Data-driven criterion for the solid-liquid transition of two-dimensional self-propelled colloidal particles far from equilibrium
aa r X i v : . [ c ond - m a t . s o f t ] F e b Data-driven criterion for the solid-liquid transition of two-dimensional self-propelledcolloidal particles far from equilibrium
Wei-chen Guo, Bao-quan Ai, ∗ and Liang He † Guangdong Provincial Key Laboratory of Quantum Engineering and Quantum Materials,School of Physics and Telecommunication Engineering,South China Normal University, Guangzhou 510006, China
We establish an explicit data-driven criterion for identifying the solid-liquid transition of two-dimensional self-propelled colloidal particles in the far from equilibrium parameter regime, wherethe transition points predicted by different conventional empirical criteria for melting and freezingdiverge. This is achieved by applying a hybrid machine learning approach that combines unsuper-vised learning with supervised learning to analyze over one million of system’s configurations inthe nonequilibrium parameter regime. Furthermore, we establish a generic data-driven evaluationfunction, according to which the performance of different empirical criteria can be systematicallyevaluated and improved. In particular, by applying this evaluation function, we identify a newnonequilibrium threshold value for the long-time diffusion coefficient, based on which the predictionsof the corresponding empirical criterion are greatly improved in the far from equilibrium parameterregime. These data-driven approaches provide a generic tool for investigating phase transitions incomplex systems where conventional empirical ones face difficulties.
For decades there have been open questions concerningthe two-dimensional (2D) solid-liquid transitions [1]. Forinstance, the well-known Kosterlitz-Thouless-Halperin-Nelson-Young theory [2–5] suggests that 2D crystals meltvia two continuous transitions to liquid with an inter-mediate hexatic phase, while first-order transitions withand without the intermediate phase are also found bothin numerical simulations and in experiments in different2D systems [6–15].The situation becomes even trickier for 2D systems farfrom equilibrium. For instance, the solid-liquid transi-tion of 2D nonequilibrium (NEQ) colloidal crystals, rang-ing from 2D living crystals of photo-activated colloidalsurfers [16], over schools of Janus colloidal particles ona flat interface [17], to active nematic liquid crystals of2D epithelial tissues [18], etc., are typical representatives[19]. In this type of systems, even though the transitionpoints predicted by different conventional empirical crite-ria [20–24] for melting and freezing agree with each otherin equilibrium, they were shown to separate away whenthe system enters the NEQ parameter regime, essentiallyresulting in a lower and an upper bound for the solidphase region and the liquid one, respectively [19]. Thisthus raises the fundamental question of how to systemat-ically establish a criterion for identifying the solid-liquidtransition of 2D systems in NEQ and a way to evaluatingthe performance of different empirical criteria.In this work, we address this question for 2D self-propelled colloidal particles with Yukawa-type interac-tion. To this end, we apply a hybrid machine learning ap-proach that combines unsupervised learning with super-vised learning to analyze over one million of system’s con-figurations and establish an explicit data-driven criterion ∗ [email protected] † [email protected] for identifying the solid-liquid transition of the system inthe far from equilibrium parameter regime (cf. Fig. 1).More specifically, after generating ∼ O (10 ) spatial dis-tributions of colloidal particles in the NEQ steady state[cf. the first row of Fig. 1(a)] via direct numerical sim-ulations of the dynamical equation (1) of the system,these spatial distributions are analyzed by an unsuper-vised learning approach [25] that is realized by a fully-connected neural network (NN) [26, 27]. As a direct re-sult, this gives an essentially unbiased (in the sense ofno prior empirical assumptions being involved) criterionto classify the spatial distributions according to whichthe solid-liquid transition boundary is extracted [cf. thesolid curve in Fig. 1(b)]. Crucially, this also promotesthe unlabeled data to the labeled data, which enablesthe direct supervised learning to concretize the criterionin the form of a set of NN-parameters that defines theNN [cf. the last row of Fig. 1(a)]. This thus establishesthe explicit data-driven criterion that is able to identifythe transition not only within, but also beyond the NEQparameter regime where the supervised learning is di-rectly performed [cf. the “ + ” marks in Fig. 1(b)], indicat-ing it indeed captures the essential feature of the solid-liquid transition in the far from equilibrium parameterregime. Furthermore, by utilizing the so-called classifi-cation accuracy generated in the unsupervised learningprocess [cf. Eq.