of the same color are always activated and inactivated together. This figure is best viewed in color.}\n\\label{fig:pipeline}\n\\end{figure*}\n\n\\subsection{Failure Cases and Interleaved Connections}\n\nWe first evaluate DARTS~\\cite{liu2018darts} and GOLD-NAS~\\cite{GoldNAS} in the search space of $\\mathbb{S}^{(4)}$. \nOn the ImageNet-1k dataset, trained for 250 epochs for each network, DARTS and GOLD-NAS report top-1 accuracy of $74.7\\%$ and $75.3\\%$, respectively, and the FLOPs of both networks are close to $600\\mathrm{M}$ (\\textit{i.e.}, the mobile setting), Tab.~\\ref{tab.main_results}. \nIn comparison, the simple chain-styled architecture (each node is only connected to its direct precursor) achieves $74.9\\%$ with merely $520\\mathrm{M}$ FLOPs.\nMore interestingly, if we only preserve one input for each node, DARTS and GOLD-NAS report completely failed results of $70.2\\%$ and $71.2\\%$ (trained for 100 epochs), which are even much worse than preserve one input randomly, Tab.~\\ref{tab.ablation_keep1}.\n\nWe investigate the searched architectures, and find that both DARTS and GOLD-NAS tend to find long-distance connections. This decreases the depth of the searched architectures, however, empirically, deeper networks often lead to better performance.\nThis drives us to rethink the reason that the algorithm gets confused. For this purpose, we randomly choose an intermediate layer from the super-network and the algorithm needs to determine the preference among its precursors.\n\nWe start from the simplest situation that all network connections are frozen (the architectural weights are fixed) besides the candidate connections. We show the trend of three (pre-$1$, pre-$3$, pre-$6$) connections in Figure~\\ref{fig:introduction}. One can observe that the pre-$1$ connection overwhelms other two connections, aligning with our expectation that a deeper network performs better. However, when we insert one connection that lies between these candidates, we observe that the advantage of the pre-$1$ connection largely shrinks, and a long training procedure is required for it to take the lead. The situation continues deteriorating if we insert more connections into this region. When two or more connections are added, the algorithm cannot guarantee to promote the pre-$1$ connection and, sometimes, the ranking of the three connections is totally reversed.\n\nThe above results inspire us that two connections are easily interfered by each other if the covering regions (\\textit{i.e.}, the interval between both ends) overlap. Hereafter, we name such pair of connections \\textbf{interleaved connections}. Mathematically, denote a connection that links the a-th and b-th nodes as $(a, b)$, where $b-a\\ge 2$. Overlook the candidate connections of the activated nodes, two connections, $(a,b)$ and $(a',b')$, interleave if and only if there exists at least one integer $d$ that satisfies $a<d+1\/2<b$ and $a'<d+1\/2<b'$.\nIntuitively, for a real number $x$, if $\\{x| a < x < b\\} \\cap \\{x | a' < x < b'\\} \\neq \\emptyset$ is satisfied, the interleaved connection occurs.\n\n\nAs a side comment, the DARTS space is also impacted by the interleaved connections, which partly cause the degradation problem observed in prior work~\\cite{DARTS+,PDARTS,Cellbased}. However, since the search space is cell-based, the degradation does not cause dramatic accuracy drop. In the experiments, we show that our solution, elaborated in the next part, generalizes well to the DARTS space.\n\n\\subsection{Interleaving-Free Sampling}\n\nFollowing the above analysis, the key to design the search algorithm is to avoid interleaved connections, meanwhile ensuring that all connections can be considered. For this purpose, we propose IF-NAS, a sampling-based approach that each time optimizes a interleaving-free sub-super-network from the super-network.\n\nThis is implemented by partitioning the layers into $L$ groups, $\\mathcal{G}_1,\\mathcal{G}_2,\\ldots,\\mathcal{G}_L$, according to their indices modulo $L$. In other words, the distance between any consecutive layers that belong to the same group is exactly $L$. When any group $\\mathcal{G}_{\\lambda}$ is chosen, we obtain an interleaving-free sub-super-network by preserving (i) the main backbone (all pre-$1$ connections) and (ii) all connections that end at any layer in $\\mathcal{G}_{\\lambda}$. We denote the sub-super-network by $\\mathbb{S}_{\\lambda}^{(L)}$ for $\\lambda=1,2,\\ldots,L$. Correspondingly, the optimization goal can be written as:\n\\begin{equation}\n\\label{eqn:loss1}\n\\min_{\\boldsymbol{\\alpha},\\boldsymbol{\\omega}}\\mathbb{E}_{\\left(\\mathbf{x},y^\\star\\right)\\in\\mathcal{D}}\\left|y^\\star-f\\!\\left(\\mathbf{x}\\mid\\mathbb{S}_{\\lambda}^{(L)}\\right)\\right|.\n\\end{equation}\n\nA straightforward search procedure repeats the loop of $L$ sub-super-networks, as shown in Figure~\\ref{fig:pipeline}. Training each sub-super-network is an approximation of optimizing the entire super-network, $\\mathbb{S}^{(L)}$, but, since the sub-super-network are fixed, the co-occurring connections may gain unexpected advantages over other ones. To avoid it, we add a warm-up step to the beginning of each loop, in which the entire super-network is trained (all connections are activated, but the architectural parameters are not updated). Throughout the remainder of the loop, we train the $L$ sub-super-networks orderly. We set the length of each step to be $100$ iterations (\\textit{i.e.}, mini-batches). Note that this number shall not be too large, otherwise the gap between the sampled and non-sampled connections will increase and harm the search performance.\n\n\\subsection{Discretization and Pruning}\n\nThe last step is to determine the final architecture. In a complicated search space, this is not simple as it seems, because many connections may have moderate weights (\\emph{i.e.}, not close to $0$ or $1$). For such a connection, either pruning it or promoting it can bring significant perturbation on the super-network. Therefore, we follow the idea of prior works~\\cite{chu2019fairdarts,GoldNAS,DAAS} to first perform a discretization procedure to push the weights towards either $0$ or $1$, after which pruning is much safer.\nWe introduce two kinds of architectural parameters, the connectivity of edges are determined by $\\boldsymbol{\\beta}$ and operators are determined by $\\boldsymbol{\\alpha}$.\n\nThe key of discretization is to add a regularization term to Eqn. (\\ref{eqn:loss1}). The term penalizes the moderate weights of edges and operators, represented by $\\boldsymbol{\\beta}$ and $\\boldsymbol{\\alpha}$ respectively. Once these weights are less than $0.01$, the corresponding edges or operators will be pruned off. The regularization term is computed by:\n\n\\begin{footnotesize}\n\\begin{equation}\n\\begin{split}\n    \\mathcal{R}(\\alpha,\\beta)&=\\mu_1\\cdot \\sum_{j\\le N}\\sum_{0\\le j-L\\le i<j}\\ln(1+g(\\beta_{i,j})\/\\overline{g(\\beta_{i,j})})\\\\\n    &+\\mu_2\\cdot \\sum_{j\\le N}\\sum_{0\\le j-L\\le i<j} \\sum_{o\\in \\mathcal{O}}\\ln(1+g(\\alpha^o_{i,j}|\\beta_{i,j})\/\\overline{g(\\alpha^o_{i,j}|\\beta_{i,j})}),\n\\end{split}\n\\end{equation}\n\\end{footnotesize}\nwhere $N$ is the number of layers, and $g(\\cdot)$ is the activate function $sigmoid$.\n$g(\\alpha|\\beta)$ means to only include operators on $\\boldsymbol{\\beta}$ that have not been pruned off. $\\overline{g(\\cdot)}$ is the average of $g(\\cdot)$.\n$\\mathcal{R}(\\alpha,\\beta)$ is multiplied by a factor of $\\mu$ and added to the cross-entropy loss, so that optimizing the overall loss will not only improve the recognition accuracy of the super-network, but also push the weights of all connections towards $0$ or $1$.\n\nWhen the weight of a connection is sufficiently small, we prune it permanently from the super-network. This merely impacts the super-network itself, but the computation of $\\overline{g(\\cdot)}$ changes and will push the next weak operator to $0$. This process iterates until the complexity (\\textit{e.g.}, FLOPs) of the super-network achieves the lower-bound.\n\n\n\n\n\n\n\n\\section{Experiments}\nIn this section, we first introduce experimental details and implementation details. Next, we introduce the results and analysis of IF-NAS and other advanced methods in our $L$-chain space under different settings.\nFinally, we compare $L$-chain spaces with 2 cell-based micro spaces using IF-NAS and other 2 search strategies. \n\n\\subsection{Implementation Details}\n\\textbf{The Dataset.}\nWe use the large-scale ImageNet dataset (ILSVRC2012) to evaluate the models. ImageNet contains 1,000 object categories, which consists of 1.3M training images and 50K validation images. The images are almost equally distributed over all classes. Unless specified, we apply the mobile setting, in which the input image is set to $224\\times224$ and the number of multiply-add operations (MAdds) does not exceed 600M. We randomly sample 100 classes from the original ImageNet dataset to perform studying and analysis experiments, ImageNet-100 for short.\n\n\n\n\\textbf{The Search Settings.}\nBefore searching, IF-NAS warm-ups the super-network for 20 epochs, with only the super-network weights updated and the architectural parameters frozen. Then we start to update $\\beta$ firstly, and after 10 epochs, start to update $\\alpha$. Super-network weights and architectural parameters are both optimized by the SGD optimizer.\nWe gradually prune off weak edges and operators with threshold of $0.01$ until the MAdds of the retained architecture meets the mobile setting, \\emph{i.e.}, 600M. \n\n\\textbf{The Training Settings.}\nWe evaluate the searched architectures following the setting of PC-DARTS~\\cite{xu2020pcdarts}. Each searched architecture is trained from scratch with a batch size of 1024 on 8 Tesla V100 GPUs. A SGD optimizer is used with an initial learning rate of 0.5, weight decay ratio of $3\\times10^{-5}$, and a momentum of 0.9. Other common techniques including label smoothing, auxiliary loss, and learning rate warm-up for the first 5 epochs are also applied.\n\n\n\\subsection{Results on $L$-chain Search Space}\n\n\n\\begin{table}\n\\fontsize{8.5}{10.0}\\selectfont\n\\renewcommand\\arraystretch{1.01}\n\\centering\n\\caption{Results of popular methods on our enlarged complicated space: comparison of classification test error (\\%) trained on ImageNet-1k for 250 epochs under the mobile setting.}\n\\label{tab.main_results}\n\\resizebox{0.48\\textwidth}{!