(2) and Fig. 2(b)], we establish a genericdata-driven evaluation function [cf. Eq. (3)], accordingto which the performance of different empirical crite-ria can be systematically evaluated and improved. Thedirect application of this evaluation function gives rise,in particular, a new NEQ threshold value D NEQthreshold forthe long-time diffusion coefficient, based on which thepredictions of the corresponding empirical criterion aregreatly improved in the far from equilibrium parameterregime [cf. Fig. 3 and the “ × ” marks in Fig. 1(b)]. More-over, since the underlying machine learning techniquesexploited in this work are completely general, we expectthat these data-driven approaches can readily providenew physical insights into phase transitions in other com-plex systems where conventional empirical approachesface difficulties. Self-propelled colloidal particles and their 2D solid-liquid transition .—The system under study consists of N self-propelled colloidal particles in a 2D space [19], whosedynamics is described by a set of overdamped Langevin-type equations with the explicit form ˙ r i = −∇ i U + f e i + ξ i . (1)Here, r i is the position of the i th particle, and ξ i is thenoise with zero mean and correlations h ξ i ( t ) ξ Tj ( t ′ ) i =2 δ ij δ ( t − t ′ ) which models the stochastic interactionswith the solvent molecules. f is the strength of theforce that propels each particle in the direction e i ≡ (cos ϕ i , sin ϕ i ) , where h ˙ ϕ i ( t ) ˙ ϕ j ( t ′ ) i = 2∆ δ ij δ ( t − t ′ ) with ∆ being the so-called rotational diffusion coefficient. U = P i
Figure 1. (a) Schematic illustration of the hybrid machinelearning approach. Spatial distributions of particles are firstprocessed by the unsupervised learning with a fully-connectedNN, after which the unlabeled data are promoted to the la-beled data. These labeled data are processed by the su-pervised learning to optimize the learnable parameter set ofthe NN and get the optimal set { ˜ w , ˜ b } that determines thedata-driven criterion. (b) Solid-liquid transition points pre-dicted by the direct unsupervised learning (marked by “ ♦ ”and solid curve), the data-driven criterion (marked by “ + ”),and the generalized empirical criterion with the NEQ thresh-old D NEQthreshold (marked by “ × ”). For comparison, the transi-tion boundaries Γ ∗ ψ , Γ ∗ D , and Γ ∗ γ L predicted by the conven-tional empirical criteria using ψ , D , and γ L , respectively, areshown by the dashed curves (reprinted from Ref. [19]). Thedata-driven criterion and the NEQ threshold D NEQthreshold areobtained by “learning” the configurations of the system with f = 0 , , , , . Their predicted transition points match verywell with the ones from the direct unsupervised learning, notonly in the parameter regime f ∈ [0 , in which the data aredirectly accessible to them, but also in the parameter regimewell beyond (the shaded area). See text for more details. system’s configurations, i.e., spatial distributions of par-ticles, which are generated by direct numerical simula-tions of the dynamical equation (1) of the system in dif-ferent parameter regimes. Here, we focus on the casewith N = 1936 , ∆ = 3 . , λ = 3 . , and all the numericalsimulations of Eq. (1) are performed in a 2D rectangularspace with aspect ratio / √ and periodic boundary con-dition imposed. In total, we generate ∼ O (10 ) spatialdistributions of particles in the steady state that corre-spond to different sets of effective interaction strength Γ and self-propel force strength f . These 2D distribu-tions are then directly transformed into images, formingthe whole data set which is directly processed by the NNthat conducts the hybrid machine learning (cf. Fig. 1) aswe shall now discuss.The hybrid machine learning starts with the unsu-pervised learning performed by employing the so-called“learning by confusion” approach [25] that is realized bya fully-connected NN [cf. Refs. [26, 27] and the last rowof Fig. 1(a)], where the transition point is extracted fromthe contrast between good and bad recognition perfor-mance when “confusing labels” are deliberately attachedto the images in the data set (cf. Fig. 2). More specif-ically, for the images corresponding to the same fixedself-propel force strength f , by imposing a testing bi-nary classification rule associated with a proposed criti-cal value Γ ∗′ ( f ) , images are labeled as “liquid” (“solid”) iftheir corresponding Γ < Γ ∗′ ( f ) ( Γ > Γ ∗′ ( f ) ). This wayof labeling generally confuses the NN when the proposedcritical value Γ ∗′ ( f ) is not equal to the physical criticalvalue Γ ∗ ( f ) . For instance, if Γ ∗′ ( f ) < Γ ∗ ( f ) , the imageswith their corresponding Γ satisfying Γ ∗′ ( f ) < Γ < Γ ∗ ( f ) are in the liquid phase but labeled as “solid” [cf. Fig. 2(a)].For a generic proposed value Γ ∗′ ( f ) , we train the NN andthen test its classification accuracy P (Γ ∗′ ( f )) ≡ M “correct” f (Γ ∗′ ( f )) M test f (2)with new testing images that have not been “seen” bythe NN in the training process (see Supplemental Ma-terial [28] for details). Here, M test f is the total numberof the testing images and M “correct” f (Γ ∗′ ( f )) is the num-ber of images that are classified “correctly” according tothe proposed labels determined by Γ ∗′ ( f ) . Noticing thatthe closer Γ ∗′ ( f ) is to the solid-liquid transition point Γ ∗ ( f ) , the fewer confusing labels exist, hence leading tothe relatively higher accuracy, the classification accuracy P (Γ ∗′ ( f )) should therefore assume a nontrivial maximumwhen Γ ∗′ ( f ) matches Γ ∗ ( f ) .Indeed, as we can see from Fig. 2(b), where the unsu-pervised learning is performed separately for the data setswith corresponding f = 0 , , , , , the classification ac-curacy P (Γ ∗′ ( f )) assumes a nontrivial maximum for each f , from which one can directly read out the correspond-ing transition point Γ ∗ ( f ) . The solid-liquid transitionboundary formed by these transition points is shown inthe phase diagram Fig. 1(b), where we can see that it lo-cates exactly between the lower bound of the solid regionset by Γ ∗ γ L and the upper bound of the liquid region setby Γ ∗ D , clearly corroborating the predictions from theseconventional dynamical criteria [19] and suggesting thereexists an underlying criterion that can capture the fea-ture for distinguishing the solid from the liquid phase inthe far from equilibrium parameter regime. This thus motivates us to extract the concrete explicit form of thisunderlying criterion revealed by the unsupervised learn-ing.To achieve this goal, we first notice that the explicitform of the criterion can be generally regarded as a mapfrom the space of the system’s configurations to two “con-fidence” values, say, y S and y L that hold the meaningof how likely a given configuration is a configuration ofthe solid and the liquid phase, respectively, i.e., a well-trained NN that takes system’s configurations as inputand outputs the correct confidence values [cf. the last rowof Fig. 1(a)]. Crucially, according to the solid-liquid tran-sition boundary predicted by the unsupervised learning,all available images with corresponding f = 0 , , , , ,can be properly labeled now, thus promoting the orig-inal unlabeled data set to the labeled one. With thelabeled data, the supervised learning realized by a fully-connected NN with the learnable parameter set { w , b } can be directly performed [28] and finally gives rise tothe optimal learnable parameter set { ˜ w , ˜ b } that com-pletely determines the well-trained NN, i.e., the explicitdata-driven criterion. From Fig. 1(b) we can see that theprediction on the solid-liquid transition points from thisdata-driven criterion [28] matches very well with the onefrom the direct unsupervised learning. In particular, forthe transition point at f = 10 , it is located way beyondthe parameter regime f ∈ [0 , in which the data aredirectly provided to the NN to be learned, clearly mani-festing that this explicit data-driven criterion indeed cap-tures the generic feature that distinguishes the solid fromthe liquid phase in the far from equilibrium parameterregime. Data-driven evaluation function for empiricalcriteria .—Besides the solid-liquid transition bound-ary in the far from equilibrium regime and its associateddata-driven criterion, another class of physical insightsthat one can obtain by making full use of the availableinformation of the system is a data-driven evaluationfunction, according to which the performance of differentempirical criteria can be systematically evaluated andimproved. From Eq. (2), we can see that for any given(empirical) criterion, denoted as C , its performancecan be naturally evaluated by the average classificationaccuracy of its predicted transition points at which asufficient amount of system’s configurations are avail-able, i.e., a criterial classification accuracy denoted as P C with the explicit form P C = 1 N f N f X i =1 P (Γ ∗C ( f i )) , (3)where Γ ∗C ( f i ) is the transition point determined by thecriterion C at the self-propel force strength f i and N f isthe total number of the transition points involved in theevaluation.Taking the empirical criterion that uses an equilibriumthreshold value D EQthreshold = 0 . for the long-time dif- Figure 2. (a) Schematic illustration of the confusing labelsinvolved in the unsupervised learning approach “learning byconfusion”. The images with their corresponding Γ satisfy-ing Γ ∗′ ( f ) < Γ < Γ ∗ ( f ) are in the liquid phase but labeledas “solid”, which confuses the NN in establishing the classifi-cation criterion. (b) Solid-liquid transition points identifiedby the NN trained via unsupervised learning. For each self-propel force strength f , three independent data sets, denotedby “A”, “B”, and “C”, are used in the unsupervised learningprocess, and the classification accuracy P (Γ ∗′ ( f )) for eachindependent