}{\n\\begin{tabular}{@{}lcccccc@{}}\n    \\toprule\n        \\multirow{2}{*}{\\textbf{Setting}}\n        & \\multirow{2}{*}{\\textbf{Method}}\n        & \\multicolumn{2}{c}{\\textbf{Test Err. (\\%)}}\n        & \\textbf{Params}\n        & $\\times+$\n     \n      \\\\\n        \\cmidrule(lr){3-4}\n        &\n        & \\textbf{top-1}\n        & \\textbf{top-5}\n        & \\textbf{(M)}\n        & \\textbf{(M)}\n       \n        \\\\\n    \\midrule\n        \\multirow{4}{*}{\\textbf{$L$ = 4}}\n        &DARTS~\\cite{liu2018darts}  &25.3  &8.1  &5.1  &589\n        \\\\\n        &PC-DARTS~\\cite{xu2020pcdarts} &24.7  &7.5  &5.3  &593 \n        \\\\\n      & GOLD-NAS~\\cite{GoldNAS}  &24.7  &7.5  &5.5  &591\n        \\\\\n   \n      & IF-NAS &\\textbf{24.3} & \\bf 7.4  &5.3  &592\n        \\\\\n       \n       \n    \\midrule\t\t\n     \\multirow{4}{*}{\\textbf{$L$ = 6}}\n        &DARTS~\\cite{liu2018darts}  &25.7  &8.1  &5.1  &587\n        \\\\\n        &PC-DARTS~\\cite{xu2020pcdarts} &24.9  &7.7  &5.2  &586\n        \\\\\n      & GOLD-NAS~\\cite{GoldNAS} &25.0  &7.8  &5.4  &596\n        \\\\\n   \n      & IF-NAS &\\textbf{24.4} & \\bf7.4  & 5.2  &598  \n        \\\\\n       \n       \n    \\midrule\t\t\n     \\multirow{4}{*}{\\textbf{$L$ = 8}}\n        &DARTS~\\cite{liu2018darts}  &25.9  &8.2  &5.2  &591\n        \\\\\n        &PC-DARTS~\\cite{xu2020pcdarts} &25.1  &7.9  &5.2  &582\n        \\\\\n      & GOLD-NAS~\\cite{GoldNAS} &25.1  &7.9 &5.3  &592\n        \\\\\n   \n      & IF-NAS &\\textbf{24.3} & \\bf7.3  & 5.4  &594  \n        \\\\\n       \n       \n    \\bottomrule\n\\end{tabular}\n}\n\\end{table}\n\n\nWe study our method IF-NAS on the proposed macro $L$-chain search space, and compare it with three other popular DNAS frameworks, including DARTS~\\cite{liu2018darts}, PC-DARTS~\\cite{xu2020pcdarts} and GOLD-NAS~\\cite{GoldNAS}.\nFor a fair comparison, the hyper-parameters are kept the same for all the studied methods.\nWithout pruning gradually, DARTS and PC-DARTS derive the final architectures by removing weak edges and operators after searching until they meet the requirements of mobile setting. All the searched architectures are re-trained from scratch on ImageNet-1k for 250 epochs.\n\n\nWe perform three sets of experiments by setting $L$, which indicates the precursors number of each layer can be connected, to 4, 6 and 8 respectively, and show the results in Tab.~\\ref{tab.main_results}.\nWhen $L$ is 4, DARTS gets the worst performance as expected. Compared with DARTS, PC-DARTS and GOLD-NAS get better results of $24.7\\%$ and $24.7\\%$ with similar MAdds. Our IF-NAS gets the best result of $24.3\\%$ with equivalent Params and MAdds. This proves that IF-NAS can work better in such a enlarged complicated space compared to the methods. \n\n\nWhen $L$ is 6, DARTS achieves the result of $25.7\\%$. PC-DARTS and GOLD-NAS get results of $24.9\\%$ and $25.0\\%$ respectively. And our IF-NAS gets the best result of $24.4\\%$ with similar Params and MAdds. Note that the results of the compared 3 frameworks get worse when $L$ increases to 6 compared to $L$ as 4. However, IF-NAS nearly maintains the performance. When $L$ increases, more interleaved connections occur, and bring more interference to the search process. The compared 3 methods, which do not suppress interleaved connections during search, are more severely effected and achieve worse results. However, IF-NAS samples interleaving-free sub-networks and updates the architectural parameters more accurately. This further confirms that our IF-NAS has great potential to handle complicated spaces.\n\n\n\nWhen set $L$ to 8, IF-NAS still achieves a promising result of $24.3\\%$, which is comparable to the former two settings. The compared ones get similar or weaker results. \nDue to the increase in complexity, the performances of PC-DARTS and GOLD-NAS continue to decrease. But the MAdds of these architectures are close to 600M, which prevents performance from declining.\n\\begin{table}\n\\fontsize{8.5}{10.5}\\selectfont\n\\centering\n\\caption{Results of reserving one input for each node. all architectures are trained on ImageNet-1k for 100 epochs.}\n\\label{tab.ablation_keep1}\n\\resizebox{0.48\\textwidth}{!}{\n\\begin{tabular}{@{}lcccccc@{}}\n    \\toprule\n        \\textbf{Setting}\n        & \\textbf{random}\n        & \\textbf{DARTS}\n        & \\textbf{GOLD-NAS}\n        & \\textbf{IF-NAS}\n      \\\\\n    \\midrule\n      \\textbf{$L$ = 4}\n      & $27.6\\pm0.3$\n      & 29.8\n      & 28.8\n      & 26.7\n      \\\\\n      \\textbf{$L$ = 6}\n      & $28.0\\pm0.3$\n      & 30.2\n      & 29.0\n      & 26.9\n      \\\\\n     \\textbf{$L$ = 8}\n      & $28.9\\pm0.4$\n      & 31.6\n      & 30.8\n      & 27.3\n      \\\\\n    \\bottomrule\t\n\\end{tabular}\n}\n\\end{table}\n\nIf we do not derive architectures with MAdds and preserve one input for each node, the results are shown in Tab.~\\ref{tab.ablation_keep1}. Interestingly, without the protection of keeping so many MAdds, DARTS and GOLD-NAS show completely failed results. When $L$=4, the results of DARTS and GOLD-NAS are $29.8\\%$ and $28.8\\%$, which are even much worse than the randomly generated architectures. When $L$ increases to 8, the performances of DARTS and GOLD-NAS further degrade. The architectures searched without suppressing interleaved connections only preserve very few nodes, which damages the final performance dramatically. On the contrary, IF-NAS still achieves comparable results even without any MAdss restriction.\n\n\nThe above experiments imply interleaving connections cause non-negligible impacts to DNAS methods when the search space grows larger and more complicated. IF-NAS is able to handle this challenge by the proposed interleaving-free sampling.\n\n\n\n\\begin{figure}[thbp]\n    \\centering\n    \\begin{minipage}{0.49\\textwidth}\n        \\centering\n        \\includegraphics[width=0.99\\textwidth]{withIF_v3.pdf}\n        \\captionof{figure}{Architectures searched with IF. The reserved connections are mainly serial pattern, which guarantees the depth of architectures.}\n        \\label{fig:architecture_withIF}\n    \\end{minipage}%\n    \\hspace{1pt}\n    \\begin{minipage}{0.49\\textwidth}\n        \\centering\n        \\includegraphics[width=0.95\\textwidth]{withoutIF_v3.pdf}\n        \\captionof{figure}{Architectures searched without IF. The reserved connections are mainly parallel pattern, resulting in a short longest-path at each stage, that is, the depth of networks are shallow.}\n        \\label{fig:architecture_withoutIF}\n    \\end{minipage}%\n   \n\\end{figure}\n\n\n\n\\begin{table}\n\\centering\n\\caption{Results of exploring the effectiveness of IF sampling on the $L$-chain space. All the architectures are trained on ImageNet-1k for 100 epochs.}\n\\label{tab.ablation_if}\n\\resizebox{0.42\\textwidth}{!}{\n\\begin{tabular}{@{}lcccccc@{}}\n    \\toprule\n        \\textbf{Setting}\n        & \\textbf{Depth}\n        & \\textbf{Params (M)} \n        & \\textbf{Err. (\\%)}\n      \\\\\n    \\midrule\n      w\/o IF\n     \n     \n     \n      & $22.5\\pm 2.0$\n      & $5.2\\pm 0.2$\n      & $26.7\\pm 0.3$\n      \\\\\n      w\/ IF\n   \n   \n   \n      & $43\\pm 3$\n      & $ 5.3 \\pm 0.1$\n      & $26.0\\pm 0.3$\n      \\\\\n    \\bottomrule\t\n\\end{tabular}\n}\n\\end{table}\n\n\n\n\\subsection{IF Sampling Effectiveness on $L$-chain Space}\n\n\nIn this part, we design a comparative experiment to verify the effectiveness of IF sampling.\nWe search for 4 times independently for each setting, and the results are shown in Tab.~\\ref{tab.ablation_if}. The \\textbf{Depth} in Tab.~\\ref{tab.ablation_if} refers to the depth summation of 3 stages, which represents the longest path length of the network. And all architectures are train on ImageNet-1k for 100 epochs.\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=8.5cm]{weight_change_v1.pdf}\n\\caption{(a): The weight changing curve of candidate inputs with IF. (b): The weight changing curve of candidate inputs without IF. The darker the color, the closer the corresponding candidate inputs are to the activated node.}\n\\label{fig:weight_change}\n\\end{figure}\n\n\nWe find the accuracy of the architectures searched without IF (``w\/o IF'') sampling is much worse than that with IF (``w\/ IF'') sampling by a margin of $0.7\\%$ with similar Params. It is worth noting that the depth of w\/o IF is much smaller than w\/ IF, because the interleaved connections produce serious interference to the search process. This verifies that our IF-NAS can effectively handle interleaved connections and guarantee the search performance.\n\n\n\nThe visualizations of searched architectures are shown in Fig.~\\ref{fig:architecture_withIF} and Fig.~\\ref{fig:architecture_withoutIF}. We can find that w\/ IF preserves many serial pattern connections and w\/o IF  mainly preserves parallel connections, as a result, the architectures of w\/ IF is much deeper than w\/o IF. Fig.~\\ref{fig:weight_change} shows some weight change curves of w\/ IF and w\/o IF. One can see that w\/ IF can select more closer inputs, which derives deeper architectures. w\/o IF prefers farther nodes, which usually leads to parallel connections and discards more nodes.\n\n\n\n\\subsection{Effects of Interleaved Connections}\n\n\\begin{table}\n\\fontsize{8.0}{10.5}\\selectfont\n\\centering\n\\caption{Results of retaining architectures searched with different number of interleaved connections (IC) involved.}\n\\label{tab.ablation_twisting}\n\\resizebox{0.45\\textwidth}{!}{\n\\begin{tabular}{@{}ccccc@{}}\n    \\toprule\n        \\textbf{IC Number}\n        & \\textbf{Depth}\n        & \\textbf{Params (M)}\n        & \\textbf{Err. (\\%)}\n      \\\\\n    \\midrule\n      0\n   \n   \n   \n      & $41.5\\pm 2.5$\n      & $5.0 \\pm 0.1$\n      & $19.1\\pm 0.2$\n      \\\\\n      1\n     \n     \n     \n      & $26.0\\pm 3.0$\n      & $4.9\\pm0.3$\n      & $19.7\\pm0.3$\n      \\\\\n       2\n     \n     \n     \n      & $24.5\\pm 2.5$\n      & $5.0 \\pm 0.3$\n      & $19.9\\pm 0.4$\n      \\\\\n       3\n      & $24.0\\pm 3.0$\n      & $5.0\\pm 0.2$\n      & $20.0\\pm 0.3$\n      \\\\\n    \\bottomrule\t\n\\end{tabular}\n} \n\\end{table}\n\nWe allow various numbers (1\/2\/3) of interleaved connections to be sampled during search to understand the effects they may bring.\nThe searched architectures are trained on ImageNet-100 for 100 epochs and the results are shown in Tab.~\\ref{tab.ablation_twisting}. We find with more interleaved connections added, the performance degrades more. This shows more interleaved connections cause greater impact on the search. It implies that Interleaving-Free is necessary in such a flexible space. Interestingly, the performance drops greatly even through one interleaved connection is sampled during searching, which demonstrates that as long as the interleaved connection is involved, interference is generated.\n\n\n\n\n\\subsection{Comparison with Cell-based Micro Spaces}\n\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=8.0cm]{acc_complex_v5.pdf}\n\\caption{Results on different spaces using 3 search strategies. DNAS indicates the differential-based search without IF. The random indicates random search.}\n\\label{fig:acc_complex}\n\\end{figure}\n\n\n\nIn this section, we test IF-NAS in two micro search spaces, DARTS space referred to as $space_D$ and GOLD-NAS space referred to as $space_G$. The two spaces adopt the same $3\\times3$ separable convolution and skip-connection operators as ours, thus that the comparison is fair. The results are shown in Fig.~\\ref{fig:acc_complex}. The horizontal axis denotes the space complexity (the exponential base of 10) and the vertical axis denotes the accuracy trained on ImageNet-1k for 100 epochs. The bubble sizes imply the complexity of spaces. The results on $space_D$ and $space_G$ are close among 3 strategies (even with random search), which implies the 2 spaces are inflexible and have a protection mechanism (the cell design) to help the search methods. The $L$-chain spaces are more flexible and difficult so that the gaps between IF-NAS and others in $L$-chain spaces are much larger. Moreover, the larger the spaces, the worse the performances of DNAS and random search. However, our IF-NAS provides all best results in the 5 spaces, which shows that IF-NAS are robust and generalize well to micro spaces.\n\nWe also test our IFNAS by training 250 epochs under the mobile setting and the results are shown in Table~\\ref{tab:micro_results}. IFNAS achieves $24.2\\%$ test error in $space_D$, which is same as PCDARTS (a specially designed method in this space). In $space_G$, our IFNAS-G1 beats GOLD-I by $0.2\\%$ with $0.2$M fewer parameters and our IFNAS-G2 surpasses GOLD-X with $0.4$M fewer parameters.\n\n\\begin{table}[]\n\\fontsize{8.5}{10.5}\\selectfont\n\\centering\n\\setlength{\\tabcolsep}{1.0mm}\n    \\caption{Results on two micro search spaces.}\n    \\begin{tabular}{ccc|ccc}\n    \\toprule\n         \\multicolumn{3}{c|}{\\textbf{$space_D$}}  &\\multicolumn{3}{c}{\\textbf{$space_G$}} \\\\ \\hline\n         arch.  & para. (M)  & Err. (\\%)  &arch.  & para. (M)  &Err. (\\%) \\\\ \\hline\n         DARTS  &4.7    &   26.7   & GOLD-I   & 5.3   &  24.7  \\\\\n         PDARTS   &4.9   & 24.4    & GOLD-X    & 6.4   &24.3\\\\\n         PCDARTS   &5.3   &24.2     & IF-NAS-G1  & 5.1   &24.5   \\\\\n         IF-NAS    &5.3   &\\textbf{24.2}     &IF-NAS-G2   &6.0   &\\textbf{24.2}  \\\\\n    \\bottomrule \n    \\end{tabular}\n    \n    \\label{tab:micro_results}\n\\end{table}\n\n\n\n\\subsection{More Examples of Interleaved Connections}\n\n\nIn Section 3.3, we show failure cases from the simplest situation. Here we perform more experiments to show the influence of interleaved connections and the weight change curves are shown in Figure~\\ref{fig:noise_all}. In Fig.~\\ref{fig:noise_all} (a), we run 4 times independently under interleaving-free setting and the nearest candidate inputs have a dominant weight at this time. In Fig.~\\ref{fig:noise_all} (b)-(d), We run 4 times independently and each time we add 1-3 different interference connections. One can see that the more interleaved connections are added, the harder it it the nearest candidate input is picked.\n\n\n\\begin{figure}[]\n    \\centering\n    \\includegraphics[width=0.8\\linewidth]{noise_all_v7.pdf}\n    \\caption{More examples of adding different numbers of interleaved connections to impact the NAS algorithm. (a) is under the interleaving-free setting and 1-3 interleaved connections are added for (b)-(d).}\n    \\label{fig:noise_all}\n\\end{figure}\n\n\n\\section{Conclusions}\n\nThis work extends the design of search space towards higher flexibility by adding lots of long-range connections to the super-network. The long-range connections raise new challenges to the search algorithm, and we locate the problem to be the interference introduced by interleaved connections during search. We propose IF-NAS based on this observation, which makes a schedule to sample sub-networks and thus guarantees that interleaved connections do not occur.\n\nSearching in a more complicated space has been of critical importance for NAS. Our research delivers the message that new properties\/challenges of NAS emerge when the search space is augmented. We advocate for more efforts in this direction to improve the ability of the search algorithms.\n\n\\vspace{0.1cm}\\noindent\n\\textbf{Limitations of this work.} First, the architectures searched in the complicated spaces do not show advantages than the existing cell-based ones, which implies that the greater flexibility we introduced does not necessarily lead to better results but greater difficulties.  Second, IF-NAS needs an extra warm up process to make up for the accuracy drop of super-network caused by sampling operation.\n\n\n{\\small\n\\bibliographystyle{ieee_fullname}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nConvolutional neural networks (CNNs) have achieved impressive success in many vision tasks such as image classification~\\cite{he2016deep, huang2017densely, sandler2018mobilenetv2}, object detection~\\cite{lin2017focal, ren2015faster, redmon2016you}, and image generation~\\cite{Goodfellow2014GenerativeAN, Liu2021CTSF}. To pursue high performance, the design of CNNs tends to be more and more complicated. These structures demand huge computation and storage resources, which are unavailable in resource-limited devices like mobile phones. In order to address this problem, many researchers propose a number of model compression techniques such as lightweight model design~\\cite{sandler2018mobilenetv2, ma2018shufflenet, hu2018squeeze}, model quantization~\\cite{wu2016quantized, jacob2018quantization, Xie2020DeepTQ}, and model pruning~\\cite{he2017channel, zhuang2018discrimination, chin2020towards}. In recent years, knowledge distillation~\\cite{hinton2015distilling, romero2014fitnets, yuan2020revisiting} has attracted widespread attention, which aims to improve the performance of a small model (student) by exploiting the knowledge of a large well-trained model (teacher).\n\nExisting knowledge distillation methods mainly focus on what kind of knowledge the teacher has learned (\\eg, logit-based~\\cite{hinton2015distilling}, feature-based~\\cite{romero2014fitnets} and relation-based~\\cite{peng2019correlation} knowledge) and how to transfer the knowledge from the teacher to the student (\\ie, proper loss function). However, the teacher-student pairs are sometimes significantly different (in structure and size), which leads to extremely learning difficulty for the student. To alleviate this problem, some works~\\cite{cho2019efficacy, mirzadeh2019improved} seek the proxy teachers to reduce the difficulty of the teacher's knowledge. Specifically, Cho \\etal~\\cite{cho2019efficacy} propose that the student can gain better performance through learning the knowledge from an early-stopped teacher model rather than final converged one. Recently, Mirzadeh \\etal~\\cite{mirzadeh2019improved} first transfer the knowledge to medium capacity assistant models, then distill knowledge from assistants to the student, which is multi-step knowledge distillation framework. \n\nHowever, although these proxy-based methods ease optimization difficulty for the student to a certain extent, there still remain the following problems. First, the core knowledge of teacher might be lost, which would prevent the student from learning useful information in the distillation process. The key idea of the early-stopped and assistant teachers is to reduce the teacher's capacity such that the teacher's knowledge can be accepted by the student. In this process, they lose part of knowledge from the original teacher, which will be discussed in detail in section~\\ref{sec:analysis}. Second, the whole training process is complicated and time-consuming. These methods require lots of grid searches and additional training to obtain an optimal early-stopped anchor or teacher assistant in different knowledge distillation scenarios.\n\nTo avoid the above issues, in this paper, we provide a new perspective to alleviate the aforementioned learning difficulty problem. Rather than rudely reduce the difficulty of the teacher's knowledge, we advocate strengthening the student's capacity. To achieve this, we exploit the idea of collaborative learning that many collaborators\/learners form a learning group in order to reach the same goal. Given multiple small-sized students, we can effectively exploit the knowledge of the teacher via ensemble learning. However, such cumbersome student models violate the purpose of model compression. Thus, how to form a learning group while avoiding incurring additional parameter overhead in inference remains a question.\n\nIn this paper, for the sake of improving the student's capacity while keeping the student be efficiency in inference, we propose an Adversarial Collaborative Knowledge Distillation (ACKD\\xspace) method that can significantly improve the performance of knowledge distillation. In the training phase, we build the student model with multiple auxiliary learners. These learners form a learning group and collaborate to achieve the same goal that fully exploits the knowledge of the teacher. For the purpose of achieving better collaboration, we introduce the attention mechanism to filter the information of auxiliary learners. Moreover, to enhance the ability of auxiliary learners to learn more comprehensive knowledge, the auxiliary learners are urged to be diverse. To this end, we distinguish their distributions using adversarial learning via discriminators. As a result, the proposed student model is more powerful to exploit the teacher's knowledge. In the inference phase, we discard all auxiliary learners such that the final student is compact and effective. Extensive experiments on four benchmarks demonstrate the effectiveness of proposed ACKD\\xspace.\n\nOur contributions are summarized as follows:\n\\begin{itemize}\n    \\item We propose an Adversarial Collaborative Knowledge Distillation (ACKD\\xspace) method that effectively alleviates the capacity gap problem in knowledge distillation. Unlike existing methods that rudely reduce the difficulty of the teacher's knowledge, we alleviate this problem from a new perspective of strengthening the student's capacity.\n    \\item We propose an adversarial collaborative learning strategy that effectively improves the student's capacity. To this end, we introduce the attention mechanism to recognize the importance of different auxiliary learners. Moreover, we introduce the adversarial learning to promote representational diversity.\n    \\item To investigate the effectiveness of our proposed method, we conduct extensive experiments on four classification datasets. These results demonstrate that our ACKD\\xspace outperforms state-of-the-art methods.\n\\end{itemize}\n\n\n\n\\section{Related Work}\n\\textbf{Knowledge Distillation.} Knowledge distillation is mainly applied to the field of model compression~\\cite{han2015deep, rastegari2016xnor, ye2019student, liu2021semi}, which trains a small and effective student model with the help of large teacher models' knowledge. As a pioneer work, Hinton \\etal~\\cite{hinton2015distilling} take output probabilities as knowledge, and the student learns these knowledge by minimizing Kullback\u2013Leibler (KL) divergence. After that, Romero \\etal~\\cite{romero2014fitnets} exploit the intermediate representations' knowledge by matching the feature activations of the teacher and the student. Besides, Zagoruyko \\etal~\\cite{Zagoruyko2017PayingMA} introduce attention mechanism to the knowledge transfer, and improve the student's performance by mimicking attention feature map. Recently, some works define the knowledge according to the relation between layers or samples. Specifically, Tung \\etal~\\cite{tung2019similarity} urge the student to learn how to preserve the pairwise similarities. Tian \\etal~\\cite{tian2019contrastive} introduce contrastive objectives by constructing positive and negative pairs. However, most existing works ignore the capacity gap between the teacher and the student, which may result in performance degradation~\\cite{cho2019efficacy, jin2019knowledge}. To address this problem, Cho \\etal~\\cite{cho2019efficacy} reduce the gap via an early-stopped teacher, and Mirzadeh \\etal~\\cite{mirzadeh2019improved} introduce teacher assistants to simplify the teacher's knowledge. Unlike these methods, in this paper we bridge the gap from the perspective of improving the student's capacity.