data set reaches the same nontrivial maximumthat corresponds to the solid-liquid transition point. The pre-dicted transition points are Γ ∗ (0) = 225 ± , Γ ∗ (2) = 265 ± , Γ ∗ (4) = 345 ± , Γ ∗ (6) = 475 ± , and Γ ∗ (8) = 605 ± . Seetext for more details. fusion coefficient D [19, 23] for example, by utilizing theavailable data at f = 0 , , , , , its criterial classificationaccuracy P D EQthreshold = P f =0 , , , , P (Γ ∗ D EQthreshold ( f )) / can be directly calculated [28], where Γ ∗ D EQthreshold ( f ) isthe transition point at f determined by this criterion[cf. upper row of “ × ” in Fig. 3(a) and the rightmostpoint in Fig. 3(b)]. Noticing this criterion employs athreshold value motivated by the equilibrium behavior[19], hence leave the space for further improvement if ageneral threshold value D threshold is chosen [cf. the lowerrow of “ × ” in Fig. 3(a)], we investigate the D threshold dependence of the corresponding criterial classificationaccuracy P D threshold . From Fig. 3(b) we notice that ata smaller threshold than the equilibrium one, P D threshold reaches the highest value, directly indicating the corre-sponding criterion using this NEQ threshold, denoted as D NEQthreshold , assumes the best performance in the NEQ pa-rameter regime f ∈ [0 , . Indeed, the transition bound-ary predicted by this improved empirical criterion is sur-prisingly close to the one obtained by the direct unsu-pervised learning [cf. Fig. 1(b)]. More remarkably, al-though D NEQthreshold is determined within the NEQ param-eter regime f ∈ [0 , , its predictive power extends wellbeyond, as manifested by its prediction of the transitionpoint at f = 10 which is still very close to the one viathe direct unsupervised learning [cf. the shaded area inFig. 1(b)]. This thus suggests that after generalizing thiscriterion [19, 22, 23] with a smaller NEQ threshold, it Figure 3. (a) Γ -dependence of the long-time diffusion coeffi-cient D of the system at different self-propel force strengths f = 0 , , , , . The upper and the lower horizontal dashedlines correspond to the equilibrium threshold D EQthreshold =0 . and the NEQ one D NEQthreshold = 0 . obtained by op-timizing P D threshold , respectively. The crossings marked by“ × ” of each horizontal dashed line with the D (Γ) curvesat different f predict the corresponding transition points Γ ∗ D threshold ( f ) . (b) Threshold value D threshold dependence ofthe criterial classification accuracy P D threshold . The right-most point corresponds to D threshold = D EQthreshold = 0 . ,and the point with the highest P D threshold corresponds to D NEQthreshold = 0 . which results in a generalized empiricalcriterion with improved performance in the NEQ scenario.See text for more details. can identify the solid-liquid transition in NEQ quite pre-cisely. In fact, this is also consistent with the naturalexpectation that due to active systems in NEQ are moredifficult to crystallize, the threshold value for the gen-eral NEQ scenario is supposed to be smaller than theone for the equilibrium case. Moreover, we also performa similar investigation on the empirical criterion basedon the threshold value of the global bond-orientationalorder parameter ψ [19] (see Supplemental Material [28]for details) and find this criterion assumes relatively poorperformance in general, which is consistent with the ex-pectation that being a static structure criterion by con-struction, it tends to ignore the effects of the positionexchange of particles, and therefore may misjudge a liq-uid as a solid. In this regard, one may expect a moredelicate static structure criterion with structural defects,for instance, dislocations, disclinations, etc., taking intoaccount could improve the performance. Although con-structing such an empirical criterion is beyond the scopeof the current work, its performance can still be straight-forwardly evaluated via the criterial classification accu-racy P C . Conclusion and outlook .—By directly analyzing overone million of system’s configurations via data-drivenapproaches, new physical insights into the solid-liquidtransition of the system in the far from equilibrium pa-rameter regime are gained: an explicit data-driven cri-terion together with its predicted solid-liquid transitionboundary is established via the hybrid machine learn-ing approach that combines unsupervised learning withsupervised learning. Furthermore, the data-driven crite-rial classification accuracy is established as a systematicway to evaluate and improve the empirical criteria, viawhich, in particular, a generalized conventional empiricalcriterion with a new NEQ long-time diffusion coefficientthreshold is found and assumes a much enhanced perfor-mance in NEQ. These data-driven approaches open up awide range of intriguing possibilities for further investi-gations to provide new physical insights into phase tran-sitions in complex systems where conventional empiricalapproaches face difficulties.