\n\n\\textbf{Collaborative Learning.} Collaborative learning refers to two or more learners work together to solve a common task~\\cite{dillenbourg1999collaborative}. For that purpose, learners learn from based on their diverse strengths such that they reach goals more efficiently. Recently, some studies have introduced this idea to improve the training efficiency of deep neural networks with knowledge distillation~\\cite{Lan2018KnowledgeDB, li2020online}. To be specific, Song \\etal~\\cite{song2018collaborative} design a hierarchical structure of multiple branches, which are trained simultaneously to improve backbone's performance. Chen \\etal~\\cite{chen2020online} introduce self-attention to improve the branch diversity and distill the knowledge of auxiliary peers into main branch. Additionally, some works apply collaborative learning across networks~\\cite{chen2020online, chung2020feature}. Zhang \\etal~\\cite{Zhang2018DeepML} propose a deep mutual learning training fashion, in which all networks distill knowledge to each other simultaneously. Guo \\etal~\\cite{DBLP:conf\/cvpr\/GuoWWYLHL20} train networks under the supervision of ensemble targets. In this paper, we follow the idea of collaborative learning to improve the imitation ability of the student network in knowledge distillation.\n\n\\textbf{Deep Supervision.} Deep supervision is proposed to alleviate the gradient vanish problem in deep neural networks by attaching auxiliary classifier to intermediate layers~\\cite{lee2015deeply, szegedy2015going}. These auxiliary classifiers will be removed in the testing phase to keep the efficiency of the model. This training mechanism has been widely proved to be effective in different applications, such as edge detection~\\cite{xie2015holistically}, object detection~\\cite{liu2016ssd}, and semantic segmentation~\\cite{zhao2017pyramid}. Recently, some works~\\cite{yao2020knowledge, zhang2020task} apply this idea to the field of knowledge distillation. In this paper, we extend this idea to construct multiple auxiliary learner to collaboratively learning knowledge from the teacher network.\n\n\\textbf{Adversarial Learning.} Generative adversarial networks~\\cite{goodfellow2014generative} is a generative model framework, which consists of a generator and a discriminator. The generator generates fake images from the noise to deceive the discriminator while the discriminator tries to distinguish the real and fake images. Recently, some works~\\cite{shen2019meal, chung2020feature, zhang2020amln} introduce this adversarial learning strategy to discriminate the features generated by different networks. In this paper, we regard the auxiliary learners as generators and use discriminators to distinguish the output features between different learners. In this way, we improve the expressive diversity of different auxiliary learners.\n\n\n\n\\section{Proposed Method}\n\n\n\n\\begin{table}[t]\n\\begin{center}\n\\resizebox{0.48\\textwidth}{!}{\n\\begin{tabular}{c|cc}\n        \\toprule \n        Method & ESKD~\\cite{cho2019efficacy} & TAKD~\\cite{mirzadeh2019improved}\\\\\n        \\midrule\n        Teacher (T) & WideResNet-28-4 & WideResNet-28-4\\\\\n        Proxy (P) & WideResNet-28-4 (65)  & WideResNet-28-2 \\\\\n        Student (S) & resnet56 & resnet56\\\\\n        \\midrule\n        Acc(T) & 78.91 & 78.91 \\\\\n        Acc(P) & 76.30 & 76.78 \\\\\n        AccGap(T, P) & 2.61 & 2.13 \\\\\n        \\midrule\n        CKA(T, P)  & 0.7761 & 0.8234\\\\\n        CKA(P, S)  & 0.8016 & 0.8113\\\\\n        CKA(T, S)  & 0.7695 & 0.7770\\\\\n        \\midrule\n        KL(T, P)  & 0.6627 & 0.7710\\\\\n        KL(P, S)  & 0.8495 & 0.8391\\\\\n        KL(T, S)  & 0.9319 & 0.9166\\\\\n\t\t\\bottomrule\n\t\t\\end{tabular}}\n\\end{center}\n\t\\caption{Comparasions of different proxy methods. Acc($\\cdot$) denotes Top-1 accuracy ($\\%$). AccGap(T, P) = Acc(T) - Acc(P). CKA($\\cdot$,$\\cdot$) and KL($\\cdot$,$\\cdot$) denote CKA similarity~\\cite{kornblith2019similarity} and Kullback\u2013Leibler divergence between two networks, respectively. The number inside the parentheses is the total number of training epochs with early-stopped strategy.}\n\t\t\\label{tab:proxy}\n\\end{table}\n\n\nIn this section, we will introduce our proposed method in detail. In section~\\ref{sec:analysis}, we analyze the advantages and shortcomings of existing proxy-based ways to motivate our method. In section~\\ref{sec:acl}, we introduce the overview of proposed method that improves the performance of the student via an adversarial collaboration learning strategy. Then, we introduce the optimization and the whole knowledge flow paradigm in our framework in sections~\\ref{sec:t-a}, \\ref{sec:ac}, and \\ref{sec:self-distillation}.\n\\subsection{Analysis on Proxy-based Approaches}\\label{sec:analysis}\nThe vanilla KD method is a two-stage process in which a high capacity teacher model is trained and then used for distillation. However, due to the high capacity divergence between the teacher and the student, it results in a common phenomenon that the student is hard to reach the teacher's performance after the distillation process.\n\nTo alleviate this divergence, many researchers try to distill knowledge from the teacher to the student in a more smooth way. One of the most well-known methods is TAKD~\\cite{mirzadeh2019improved}, which utilizes proxy models to alleviate the gap, aiming to provide an easier learning curve for the student. Another insightful method is to reduce the teacher's capacity by using early-stop teacher regularization~\\cite{cho2019efficacy}, in this way, the student can gain better performance. \n\nA major advantage of these methods is to smooth the student's learning curve and mitigate the mismatched capacity between teacher and student. However, there are certain drawbacks associated with the use of the proxy model and early-stop teacher model. Firstly, although these approaches alleviate the student's learning burdens, they also lose abundant knowledge of the original teacher in the distillation process, which may cause that the student model can not \ndraw some key information. As Table~\\ref{tab:proxy} showed, we use qualitative analysis in order to gain insights into the information loss in the distillation process. We conduct experiments with the teacher-student pair of WRN-28-4 and resnet56 on CIFAR-100. Specifically, we use both CKA~\\cite{kornblith2019similarity} and KL divergence to measure the representational similarity among the teacher, the proxy, and the student. The larger the CKA, the more similar the representations among the teacher, the proxy, and the student. The larger the KL divergence, the greater the difference between the two distributions. Besides, we use the relative accuracy (AccGap) to evaluate the capacity gap among the teacher and the proxy. From Table~\\ref{tab:proxy}, the CKA of the teacher-proxy pair is higher than the teacher-student pair while KL is quite the reverse, which means that the student model suffers the loss of information during the proxy distillation process. Meanwhile, both AccGap of ESKD and TAKD are more than 2\\%, also proving the existence of capacity gap and information loss between teacher and proxy model. Secondly, these methods require lots of grid searches to obtain an optimal proxy model (\\ie, early-stopped anchor or teacher assistant), which is time-consuming and inefficient.\n\nThus, how to maintain the teacher's capacity (original abundant information) while improving the students' capacity has become a prisoner's dilemma problem: reducing the difficulty of student learning would hurt teachers' capacity while a large and well-trained teacher would mismatch its student capacity.  \n\n\\begin{figure*}[t]\n    \\centering\n\t\\includegraphics[width=0.9\\linewidth]{framework2.pdf}\n\t\\caption{An overview of the proposed ACKD\\xspace. In the training phase, we first construct the student model with auxiliary learners (gray boxes). Then, we improve the student's capacity via an adversarial collaborative learning strategy. In the inference phase, we only use the target learner and remove all auxiliary learners. (Best viewed in color)}\n\t\\label{fig:framework}\n\\end{figure*}\n\n\\subsection{Adversarial Collaborative Learning}\\label{sec:acl}\nImagine a real-world scenario where incomprehensible (complex) knowledge is given by the teacher model, and the student is asked to absorb these contents by its own small-capacity network. The capacity gap discussed in the previous subsection occurs and makes the knowledge distillation less effective than it could be. To alleviate this issue, we seek to exploit the idea of collaborative learning that different learners work together to solve a common task. Inspired by this, we propose to strengthen the student's capacity by forming a learning group rather than rudely reduce the difficulty of the teacher's knowledge. In this way, it is possible to achieve better distillation performance without damaging the teacher's original capacity and semantic information.\n\nTo form a learning group that exploits the knowledge of the teacher, we can build a multi-students~\\cite{you2018learning} ensemble learning framework~\\cite{dietterich2000ensemble}. However, such a cumbersome model is inconsistent with our goal of model compression. In order to address this problem, in this paper, we propose to construct the student model with multiply branches. For convenience, we call these branches as auxiliary learners, and use them to form a learning group. The backbone model without auxiliary branches is named as target learner. In practice, we select different layers branch from the target learner as our auxiliary learners' inputs. The auxiliary learners are composed of different high-level feature extractor blocks. In this way, the auxiliary learners share the low-level feature information but have their own high-level semantic information. \n\nThe auxiliary learners are asked to reach the same objective that exploits the knowledge of the teacher in the greatest degree. To fully extract the representation information of the auxiliary learners, we introduce an attention mechanism to collaborative learning. In this way, each auxiliary is urged to focus on what it does well. However, auxiliary learners share the same low-level layers, which might lead to a lack of diversity in expressivity. To alleviate this issue, we introduce adversarial learning to promote the diversification of auxiliary learners. In the whole training phase, we hope that the auxiliary learners are in the state of \\textit{collaborative learning} while exploring unique representation via \\textit{adversarial learning}. In the inference phase, we discard all auxiliary learners to obtain a compact and effective model. For simplicity, we call our training framework as Adversarial Collaborative Knowledge Distillation (ACKD\\xspace).\n\nFigure~\\ref{fig:framework} shows the overall architecture. In practice, the teacher's knowledge first flows to auxiliary learners. Then, the auxiliary learners mine this knowledge via an adversarial collaborative manner. In this process, the knowledge flows to shared low-level layers of the target learner. Besides, the target learner exploits the knowledge from auxiliary learners, which is a more smooth learning manner than directly learning from the teacher. In the following, we will describe in detail how the knowledge flows and the parameters update in our framework.\n\n\\subsection{Teacher-Auxiliary Learners Knowledge Flow}\\label{sec:t-a}\nWe follow the common KD approach to distill teacher's knowledge into auxiliary learners. Suppose that we train the models on the dataset $\\mD = \\{(x_i, y_i)\\}_{i=1}^{N}$, where $N$ is the number of samples. We can obtain the logits $z_i$ of $i^{th}$ sample. Then we can get the softened output probability of a model by applying a temperature factor as follows:\n\\begin{equation}\\label{eq:soft_target}\n    p_i=\\frac{exp(z_i^j\/\\tau)}{\\sum_{j=1}^{C}exp(z_i^j\/\\tau)}, \n\\end{equation}\nwhere $\\tau$ is a temperature factor. In this way, we can get $p_i^t$ of the teacher and $p_i^a$ of auxiliary learners. The auxiliary learners learn the knowledge of the teacher by minimizing Kullback-Leibler (KL) divergence between these two output distributions as follows:\n\\begin{equation}\\label{eq:kl}\n    \\mL_{kd}^a=KL(p^a,p^t)=\\sum_{x\\sim \\mD}p^a\\log\\frac{p^a}{p^t}.\n\\end{equation}\n\nTaking advantage of collaborative learning, the output distribution of auxiliary learners is easier to fit that of the teacher. Through backward propagation, we are able to get more robust low-level representations of the target learner, which is conducive to the target learner's optimization.\n\n\\subsection{Adversarial Collaborative Module}\\label{sec:ac}\nIn this subsection, we introduce how the auxiliary learners collaborate to exploit the teacher's knowledge via an Adversarial Collaborative Module (ACM), as shown in Figure~\\ref{fig:acm}. For simplicity, we use two auxiliary learners as examples for illustration. In ACM, there are two complementary learning steps: attention-based collaborative learning and adversarial-based diversity learning.\n\n\\subsubsection{Attention-Based Collaborative Learning}\nTo take advantage of different auxiliary learners' ability, we present the attention-based collaborative module to enhance the representations of auxiliary learners. First, we concatenate all feature maps from different auxiliary learners. Then, we input the fused feature into a two-layer perceptron with a softmax to recognize the importance of different learners. Formally, the attention weight vector and output logits of the learning group can be computed as follows:\n\\begin{equation}\\label{eq:attention}\n\\begin{split}\n    \\bx &= g({\\rm Concat}(h^{(1)}, .., h^{(M)})), \\\\\n    \\ba^{(k)} &= {\\rm Softmax}({\\rm MLP}(\\bW_m, \\bx)), \\\\\n    z &= \\sum_{k=1}^M\\ba^{(k)}z^{(k)}, \n\\end{split}\n\\end{equation}\nwhere $M$ is total number of auxiliary learners; $h^{(k)}$ denotes the $k^{th}$ auxiliary learner's feature map; $g(\\cdot)$: $\\mathbb{R}^{C \\times H \\times W} \\rightarrow \\mathbb{R}^{C \\times HW}$ flattens the feature maps in spatial dimension; ${\\rm \\bW}_{m}$ denotes the parameters of ${\\rm MLP}$; $\\ba^{(k)}$ is the attention weight of $k^{th}$ auxiliary learner. After we get attention-based logit, we compute output probability in Eqn~\\ref{eq:soft_target}. We update the parameters of auxiliary learners and ${\\rm MLP}$ in Eqn~\\ref{eq:kl}.\n\n\\subsubsection{Adversarial-Based Diversity Learning}\n\nAlthough each auxiliary learner has its own branch structure, their feature inputs come from the target learner and the knowledge learned in the auxiliary learners is easy to overlap between each other. It is still difficult to avoid the homogeneity of their network parameters, which may hurt the performance of the ensemble result and even distillation performance. To alleviate this problem, we leverage the adversarial training to force auxiliary learners to learn diverse feature distribution. As Figure~\\ref{fig:acm} illustrated, each learner has a corresponding discriminator aiming at distinguishing whether the feature output belongs to their own. \n\nFollowing the traditional adversarial training definition, we regard the auxiliary learners' network feature as generative distribution and name the $i^{th}$ learner's discriminator as $D_i$. The architecture of the discriminator is a simple stack of $1\\times1$ Conv with ReLU. Each auxiliary learner tries to fool the discriminator while the discriminator need to discriminate the right source of feature. Our overall adversarial loss for $i^{th}$ discriminator is as follows:\n\\begin{equation}\\label{eq:adi}\n    \\mL_{ad}^i=\\frac{1}{M-1}\\sum_{j\\neq i}\\mathop{\\mathbb{E}}\\limits_{x\\sim p(i)}{\\rm log}D_i(x)+\\mathop{\\mathbb{E}}\\limits_{x\\sim p(j)}{\\rm log}(1-D_i(x)),\n\\end{equation}\nwhere $p(i)$ denotes the output feature produced by $i^{th}$ auxiliary learner, $M$ is total number of auxiliary learners. Final adversarial loss is a average of the individual ones:\n\\begin{equation}\\label{eq:ad}\n    \\mL_{ad}=\\frac{1}{M}\\sum_{i=1}^M\\mL_{ad}^i.\n\\end{equation}\n\nThrough adversarial training manners, we can make better use of the diversity of semantic information contained in the auxiliary learners. Meanwhile, the parameters of target learner also are updated in this process.\n\n\\begin{figure}[t]\n    \\centering\n\t\\includegraphics[width=1\\linewidth]{acm.pdf}\n\t\\caption{An overview of the proposed ACM. }\n\t\\label{fig:acm}\n\\end{figure}\n\n\\begin{table*}[t]\n\\begin{center}\n\\begin{tabular}{c|ccccc}\n\t\t\t\\toprule \n\t\t\tTeacher & WideResNet-40-2 & resnet110 & MobileNetV2 (1.4) & ShuffleNetV1 & WideResNet-28-4  \\\\\n\t\t\tStudent & WideResNet-40-1 & resnet20 & MobileNetV2 (0.5) & MobileNetV2 (0.5) & resnet56 \\\\\n\t\t\t\\midrule\n\t\t\tTeacher & 76.22 & 74.00 & 69.57 & 71.85 & 78.91 \\\\\n\t\t\tStudent & 71.78 & 69.20 & 65.17 & 65.17 & 73.05 \\\\\n\t\t    \\midrule\n\t\t\tAT~\\cite{Zagoruyko2017PayingMA} & 72.94 & 70.78 & 66.26 & 65.55 & 73.64 \\\\\n\t\t\tSP~\\cite{tung2019similarity} & 73.18 & 70.69 & 61.52 & 67.67 & 74.28 \\\\\n\t\t\tCC~\\cite{peng2019correlation} & 72.22 & 70.30 & 65.39 & 65.16 & 73.16 \\\\\n\t\t\tVID~\\cite{ahn2019variational} & 72.52 & 70.26 & 65.15 & 65.49 & 73.24 \\\\\n\t\t\tCRD~\\cite{tian2019contrastive} & 73.57 & 70.88 & 66.47 & 67.32 & 74.34\\\\\n           \tSemCKD~\\cite{chen2020cross}& 73.32 & 70.41 & 67.70 & 67.19 & 73.13\\\\\n\t\t\tKD~\\cite{hinton2015distilling} & 73.54 & 70.73 & 67.94 & 68.19 & 74.21 \\\\\n\t\t\tESKD~\\cite{cho2019efficacy} & 73.21 & 70.69 & 67.61 & 67.82 & 74.61\\\\\n\t\t\tTAKD~\\cite{mirzadeh2019improved} & 73.52 & 71.42 & 68.88 & 68.71 & 75.29 \\\\\n\t\t\t\\midrule\n\t\t    ACKD\\xspace (Ours) & \\textbf{75.29} & \\textbf{72.38} & \\textbf{69.43} & \\textbf{69.48} & \\textbf{75.91}\\\\\n\t\t\n\t\t    \\bottomrule\n\t\t\\end{tabular}\n\\end{center}\n\t\\caption{Comparison with state-of-the-art models on CIFAR-100.}\n\t\t\\label{tab:cifar-100}\n\\end{table*}\n\n\\begin{table}[t]\n\\begin{center}\n\\begin{tabular}{c|cc}\n\t\t\t\\toprule \n\t\t\tTeacher & resnet110 &  WideResNet-28-4  \\\\\n\t\t\tStudent & resnet20 & resnet56 \\\\\n\t\t\t\\midrule\n\t\t\tTeacher & 84.62 & 87.74\\\\\n\t\t\tStudent & 82.29 & 83.95\\\\\n\t\t    \\midrule\n\t\t\tAT~\\cite{Zagoruyko2017PayingMA} & 83.84 & 85.16 \\\\\n\t\t\tSP~\\cite{tung2019similarity} & 79.75 & 84.34 \\\\\n\t\t\tCC~\\cite{peng2019correlation} & 83.89 & 85.14 \\\\\n\t\t\tCRD~\\cite{tian2019contrastive} & 84.11 & 84.99 \\\\\n            SemCKD~\\cite{chen2020cross} & 84.13&85.12\\\\\n\t\t\tKD~\\cite{hinton2015distilling} & 84.03 & 85.18\\\\\n\t\t\tESKD~\\cite{cho2019efficacy} & 83.78 & 85.44 \\\\\n\t\t\tTAKD~\\cite{mirzadeh2019improved} & 84.21 & 85.37\\\\\n\t\t\t\\midrule\n\t\t    ACKD\\xspace (Ours) & \\textbf{84.42} & \\textbf{85.63}\\\\\n\t\t\n\t\t    \\bottomrule\n\t\t\\end{tabular}\n\\end{center}\n\t\\caption{Comparison with state-of-the-art models on CINIC-10.}\n\t\t\\label{tab:cinic-10}\n\\end{table}\n\n\\subsection{Auxiliaries-Target Learner Knowledge Flow}\\label{sec:self-distillation}\n\nAfter the adversarial collaborative training, the auxiliary learners gain diverse and rich semantic information. Thus, it benefits the optimization of the target learner via shared low-level layers. Meanwhile, we dig these knowledge via a self-distillation manner (\\ie, minimizing the distribution divergence between the target learner and auxiliary learners). We formulate it as follows:\n\\begin{equation}\\label{eq:kl_S}\n    \\mL_{kd}^s=\\sum_{x\\sim \\mD}p^s\\log\\frac{p^a}{p^s},\n\\end{equation}\nwhere $p^s$ and $p^a$ are the output probability of the target learner and auxiliary learners, respectively. Such optimization is much easier than the vanilla KD process. The main reason is that the auxiliary learners have same structure with the target learner, and the target learner shares low-level layers with auxiliary learners. This special characteristic of structure makes the target learner be easier to understand the knowledge of auxiliary learners.\n\nThe overall training objective of ACKD\\xspace is as follows:\n\\begin{equation}\\label{eq:total}\n    \\mL=\\lambda_1\\mL_{kd}^a + \\lambda_2\\mL_{kd}^s + \\lambda_3\\mL_{ad} + \\lambda_4\\mL_{ce},\n\\end{equation}\nwhere $\\lambda_1$, $\\lambda_2$, $\\lambda_3$, $\\lambda_4$ are balanced hyperparameters, we set them to 1 in our experiments. $\\mL_{ce}$ denotes typical cross entropy loss for the target learner and auxiliary learners. \n\n\n\n\\section{Experiments}\n\\subsection{Datasets and Networks}\n\\textbf{Dataset.} We conduct experiments on four widely used benchmarks. 1) \\textit{CIFAR-100} is composed of 50K training images and 10K testing images, which are divided into 100 fine-grained categories. 2) \\textit{CINIC-10} consists of 90K training and 90K testing images, which are from CIFAR and ImageNet datasets. This dataset is more complex than CIFAR dataset. 3) \\textit{Tiny-ImageNet} has 200 classes, each of which contains 500 images for training and 50 images for validation. All images are at the resolution of $64\\times64$. 4) \\textit{ImageNet} is a large-scale dataset with 1000 classes, which contains 1.2 million images for training and 50k images for validation. All images are cropped at the resolution of $224\\times224$ for training and validation.\n\n\\begin{table*}[h]\n\\begin{center}\n\\resizebox{0.98\\textwidth}{!}{\n\\begin{tabular}{c|cc|cccccccc|c}\n\\toprule \n        & teacher & student &\n          AT~\\cite{Zagoruyko2017PayingMA} &  CC~\\cite{peng2019correlation} &SP~\\cite{tung2019similarity} &\n          CRD~\\cite{tian2019contrastive} &\n       \n        SemCKD~\\cite{chen2020cross}&\n          KD~\\cite{hinton2015distilling} & \n          ESKD~\\cite{cho2019efficacy} & \n          TAKD~\\cite{mirzadeh2019improved} &\n          ACKD\\xspace (Ours)\\\\\n\t\t\t\\midrule\n            Top-1 & 52.98 & 51.60 & 51.59 & 51.82 & 53.71 & 54.33 & 54.19 & 54.20 & 52.58 & 54.95 & \\textbf{55.97} \\\\\n\t\t\tTop-5 & 75.97 & 74.37 & 76.96 & 75.13 & 76.83 & 77.52 & 77.48 & 77.45 & 76.77 & 77.44 & \\textbf{78.15} \\\\\n\t\t\t\\bottomrule\n\t\t\\end{tabular}}\n\\end{center}\n\t\\caption{Test Accuracy ($\\%$) of different methods on Tiny-ImageNet.}\n\t\t\\label{tab:tiny-imagenet}\n\\end{table*}\n\n\\begin{table*}[h]\n\\begin{center}\n\\resizebox{0.98\\textwidth}{!