ACKNOWLEDGMENTS
We thank Danbo Zhang and Fujun Lin for use-ful discussions. This work was supported by NSFC(Grant No. 11874017 and No. 12075090), GDSTC(Grant No. 2018A030313853 and No. 2017A030313029),GDUPS (2016), Major Basic Research Project ofGuangdong Province (Grant No. 2017KZDXM024), Sci-ence and Technology Program of Guangzhou (GrantNo. 2019050001), and START grant of South China Nor-mal University. [1] K. J. Strandburg, Rev. Mod. Phys. , 161 (1988).[2] J. M. Kosterlitz and D. J. Thouless, J. Phys. C , 1181(1973).[3] B. I. Halperin and D. R. Nelson, Phys. Rev. Lett. ,121 (1978).[4] D. R. Nelson and B. I. Halperin, Phys. Rev. B , 2457(1979).[5] A. P. Young, Phys. Rev. B , 1855 (1979).[6] E. P. Bernard and W. Krauth, Phys. Rev. Lett. ,155704 (2011).[7] S. C. Kapfer and W. Krauth, Phys. Rev. Lett. ,035702 (2015).[8] J. A. Anderson, J. Antonaglia, J. A. Millan, M. Engel,and S. C. Glotzer, Phys. Rev. X , 021001 (2017).[9] R. E. Guerra, C. P. Kelleher, A. D. Hollingsworth, andP. M. Chaikin, Nature , 346 (2018).[10] J. Russo and N. B. Wilding, Phys. Rev. Lett. , 115702(2017).[11] P. Digregorio, D. Levis, A. Suma, L. F. Cugliandolo, G.Gonnella, and I. Pagonabarraga, Phys. Rev. Lett. ,098003 (2018).[12] L. F. Cugliandolo, P. Digregorio, G. Gonnella, and A.Suma, Phys. Rev. Lett. , 268002 (2017).[13] Y. Komatsu and H. Tanaka, Phys. Rev. X , 031025(2015).[14] M. Zu, J. Liu, H. Tong, and N. Xu, Phys. Rev. Lett. ,085702 (2016).[15] Y.-W. Li and M. P. Ciamarra, Phys. Rev. Lett. ,218002 (2020).[16] J. Palacci, S. Sacanna, A. P. Steinberg, D. J. Pine, andP. M. Chaikin, Science , 936 (2013).[17] K. Dietrich, G. Volpe, M. N. Sulaiman, D. Renggli, I. Buttinoni, and L. Isa, Phys. Rev. Lett. , 268004(2018).[18] T. B. Saw, W. Xi, B. Ladoux, and C. T. Lim, Adv. Mat. , e1802579 (2018).[19] J. Bialké, T. Speck, and H. Löwen, Phys. Rev. Lett. ,168301 (2012).[20] P. J. Steinhardt, D. R. Nelson, and M. Ronchetti, Phys.Rev. B , 784 (1983).[21] P. Hartmann, G. J. Kalman, Z. Donkó, and K. Kutasi,Phys. Rev. E , 026409 (2005).[22] H. Löwen, T. Palberg, and R. Simon, Phys. Rev. Lett. , 1557 (1993).[23] H. Löwen, Phys. Rev. E , R29 (1996).[24] K. Zahn and G. Maret, Phys. Rev. Lett. , 3656 (2000).[25] E. P. L. van Nieuwenburg, Y.-H. Liu, and S. D. Huber,Nat. Phys. , 435 (2017).[26] M. A. Nielsen, Neural Networks and Deep Learning , De-termination Press (2015).[27] I. Goodfellow, Y. Bengio, and A. Courville,
Deep Learn-ing , MIT Press (2016).[28] See Supplemental Material for discussions on relevanttechnical details.[29] The empirical criterion associated with the Lindemann-like parameter γ L is a dynamical criterion for meltingonly, hence essentially determines a lower bound of Γ forthe solid phase region [19], above which the system is in asolid phase. While the one associated with the long-timediffusion coefficient D is a dynamical criterion for freezingonly, hence essentially determines an upper bound of Γ for the liquid phase region [19], below which the systemis in a liquid phase. r X i v : . [ c ond - m a t . s o f t ] F e b Supplemental Material for “Data-driven criterion for the solid-liquid transition oftwo-dimensional self-propelled colloidal particles far from equilibrium”
Wei-chen Guo, Bao-quan Ai, ∗ and Liang He † Guangdong Provincial Key Laboratory of Quantum Engineering and Quantum Materials,School of Physics and Telecommunication Engineering,South China Normal University, Guangzhou 510006, China.
NN’S CONFIDENCE VALUES DETERMINED BYTHE LEARNABLE PARAMETER SET