}{\n\\begin{tabular}{c|cc|cccccccc|c}\n\\toprule \n        & teacher & student &\n          AT~\\cite{Zagoruyko2017PayingMA} & SP~\\cite{tung2019similarity} & CC~\\cite{peng2019correlation} & ONE~\\cite{Lan2018KnowledgeDB} &\n          CRD~\\cite{tian2019contrastive} & SemCKD$^\\dagger$~\\cite{chen2020cross} & KD~\\cite{hinton2015distilling}&ESKD$^\\dagger$~\\cite{cho2019efficacy} & \n          ACKD\\xspace (Ours)\\\\\n\t\t\t\\midrule\n\t\t\tTop-1 & 26.69 & 30.25 & 29.30 & 29.38 & 30.04 & 29.45 & 28.83 & 29.13 & 29.34 &29.12 &\\textbf{28.67} \\\\\n\t\t\tTop-5 & 8.58 & 10.93 & 10.00 & 10.20 & 10.83 & 10.41 & 9.87 & -&10.12&- & \\textbf{9.57} \\\\\n\t\t\t\\bottomrule\n\t\t\\end{tabular}}\n\\end{center}\n\t\\caption{Test error ($\\%$) of different methods on ImageNet. $^\\dagger$The authors only report Top-1 error.}\n\t\t\\label{tab:ImageNet}\n\\end{table*}\n\n\\textbf{Network Constructions.} We perform experiments with four different types of networks, including WideResNet~\\cite{zagoruyko2016wide}, resnet~\\cite{he2016deep}, MobileNetV2~\\cite{sandler2018mobilenetv2}, and ShuffleNetV1~\\cite{zhang2018shufflenet}. Specifically, WideResNet-d-w represents the WideResNet with depth $d$ and width factor $w$. MobileNetV2 (w) denotes MobileNetV2 with a width multiplier of $w$. To investigate the generalization ability of our method, we construct teacher-student pairs with similar structures (\\eg, resnet110-resnet20) and different structures (\\eg, ShuffleV1-MobileNetV2).\n\n\\textbf{Compared Methods.} We compare our proposed ACKD\\xspace with many state-of-the-art methods, including KD~\\cite{hinton2015distilling}, \tAT~\\cite{Zagoruyko2017PayingMA}, SP~\\cite{tung2019similarity},\nCC~\\cite{peng2019correlation},\nVID~\\cite{ahn2019variational}, CRD~\\cite{tian2019contrastive}, and\nSemCKD~\\cite{chen2020cross}. Besides, we compare our method with ESKD~\\cite{cho2019efficacy} and TAKD~\\cite{mirzadeh2019improved}, which also aims to alleviate the capacity gap. The implementations of compared methods are mainly based on author-provided codes and a open-resource benchmark~\\cite{tian2019contrastive}. As the authors' suggestions, we set the number of early-stopped training epochs to 65 (on CIFAR-100 and CINIC-10) and 35 (on Tiny-ImageNet and ImageNet) for ESKD. We choose the model, whose performance is close to average performance of the teacher and the student, as teacher assistant for TAKD (\\eg, we regard resnet44 (71.98\\%) as assistant model for the pair of resnet110 (74.00\\%) and resnet20 (69.20\\%) on CIFAR-100). \n\n\n\\textbf{Implementation Details.} On CIFAR-100 and CINIC-10 datasets, we run a total of 200 epochs for all methods with SGD optimizer. We set batch size to 128, momentum to 0.9, and weight decay to 5e-4. We initialize the learning rate to 0.1 (MobileNetV2 and ShuffleNetV1 to 0.05) and decay it by 0.1 at 100 and 150 epochs. On Tiny-ImageNet and ImageNet datasets, we train models for 100 epochs. Learning rate is initialized to 0.1 and decayed every 30 epochs.\n\n\\subsection{Comparasions with State-of-the-art Methods}\n\\textbf{Results on CIFAR-100.} We evaluate our method on three similar and two different structures of teacher-student pairs. We show the results on Table~\\ref{tab:cifar-100}. From this table, we have the following observations. 1) The methods that exploit the logit-based knowledge (\\ie, KD, ESKD, TAKD) show superiority over those feature-based methods. Such a phenomenon has also been found in CRD~\\cite{tian2019contrastive}. We claim that logit-based knowledge has not yet been fully exploited, which is overlooked in recent\nstate-of-the-art feature-based methods. 2) Our ACKD\\xspace is well generalized to different teacher-student pairs. On average, our ACKD\\xspace achieves higher accuracy than the vanilla student by 3.62\\%, ranging from 2.86\\% to 4.31\\%. 3) Our ACKD\\xspace performs better than all of the compared methods. Specifically, our method evenly outperforms vanilla KD and two related methods (\\ie, ESKD and TAKD) by 1.58\\%, 1.71\\%, 0.93\\% on all teacher-student pairs. Note that ESKD is inferior to vanilla KD in many teacher-student combinations. The main reason is that they lost a lot of teacher's knowledge. On the contrary, we improve the student's capacity to effectively exploit the teacher's knowledge.\n\n\\textbf{Results on CINIC-10.} In this experiment, we evaluate our method on two teacher-student pairs. As shown in Table~\\ref{tab:cinic-10}, our method achieves the best results on all pairs. Specifically, our ACKD\\xspace outperforms vanilla student by 1.91\\% on average, which is a relatively large improvement on CINIC-10 dataset. These results demonstrate the effectiveness of our method on more complex dataset.\n\n\\textbf{Results on Tiny-ImageNet.} In this experiment, we evaluate ACKD\\xspace with the pair of ResNet50 and ResNet18. From Table~\\ref{tab:tiny-imagenet}, we have the following observations. First, most distillation methods outperform the vanilla student by a large margin. It is worth noting that the students in many distillation methods even achieve higher accuracy than the teacher model. Such a phenomenon shows the importance and effectiveness of knowledge distillation on a more complex and challenging dataset. Second, our ACKD\\xspace performs better than all of the compared methods by at least 1.02\\% in terms of Top-1 accuracy. These results show the effectiveness and generalization of our ACKD\\xspace on more challenging datasets.\n\n\\textbf{Results on ImageNet.}\nIn this experiment, we compare our ACKD\\xspace with state-of-the-art methods by using the pair of ResNet34 and ResNet18. Note that TAKD~\\cite{mirzadeh2019improved} does not provide comparable ImageNet results. From Figure~\\ref{tab:ImageNet}, our proposed ACKD\\xspace achieves highest accuracy. Specifically, our ACKD\\xspace reduces the performance gap between the teacher and the student from 3.56\\% to 1.98\\%, a 44\\% relative improvement. These results demonstrate the scalability of our proposed ACKD\\xspace.\n\n\n\\begin{figure}[t]\n    \\centering\n\t\\includegraphics[width=1\\linewidth]{cam.pdf}\n\t\\caption{Gram-CAM visualization on Tiny-ImageNet. }\n\t\\label{fig:cam}\n\\end{figure}\n\n\\subsection{Effect of Adversarial Collaborative Module}\n\\textbf{Qualitative Analysis.} In this experiment, we analyze how the adversarial collaborative module works by Grad-CAM visualization. We randomly select several samples from the testing set on Tiny-ImageNet. We show the qualitative results of different parts (\\ie, teacher, target learner and auxiliary learners) of ACKD\\xspace framework in Figure~\\ref{fig:cam}. We observe that 1) The regions highlighted by different auxiliary learners are somewhat different. Interestingly, there always exists one learner activates similar region to the teacher (\\eg, auxiliary learner1 in the $1st$ and $4th$ column). 2) The target learner captures more semantic-related information than auxiliary learners even the teacher model. For example, in the $3rd$ column, the target learner successfully puts its attention on the whole target objective while the teacher distracts part of its attention to the background. The main reason is that the auxiliary learners capture diverse information, which helps the target learner learn more accurate and robust class-discriminative representation.\n\n\\textbf{Effect of proposed modules.} In this experiment, we conduct experiments to investigate the effectiveness of proposed modules. We show the results in Table~\\ref{tab:ablation-loss}. 1) The performance significantly boost when we apply the auxiliary learners to collaboratively learn the knowledge from the teacher and then transfer to the student (\\ie, collaboration). 2) Equipping with the attention or adversarial module alone improves the performance, which means that the attention and the adversarial module effectively strength the collaboration and diversity, respectively. 3) When equipped with all proposed modules, the student achieves highest accuracy. These results demonstrate the effectiveness of our proposed adversarial collaborative module.\n\n\\begin{table}[t]\n\\begin{center}\n\\resizebox{0.45\\textwidth}{!}{\n\\begin{tabular}{ccc|c}\n            \\toprule\n            Collaboration & Attention module & Adversarial module & Accuracy (\\%) \\\\\n            \\midrule\n             &  &  & 70.73 \\\\\n            $\\surd$ &  &  & 71.67\\\\\n            $\\surd$ & $\\surd$ &  & 72.02 \\\\\n            $\\surd$ &  & $\\surd$ & 71.99 \\\\\n            $\\surd$& $\\surd$ & $\\surd$ & \\textbf{72.38} \\\\\n\t\t\t\\bottomrule\n\t\t\\end{tabular}}\n\\end{center}\n\t\\caption{Effect of different modules.}\n\t\t\\label{tab:ablation-loss}\n\\end{table}\n\n\n\\begin{table}[t]\n\\begin{center}\n\\begin{tabular}{cc|c}\n            \\toprule\n            learner1 & learner2 & Accuracy (\\%) \\\\\n            \\midrule\n            D3B1 & D3B1 & 72.02 \\\\ \n            D3B1 & D2B2 & \\textbf{72.38} \\\\\n            D2B2 & D2B2 & 71.64 \\\\\n            D2B2 & D1B3 & 72.16 \\\\\n            D1B3 & D1B3 & 71.40 \\\\\n\t\t\t\\bottomrule\n\t\t\\end{tabular}\n\\end{center}\n\t\\caption{Effect of different learners. \"D$N$B$M$`` denotes that we add an auxiliary learner with $M$ increased feature extractor blocks to the $Nth$ down-sampling layer.}\n\t\t\\label{tab:branch-position}\n\\end{table}\t\n\n\\textbf{Effect of different learners.} \nIn this experiment, we investigate the effect of different learners. We use \"D$N$B$M$`` to denote that we add a learner with $M$ increased feature extractor blocks to the $Nth$ down-sampling layer. From Table~\\ref{tab:branch-position}, we have the following observations. 1) Different learners all lead to performance improvement compared with vanilla KD (70.73\\%), which shows the effectiveness and robustness of collaborative module. 2) Attaching learners to different down-sampling layers shows superiority \nover same layers. The main reason is that they effectively utilize different semantic information of the student, improving the diversity of learners. 3) ``D3B1 with D2B2\" achieves higher accuracy than ``D2B2 with D1B3\". The main reason is that the student shares more low-level layers with learners, leading to a more stable training process.\n\n\n\\section{Conclusion}\nIn this paper, we have proposed an Adversarial Collaborative Knowledge Distillation (ACKD\\xspace) method that effectively improves the performance of the student model in the knowledge distillation task. To this end, we have constructed a student model with multiple auxiliary learners. Based on the reinforced student model, we have proposed an adversarial collaborative learning strategy. Specifically, we have proposed an attention module to promote collaboration between auxiliary learners and used discriminators to distinguish the output distribution of auxiliary learners to enhance representational diversity. Extensive experiments on four benchmarks have shown that our ACKD\\xspace obtains state-of-the-art performance.\n\n{\\small\n\\bibliographystyle{ieee_fullname}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{isec}\n\t\n\tTime-dependent spatially localized states in dissipative systems, such as action potentials in\n\tphysiology~\\cite{izhikevich2007dynamical,murray2007mathematical,ksp08,allard2013traveling} and oscillons in Faraday waves~\\cite{umbanhowar1996localized,lioubashevski1999oscillons,blair2000patterns}, have attracted considerable interest in the past half-century~\\cite{ankiewicz2008dissipative,knobloch2015spatial}. Work in these fields not only provided insights into the original model (e.g., neural and cardiac) systems but also stimulated understanding of pattern formation mechanisms, which in turn proved applicable in other settings, such as in chemical reactions~\\cite{VE07}, nonlinear optics~\\cite{firth2007homoclinic,barbay2008homoclinic,firth2007snaking},\n\tfluid convection~\\cite{jacono2011magnetohydrodynamic,alonso2011convectons,mercader2011convectons,watanabe2012spontaneous}, and even sound discrimination in the inner ear~\\cite{edri2018molding}. Importantly, advances in pattern formation theory require the development of nonlinear methodologies and, specifically, numerical (continuation) methods for differential equations~\\cite{knobloch2015spatial}, such as the packages AUTO~\\cite{doedel2012auto}, MatCont~\\cite{bindel14}, and pde2path~\\cite{p2pbook}. These assist in revealing complex pattern selection mechanisms via computation of both stable and unstable solutions {upon variation of control parameters}, and so provide insights that are impossible to uncover otherwise. Thus, continuation methods are essential means of analysis for spatially extended complex systems regardless of the nature of the model equations.\n\t\n\n\t\\begin{figure}[bp]\n\t\t\\centering\n\t\t(a)\\includegraphics[width=0.21\\textwidth]{fig1_1JO_3D-eps-converted-to}\n\t\t(d)\\includegraphics[width=0.21\\textwidth]{fig1_1JO1P_3D-eps-converted-to}\n\t\t(b)\\includegraphics[width=0.21\\textwidth]{fig1_2JO-eps-converted-to}\n\t\t(e)\\includegraphics[width=0.21\\textwidth]{fig1_2JO1P-eps-converted-to}\n\t\t(c)\\includegraphics[width=0.21\\textwidth]{fig1_3JO-eps-converted-to}\n\t\t(f)\\includegraphics[width=0.21\\textwidth]{fig1_1JO2P-eps-converted-to}\n\t\t\\caption{Space-time plots showing a selection of states obtained through DNS of \\reff{bm0} with periodic boundary conditions. (a) JO at $k_1=-8.25$, (b) bound state of two JOs at $k_1=-9$, (c) 3JPP state at $k_1=-8.5$, (d) bound 1JO1TP state at $k_1=-7.8$, (e) 2JO1TP bound state at $k_1=-8.52$, and (f) 1JO2TP bound state at $k_1=-8.26$.  Other parameters are as in \\reff{opar}. All plots show $u(x,t)$, with colorscale $-1.7$ to $1.5$.} \n\t\t\\label{nfa}\n\t\\end{figure}\n\n\t\n\tIn 2006,~\\citet{YZE06} reported a {hitherto unseen spatially localized state they called a {\\it jumping oscillon} (JO). Related behavior, a \\textit{jumping wave}, was subsequently observed in the Belousov\u2013Zhabotinsky reaction in a microemulsion~\\cite{cherkashin2008discontinuously}.} JOs resemble the oscillons observed in parametrically driven systems~\\cite{umbanhowar1996localized,lioubashevski1999oscillons} but translate at the same time, whereby they disappear and then reappear at a shifted location. The process repeats, generating a time-periodic state in a frame moving with the average translation speed, as shown in Fig.~\\ref{nfa}(a). \\citet{YZE06} and discovered these JOs in direct numerical simulations (DNS) of a three-variable FitzHugh--Nagumo (FHN) reaction--diffusion (RD) equation but provided little explanation of the origin of these states and the plethora of states associated with them. The simulations also show that these states can collide and either annihilate or combine in a uniformly translating localized pulse and that they can organize themselves into traveling rafts with a spatially periodic or crystalline structure. Similar structures have also been recently found in an active phase-field crystal model~\\cite{ophaus2021} and in nonlinear optics~\\cite{schelte2020dispersive}, implying broader applicability. Tailored patterns with spatiotemporal modulation, some of which are exhibited in Fig.~\\ref{nfa}, can be envisioned as building blocks of information and storage handling~\\cite{coullet2000stable} especially in the context of chemo-liquid\n\tcomputers~\\cite{borresen2009neuronal,hiratsuka2009toward,adamatzky2011topics,gorecki2015chemical}.\n\t\n\tIn this Letter, we focus on the origin and the bifurcation structure of stable and unstable JO branches, as well as traveling wave (TW) and traveling pulse (TP) branches. In view of the abundance of solutions of each type (Fig.~\\ref{nfa}), numerical continuation must overcome possible branch jumping (undetected switching to a nearby branch in the predictor--corrector continuation setup). Nevertheless, the method enables us to identify the origin of the JOs through a modulational (Hopf) instability {of already unstable} TPs, and to obtain the bifurcation structure of accompanying multi-JOs [Fig.~\\ref{nfa}(b)] and jumping periodic patterns (JPP) [Fig.~\\ref{nfa}(c)] as well as the multitude of mixed JO and TP states [Figs.~\\ref{nfa}(d-f)].\\\\ \n\t\n\t\\noindent \\textit{Model equations and wave instability} -- We employ the Purwins system originally developed~\\cite{schenk1997interacting} as a phenomenological model of an electrical discharge system exhibiting multiple stationary and moving localized states~\\cite{purwins2010dissipative}. The system is a three-variable FHN system with one activator and two inhibitors acting on distinct timescales, and has broad applicability in studies of dissipative solitons in excitable RD systems in both one (1D) and two (2D) spatial dimensions (pulse interactions in 1D~\\cite{doelman2009pulse,van2011planar,van2014bifurcations,teramoto2021traveling,nishiura2021matched} and time-dependent spots in\n\t2D~\\cite{schenk1997interacting,or1998spot,bode2002interaction,gurevich2006breathing}). The closely related models studied\n\tin~\\cite{VE04,nishiura2005scattering,yochelis2008generation,stich2009self,marasco2014vegetation,yochelis2015origin} exhibit similarly rich dynamics.\n\t\n\tThe Purwins system in 1D reads \n\t\\begin{eqnarray}\\label{bm0}\n\t\t\\nonumber  \\pa_t u&=&D_u\\pa_x^2 u+k_1+k_2u-u^3-k_3v-k_4w,\\\\\n\t\t\\tau \\pa_t v&=&D_v\\pa_x^2 v+u-v,\\\\\n\t\t\\nonumber  \\vartheta}\\def\\Id{\\mbox{Id}}\\def\\pa{{\\partial}}\\def\\lam{\\lambda\\pa_t w&=&D_w\\pa_x^2 w+u-w, \n\t\\end{eqnarray} \n\twhere $k_4$, $\\tau$, $\\vartheta}\\def\\Id{\\mbox{Id}}\\def\\pa{{\\partial}}\\def\\lam{\\lambda$ are parameters and $D_u$, $D_v$, $D_w$ are diffusion coefficients. The JOs in \\cite{YZE06} were obtained with $(k_2,k_3,k_4,\\tau,\\vartheta}\\def\\Id{\\mbox{Id}}\\def\\pa{{\\partial}}\\def\\lam{\\lambda,D_u,D_v,D_w)=(2,10,2,50,1,1,50,60)$ and near\n\t$k_1=-8.5$. Here we consider a similar parameter regime but with \n\t\\begin{equation}\\label{opar}\n\t\t(\\vartheta}\\def\\Id{\\mbox{Id}}\\def\\pa{{\\partial}}\\def\\lam{\\lambda,D_u,D_v,D_w)=(0.5,2,25,100) \n\t\\end{equation}\n\tin order to separate out the branches in the bifurcation diagrams, and also employ $k_1$ as a control parameter. We supplement~\\reff{bm0} with periodic boundary conditions on the 1D domain $x\\in [-L\/2,L\/2]$, where $L=80$, and let $U\\equiv(u,v,w)$. In our parameter regime, \\reff{bm0} has a unique spatially homogeneous steady state $U_*=(u_*,u_*,u_*)$ (see SM~\\ref{SM:disp}), which loses stability to a wave (finite wave number Hopf) instability at $k_1=k_{1c}\\approx -7.6$ with a critical wave number $q=q_c$ associated with the critical wavelength $\\lambda_c=2\\pi\/q_c \\approx 20$; the dispersion relation at the onset\n\tis shown in SM~\\ref{SM:disp}.\n\t\n\n\t\\begin{figure*}[tp]\n\t\t\\centering\n\t\t({\\color{orange}{\\large$\\blacktriangle$}})\\includegraphics[height=2.6cm]{fig3_1JO_28-eps-converted-to}\n\t\t({\\color{orange}{\\small$\\blacklozenge$}})\\includegraphics[height=2.6cm]{fig3_1JO_48-eps-converted-to}\n\t\t({\\color{orange}{\\small$\\blacksquare$}})\\includegraphics[height=2.6cm]{fig3_1JO_75-eps-converted-to}\n\t\t({\\color{orange}{\\large$\\blacktriangledown$}})\\includegraphics[height=2.6cm]{fig3_1JO_94-eps-converted-to}\n\t\t\\includegraphics[width=0.9\\textwidth]{fig3_bif_F-eps-converted-to}\n\t\t\\caption{Partial bifurcation diagram showing $||U||_2$ as a function of $k_1$ for the uniform branch (black line), traveling waves (blue), traveling 1--pulse (green), time-modulated (including JO, orange) pulses, and further TW branches as indicated, with stability indicated by solid lines. The branch of modulated traveling pulses becoming JOs bifurcates from a Hopf point near $k_1{=}k_1^{\\text{fp}1}$ (green bullet) and the JOs are stable for $k_1^{\\text{fp}2}{<}k_1{\\lesssim}k_{1c}$; the locations $k_1^{\\text{fp}j}$ indicate other folds. Selected traveling (right) and time-modulated (top) solutions at different locations are also shown. The black bullet labeled (a) on the orange branch corresponds to the JO shown in Fig.~\\ref{nfa}(a). Other parameters are as in \\reff{opar}.} \n\t\t\\label{jof3}\n\t\\end{figure*}\n\n\t\n\tThe primary wave instability generates branches of spatially \n\tperiodic traveling and standing waves~\\cite{knobloch1986oscillatory}, hereafter 4TW and 4SW, with four wavelengths in the domain (see SM~\\ref{SM:disp}). Subsequent primary bifurcations generate 3TW, 5TW, 6TW and eventually 2TW. In what follows, we focus on traveling solutions that coexist with stable $U_*$, i.e.~on the \\textit{excitable} regime, and show that the JOs are related to a temporal modulation of traveling solutions.\\\\ \\\\\n\n\t\\noindent \\textit{Continuation methodology} -- To compute solutions that are steady in a comoving frame (TW or TP), we rewrite (\\ref{bm0}) in a reference frame moving with speed $s$ to the right, i.e.,~we add $-s\\pa_x(u,v,w)$ to the right hand side of (\\ref{bm0}), set the time derivatives to zero and solve the resulting nonlinear eigenvalue problem for $(U,s)$ using the phase condition\n\t\\begin{equation}\\label{se1} \n\t\t\\spr{\\pa_x\\Uold,U}\\stackrel!=0, \n\t\\end{equation} \n\twhere $\\Uold$ is the solution from the last continuation step, and \n\t$\\spr{f,g}\\equiv\\int_{-L\/2}^{L\/2}f(x)g(x)\\, {\\rm d}}\\def\\ri{{\\rm i}}\\def\\sech{{\\rm sech} x$. {This minimizes the \n\t\t$L^2$ distance of the current step to translates of the previous step}~\\footnote{\\label{cfoot}{ {Additionally, $k_1$ is solved for in an arclength continuation setting, where the independent parameter is the arclength along a branch; this setting allows one to follow branches that oscillate back and forth in $k_1$, and in particular to pass folds.}}}. The continuation is thus orthogonal to the group orbit of translations $u(x{-}\\xi)$, with $\\pa_x$ as generator of the associated Lie algebra. We represent the traveling solutions using the norm $||U||_2\\equiv\\sqrt{L^{-1}\\int [u(x)-u_*]^2\\, {\\rm d}}\\def\\ri{{\\rm i}}\\def\\sech{{\\rm sech} x}$, see the TW branches in Fig.~\\ref{jof3}.\n\t\n\tFor solutions of JO type or more generally modulated TW (mTW), we retain the time derivatives and solve for both the (mean) frame speed $s$ and the oscillation period $T$. To do so, we extend the phase condition to \n\t\\begin{equation}\\label{avs} \n\t\tq_H(U):=\\sum_{i=1}^{m-1} \\spr{\\pa_x U_*, U(t_i)}\\stackrel!