In this work, we employ a fully-connected NN [1, 2]consisting of an input layer x ≡ ( x , x , · · · x k , · · · , x K ) T with K = 3 × × neurons, two fully-connected hid-den layers h ≡ ( h , h , · · · h k , · · · , h K ) T and h ≡ ( h , h , · · · , h k , · · · h K ) T with K = K = 2 × neurons, and an output layer with neurons y S and y L .When a sample of our data, i.e., an image I of theself-propelled colloidal particles’ spatial distribution, isfed to the fully-connected NN, each neuron x k in the in-put layer x collects a single raw pixel of I and deliver itsvalue to the next layer. Each neuron h k in the next layer,namely, the first hidden layer h , receives x and takes thevalue h k = ReLU ( w Th k x + b h k ) ≡ max(0 , w Th k x + b h k ) ,where the rectified linear unit (ReLU) is a nonlinearactivation function [1, 2]. Similarly, each neuron h k in the second hidden layer h takes the value h k = ReLU ( w Th k h + b h k ) , and, in the end, y S (L) = Sigmoid ( w Ty S (L) h + b y S (L) ) ≡ (1 + exp( − ( w Ty S (L) h + b y S (L) ))) − . (S-1)The sigmoid function is another nonlinear activationfunction [1, 2], whose return value is within [0 , andtherefore can be regarded as the “confidence” value.These two confidence values y S and y L hold the mean-ing of how likely any given configuration is a con-figuration of the solid and the liquid phase, respec-tively, and the learnable parameter set { w , b } ≡ (cid:8) w h k , w h k , w y S (L) , b h k , b h k , b y S (L) (cid:9) completely deter-mines the map that takes system’s configurations as in-put and gives the confidence values y S and y L as output. TRAINING PROCESS OF NN, SUPERVISEDLEARNING AND UNSUPERVISED LEARNING
Here, we briefly outline how to train the NN, i.e., tooptimize the learnable parameter set { w , b } with respectto the labels. For more thorough discussions on the ma-chine learning techniques involved in the training process,we refer the reader to Refs. [1, 2]. ∗ [email protected] † [email protected] Figure S1. Solid-liquid transition points predicted bythe data-driven criterion. The predicted transition points(marked by “ + ”) are (a) Γ ∗ (5) = 415 and (b) Γ ∗ (10) = 754 ,which match very well with the predictions from the directunsupervised learning. Here, the self-propelled force strength f = 5 is within the parameter regime f ∈ [0 , where thesupervised learning is directly performed, while f = 10 is waybeyond f ∈ [0 , . See text for more details. To train the NN, one shall first label the data, where alabel ˜ y = (˜ y S , ˜ y L ) T is a suggestion of the expected y S and y L . For instance, while labeling an image I as “liquid”,we suggest that I is likely to be a configurationof the liquid phase, namely, ˜ y = (0 , T . In both thesupervised learning process and the unsupervised learn-ing process in this work, every sample of data used fortraining has a label. The major difference between thesetwo learning process is whether the labels are physical.More specifically, labels in the supervised learning pro-cess are always expected to be physically correct, whilelabels in the unsupervised learning process are not alwaysthe case.After the labeling, the error of the confidence valuescompared to ˜ y can be quantified by the cross-entropycost function S = − (˜ y S ln y S + ˜ y L ln y L ) [1, 2]. Wheneverthe NN is “trained” in this work, { w , b } is optimized byminimizing the cost function S traversing the trainingsamples for epochs using the Adam method [3] withthe learning rate × − . Figure S2. (a) Γ -dependence of the global bond-orientationalorder parameter ψ of the system at different self-propel forcestrengths f = 0 , , , , . The horizontal dashed line corre-sponds to the equilibrium threshold ψ EQ ,threshold = 0 . . Thecrossings marked by “ × ” of the horizontal dashed line with the D (Γ) curves at different f predict the corresponding transi-tion points Γ ∗ ψ ( f ) . (b) Threshold value ψ ,threshold depen-dence of the criterial classification accuracy P ψ ,threshold . Theleftmost point corresponds to ψ ,threshold = ψ EQ ,threshold = 0 . ,and the point with the highest P ψ ,threshold corresponds to ψ ,threshold = 0 . which is a general threshold value for ψ inthe NEQ scenario. See text for more details. TRANSITION POINTS PREDICTED BY THEDATA-DRIVEN CRITERION
The data-driven criterion which captures the genericfeature that distinguishes the solid from the liquid phaseis expected to give equal confidence values y S and y L at the solid-liquid transition point, since the system canbe either in the solid phase or the liquid phase. There-fore, after the supervised learning process with all theproperly labeled data at f = 0 , , , , in three in-dependent data sets “A”, “B”, and “C”, the solid-liquidtransition point for a fixed self-propel force strength f can be predicted by calculating the average classifi-cation confidence values [4], whose explicit form reads C S (L) (Γ , f ) = P N Γ ,f n =1 y S (L) ( I Γ ,f ; n ) / N Γ ,f , where I Γ ,f ; n is the n th testing sample with its corresponding effectiveinteraction strength being Γ and N Γ ,f is the total num-ber of these samples. Here, the intersection points ofthe Γ -dependence curves of C S and C L in Fig. S1 indi-cate Γ ∗ (5) = 415 and Γ ∗ (10) = 754 , respectively, whichmatch very well with the predictions from the direct un-supervised learning. CRITERIAL CLASSIFICATION ACCURACYAND THE NEQ THRESHOLD VALUE
For a given criterion C , the transition point Γ ∗C ( f i ) at the self-propel force strength f i is determined, andit corresponds to a binary classification rule associatedwith the proposed critical value Γ ∗′ ( f i ) = Γ ∗C ( f i ) . There-fore, after the unsupervised learning process, the criterialclassification accuracy P C = P N f i =1 P (Γ ∗C ( f i )) /N f , can bedirectly calculated according to the Γ ∗′ ( f i ) dependencecurves of the classification accuracy P (Γ ∗′ ( f i )) .Concerning the empirical criterion associated withthe long-time diffusion coefficient D , we moni-tor the criterial classification accuracy P D threshold = P f =0 , , , , P (Γ ∗ D threshold ( f )) / by utilizing all the avail-able data at f = 0 , , , , in three independent datasets “A”, “B”, and “C”, and indeed find an optimal choiceof a smaller NEQ threshold D NEQthreshold = 0 . . How-ever, for the empirical criterion associated with the globalbond-orientational order parameter ψ , it assumes rela-tively poor performance in general. As we can see fromFig. S2(b), for this empirical criterion, the highest cri-terial classification accuracy P ψ ,threshold with a generalthreshold value ψ ,threshold = 0 . for ψ is just roughlyaround P D EQthreshold [cf. Fig. 3(b) and the lower horizontaldashed line in Fig. S2(b)], and much lower than P D NEQthreshold [cf. Fig. 3(b) and the upper horizontal dashed line inFig. S2(b)]. [1] M. A. Nielsen,
Neural Networks and Deep Learning (De-termination Press, 2015).[2] I. Goodfellow, Y. Bengio, and A. Courville,
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