=0, \n\t\\end{equation} \n\twhere $U_*=U_*(x)$ is a reference profile (usually $U_H(x)$, the spatial profile at the Hopf point) and\n\t$t_1, t_2\\ldots, T$ are the grid points of the time discretization. Consequently, for mTW we have the three unknowns $(U,s,T)$ and solve the two equations \\reff{bm0} and \\reff{avs} together with the additional temporal phase condition\n\t\\begin{equation}\n\t\t\\int_0^T \\spr{\\ddt \\Uold(t'),U(t')}\\, {\\rm d}}\\def\\ri{{\\rm i}}\\def\\sech{{\\rm sech} t\\stackrel!=0, \n\t\\end{equation} \n\t{to make the continuation orthogonal to the group orbit of time \n\t\ttranslates.} \n\tWe also modify the norm $\\|\\cdot\\|_2$ to \n\t\\begin{equation}\\label{phase}\n\t\t\\|U\\|_2{=}\\sqrt{\\frac 1{TL} \\int_0^T \\int_{-L\/2}^{L\/2}\n\t\t\t\\Bra{u(x,t){-}u_*}^2\\, {\\rm d}}\\def\\ri{{\\rm i}}\\def\\sech{{\\rm sech} x\\, {\\rm d}}\\def\\ri{{\\rm i}}\\def\\sech{{\\rm sech} t}.  \n\t\\end{equation} \n\tFor the TW (TP) branches we thus have $n_u+2$ unknowns (including $k_1$, see \\cite{Note1}) and $n_u+1$ equations, where $n_u=3n_p$ and $n_p$ is the number of spatial discretization points, while for the mTW (and JO) branches we have $mn_u+3$ unknowns (again including $k_1$) and $mn_u+2$ equations, where $m$ is the number\n\tof temporal discretization points. For our domain, we typically use $n_p\\approx 1000$ discretization points in space and $m=50$ in time (for mTW), yielding $\\approx 150000$ degrees of freedom. The predictor\/corrector continuation method uses a corrector based on Newton's method and carries the danger of branch--jumping when many solutions are close together. To mitigate this, we monitor the convergence speed of the Newton loops, cf.~\\cite[\\S3.6]{p2pbook}. We also monitor selected eigenvalues of the linearizations to check stability and detect possible branch points, which are then localized, for subsequent branch switching \n\t\\footnote{\\label{flfoot}For (relative) time--periodic orbits (PO), the role of eigenvalues (for stability, bifurcation detection, and branch switching) is played by Floquet multipliers. {These multipliers (in particular for PDE discretizations) may differ by many orders of magnitude and their (stable) numerical \n\t\t\tcomputation therefore requires advanced methods such as periodic Schur decomposition \n\t\t\tand hence is numerically expensive \\cite[\\S3.5]{p2pbook}}. Moreover, for some \n\t\tpoints on our PO branches the multiplier computations do not converge. \n\t\tWe therefore refrain from computing bifurcations from POs, and instead check their stability {\\it a posteriori} via DNS. See \\cite{BBC15} for a review of the numerical methods related to POs in the presence of symmetry.}.\\\\\n\t\n\n\t\\noindent \\textit{Modulational instability of excitable pulses} -- Our first aim is to understand the origin of the JO states. Figure \\ref{jof3} illustrates our results. The primary 4TW (blue) branch bifurcates subcritically from the primary Hopf point. Additional TW states are also shown: primary branches of 3TW (brown) and 6TW (gray), and two secondary TW branches that arise from period doubling, namely the 3TW\n\t(light green, from 6TW) and 2TW (pink, from 4TW). The dark green branch corresponds to TP states obtained by continuation from a stable 1TP state found in DNS starting from an initial condition obtained by cutting out a part of the 4TW solution (dark green square). The 1TP states are stable until the left-most fold at\n\t$k_1=k_1^{\\text{fp}1}\\approx -9.9$ where they lose stability. Beyond the second fold, near $k_1= k_{1c}$, the branch begins to {\\it snake} \\cite{knobloch2015spatial} forming 2TP and then 3TP states. The latter fail to connect to a 3TW state and begin to snake downwards until they connect at the pink bullet to the 2TW branch. In the opposite direction the 1TP undergo complex behavior before terminating back on the same 2TW branch (SM~\\ref{SM:TP}). Thus, both the {1TP} branch and the 2TW branch represent branches that start and end on the same branch in secondary bifurcations. \n\t\n\t{The properties of the JOs are closely related to these background states. We find that the JO branch (orange) emerges from the first Hopf bifurcation located on the \\textit{unstable} portion of the 1TP branch (green bullet in Fig.~\\ref{jof3}). Consequently, the JOs start out as an unstable \\textit{small amplitude modulation} of the 1TP state, and turn into stable, fully developed JOs (orange triangle in Fig.~\\ref{jof3}) only after the fold near $k_1=k_{1c}$, before losing stability again at the next fold at $k_1=k_1^{\\text{fp}2}$}; the time $t=0$ state at the black bullet (a) was used as initial condition for the DNS in Fig.~\\ref{nfa}(a). The stable JOs have the highest travel speed $s$ and shortest oscillation period $T$ along the whole orange branch (SM~\\ref{SM:speed}).\n\t\n\tAs the branch snakes below the fold at $k_1=k_1^{\\text{fp}2}$, the solutions become unstable (orange diamond in Fig.~\\ref{jof3}), before turning into a (smaller amplitude) 2JO bound state (at $k_1\\approx k_1^{\\text{fp}4}$,\n\torange square in Fig.~\\ref{jof3}), and then into a (yet smaller amplitude) 3JO bound state (orange down-triangle in Fig.~\\ref{jof3}). Beyond this point, the continuation becomes unreliable in the sense that different numerical settings (finer discretizations\/tolerances) may lead to different behavior, but the\n\tbranch continues. \\\\\n\t\n\n\t\\noindent\\textit{Bound states and mixed states} -- In Fig.~\\ref{jof4}, we plot several branches that are associated with multi-JO solutions and mixed JO\/TP states. The starting point for finding these states is a domain-filling 3JPP state (red branch in Fig.~\\ref{jof4}). These extended 3JPP states are characterized by a pronounced phase gradient and bifurcate from a 3TW (light green) branch via a Hopf bifurcation just above $k_1=k_1^{\\text{fp}3}\\approx -9.5$ and are initially unstable. Stable states are found on the segment between the first two folds, $k_1^{\\text{fp}8}<k_1<k_1^{\\text{fp}7}$. Stability is again checked using DNS, yielding for instance Fig.~\\ref{nfa}(c). There are many other Hopf points along the 1TP branch, along the 3TW branch, and along similar branches, which give rise to further mTP branches. Many of these are pairwise connected via small-amplitude mTW branches. We show one example of such a branch in SM~\\ref{SM:mTW}.\n\t\n\tThe overall picture of localized and extended JO states is quite robust with respect to parameter changes, although details of the connections between branches depend sensitively on parameters, as shown in SM~\\ref{SM:mTW} for $D_v<25$. One quantitative effect of decreasing $D_v$ is the loss of the first two Hopf points on the 1TP branch (see SM~\\ref{SM:mTW} for details), resulting in a shift of the Hopf point from which the JO branch originates.\n\n\t\\begin{figure}[tp]\n\t\t\\centering\n\t\t({\\small$\\blacklozenge$})\\includegraphics[height=2.6cm]{fig4_2JO1P-eps-converted-to}\n\t\t({\\small$\\blacksquare$})\\includegraphics[height=2.6cm]{fig4_2JO-eps-converted-to}\n\t\t\\includegraphics[width=0.95\\columnwidth]{fig4_bif_B-eps-converted-to}\n\t\t\\caption{Partial bifurcation diagram showing $||U||_2$ for the 3TW branch (green), the associated 3JPP branch (red) and branches of bound states 1JO1TP (brown) and 1JO2TP (purple), with the JO branch kept in the background in gray; solid lines indicate stability. The 3JPP branch bifurcates from the Hopf point near $k_1=k_1^{\\text{fp}3}$; other $k_1^{\\text{fp}j}$ points mark folds designating the limits of stability. Selected time-modulated solutions at different locations are shown in the top panels (square\/diamond symbols) and labeled bullets correspond to states shown in Fig.~\\ref{nfa}. Other parameters are as in \\reff{opar}.}\\label{jof4}\n\t\\end{figure}\n\t\n\tFigure~\\ref{jof4} demonstrates that the continuation method can also be used to identify the bifurcation structure of more complex states, including mixed JO\/TP states. Starting from (stable) bound states of JO and TP obtained from DNS, e.g., 1JO1TP [Fig.~\\ref{nfa}(d)] and 1JO2TP [Fig.~\\ref{nfa}(f)] and identifying the period $T$ and frame speed $s$ we can use continuation to compute both stable and unstable regimes associated with such solutions. Stable 1JO1TP (brown branch) states exist for $k_1>k_1^{\\text{fp}8}$ while stable 1JO2TP (purple branch) are present between $k_1^{\\text{fp}11}<k_1<k_1^{\\text{fp}10}$. Moreover, unstable states (see the square and diamond symbols in Fig.~\\ref{jof4}) provide initial conditions that converge, via DNS, to stable 2JOs [Fig.~\\ref{nfa}(b)] and 2JO1TP [Fig.~\\ref{nfa}(e)], respectively. The continuation of these states (not shown) follows the same procedure as used for single \n\tJOs. \\\\\n\t\n\n\t\\noindent\\textit{Discussion} --\n\tWe have shown that a successful understanding of the origin of the 1JO and associated states, such as the 2JO and 1JO1TP bound states or the JPP state discovered in~\\cite{YZE06}, can be achieved within a careful continuation\/bifurcation setting -- the only currently existing technique for such purposes. For the three-variable FHN model~\\reff{bm0}, the continuation of excitable states turned out to hold significant challenges: while for the TW and TP we only have spatial degrees of freedom to deal with, for the mTP and mTW, as relative time-periodic orbits, we need a fine temporal resolution as well, leading to $10^5$ (and more) necessary degrees of freedom, even on the small domain used here. We have checked that our results persist to larger domains with more turns on the 1TP branch as it snakes (not shown), and similar behavior of the JO state, although some differences are inevitably present. The numerical difficulties mount, however, due to the abundance of states and possible branch jumping, issues that (mostly) do not arise on smaller domains or in DNS. We showed here that the origin of stable JOs is highly subtle, requiring first and foremost an understanding of the underlying 1TP states and their stability, but that our approach is up to the task. {Moreover, 2D DNS in the corresponding parameter range yield target-like jumping waves similar to those observed experimentally in the Belousov\u2013Zhabotinsky reaction in a microemulsion~\\cite{cherkashin2008discontinuously}, as shown in SM~\\ref{SM:JW}. Thus our approach yields} useful insight into the extremely rich solution structure of \\reff{bm0} in the excitable regime, and in particular into the origin of JOs and their subsequent snaking forming 2JOs, 3JOs,..., and ultimately domain-filling JPP arrays.\n\t\n\tLike the wide interest stimulated by the discovery of oscillons~\\cite{umbanhowar1996localized,lioubashevski1999oscillons,blair2000patterns}, we expect that the methodology developed here for JOs will also lead to new theoretical questions as well as potential applications in other multi-variable excitable RD media. The rich yet programmable JO patterns strengthen the suggestion in~\\cite{coullet2000stable} that localized states could be useful for data storage in computers with RD kinetics~\\cite{borresen2009neuronal,hiratsuka2009toward,adamatzky2011topics,gorecki2015chemical}.\n\t\\\\ \\\\\n\t\\noindent {\\bf Acknowledgement}. We thank Svetlana Gurevich (M\\\"{u}nster) for helpful discussions. This work was supported in part by the National Science Foundation under Grant No. DMS-1908891 (EK).                 \n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}