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We develop the neural network based “learning from regression uncertainty” approach for the automatic detection of phases of matter in nonequilibrium active systems. Taking the flocking phase transition of self-propelled active particles described by the Vicsek model for example, we find that after training a neural network for solving the inverse statistical problem, i.e. for performing the regression task of reconstructing the noise level from given samples of such a nonequilibrium many-body complex system’s steady state configurations, the uncertainty of regression results obtained by the well-trained network can actually be utilized to reveal possible phase transitions in the system under study. The noise level dependence of regression uncertainty is assumed to be in a non-trivial M-shape, and its valley appears at the critical point of the flocking phase transition. By directly comparing this regression-based approach with the widely-used classification-based “learning by confusion” and “learning with blanking” approaches, we show that our approach has practical effectiveness, efficiency, good generality for various physical systems across interdisciplinary fields, and a greater possibility of being interpretable via conventional notions of physics. These approaches can complement each other to serve as a promising generic toolbox for investigating rich critical phenomena and providing data-driven evidence on the existence of various phase transitions, especially for those complex scenarios associated with first-order phase transitions or nonequilibrium active systems where traditional research methods in physics could face difficulties.
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Keywords:
- machine learning /
- phase transition /
- nonequilibrium many-body system /
- inverse statistical problem
[1] Melko R G, Carrasquilla J 2017 Nat. Phys. 13 431
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[2] van Nieuwenburg E P L, Liu Y H, Huber S D 2017 Nat. Phys. 13 435
Google Scholar
[3] Guo W C, Ai B Q, He L 2021 EPL 136 48002
Google Scholar
[4] Venderley J, Khemani V, Kim E A 2018 Phys. Rev. Lett. 120 257204
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[5] Beach M J S, Golubeva A, Melko R G 2018 Phys. Rev. B 97 045207
Google Scholar
[6] Lee S S, Kim B J 2019 Phys. Rev. E 99 043308
Google Scholar
[7] Ch’ng K, Carrasquilla J, Melko R G, Khatami E 2017 Phys. Rev. X 7 031038
[8] Broecker P, Carrasquilla J, Melko R G, Trebst S 2017 Sci. Rep. 7 1
Google Scholar
[9] Carrasquilla J, 2020 Adv. Phys. X 5 1797528
[10] Yu L W, Zhang S Y, Shen P X, Deng D L 2023 Fundamental Research (In Press
[11] Rem B S, Käming N, Tarnowski M, Asteria L, Fläschner N, Becker C, Sengstock K, Weitenberg C 2019 Nat. Phys. 15 917
Google Scholar
[12] Gökmen D E, Ringel Z, Huber S D, Koch-Janusz M 2021 Phys. Rev. Lett. 127 240603
Google Scholar
[13] Gökmen D E, Ringel Z, Huber S D, Koch-Janusz M 2021 Phys. Rev. E 104 064106
[14] Miles C, Bohrdt A, Wu R, Chiu C, Xu M, Ji G, Greiner M, Weinberger K Q, Demler E, Kim E A 2021 Nat. Commun. 12 3905
Google Scholar
[15] Nguyen H C, Zecchina R, Berg J 2017 Adv. Phys. 66 197
Google Scholar
[16] Udrescu S M, Tegmark M 2021 Phys. Rev. E 103 043307
[17] Liu Z, Tegmark M 2022 Phys. Rev. Lett. 128 180201
Google Scholar
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Google Scholar
[19] Udrescu S M, Tegmark M 2020 Sci. Adv. 6 eaay2631
Google Scholar
[20] Guo W C, He L 2023 New J. Phys. 25 083037
[21] Binder K 1987 Rep. Prog. Phys. 50 783
Google Scholar
[22] Falkovich G, Gawȩdzki K, Vergassola M 2001 Rev. Mod. Phys. 73 913
Google Scholar
[23] Jarzynski C 2015 Nat. Phys. 11 105
Google Scholar
[24] Vicsek T, Czirók A, Ben-Jacob E, Cohen I, Shochet O 1995 Phys. Rev. Lett. 75 1226
Google Scholar
[25] Toner J, Tu Y, Ramaswamy S 2005 Ann. Phys. 318 170
Google Scholar
[26] Grégoire G, Chaté H 2004 Phys. Rev. Lett. 92 025702
Google Scholar
[27] Chaté H, Ginelli F, Grégoire G, Raynaud F 2008 Phys. Rev. E 77 046113
Google Scholar
[28] He K, Zhang X, Ren S, Sun J 2016 Proceedings of 2016 IEEE Conference on Computer Vision and Pattern Recognition (CVPR) Las Vegas, USA, June 27–30, 2016 p770
[29] Nguyen H C, Berg J 2012 Phys. Rev. Lett. 109 050602
Google Scholar
[30] Jo J, Hoang D T, Periwal V 2020 Phys. Rev. E 101 032107
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图 1 数值模拟生成的对应于不同噪声强度
$ \eta $ 的典型样本. 样本中的每个圆形标记表示二维空间中的一个自驱动粒子, 其空间分布表示自驱动粒子的瞬时空间分布, 其颜色分布表示自驱动粒子的运动方向的瞬时角度分布. 此处作为示例的样本中, 左边的5个样本处于群集相, 最右边的样本处于无序相Figure 1. Typical samples corresponding to different noise levels that are generated by numerical simulations. In every sample, each of the circular markers represents a single self-propelled particle in the two-dimensional space, with their spatial distribution representing the instantaneous spatial distribution of self-propelled particles, and their color distribution representing the instantaneous angular distribution of directions of motion of these self-propelled particles. Among the samples shown here for instance, the five samples in the left are in the flocking phase, and the rightmost one is in the disordered phase.
图 2 自驱动活性粒子系统中的ISP (a) 系统的群速度
$ \bar v $ 关于噪声强度$ \eta $ 的依赖关系,$ \bar v $ 在$ {\eta _{\text{c}}} = {0}{.626} \pm {0}{.006} $ 的突变表明系统在该噪声强度处发生一阶相变; (b)训练完成的ANN给出的重构噪声强度$ {\eta _{\text{R}}} $ 关于实际噪声强度$ \eta $ 的依赖关系, 误差棒表示回归不确定性$ U(\eta ) $ , 对角线表示理想的回归结果$ {\eta _{\text{R}}}{=}\eta $ Figure 2. Inverse statistical problem in a self-propelled active particle system: (a) Noise level dependence of the system’s global group velocity, whose jump at
$ {\eta _{\text{c}}} = {0}{.626} \pm {0}{.006} $ characterizes the first-order flocking phase transition; (b) noise level dependence of the reconstructed noise level predicted by the well-trained ANN. The error bars represent the regression uncertainty$ U(\eta ) $ , and the diagonal line represent the ideal regression result$ {\eta _{\text{R}}}{=}\eta $ .
图 3 三种机器学习方法揭示自驱动活性粒子的群集相变 (a) 基于回归不确定性的LFRU方法; (b) “混淆法”; (c) “留白法”
Figure 3. Revealing the flocking phase transition of self-propelled active particles via applying three different machine learning approaches: (a) The LFRU approach; (b) the “learning by confusion” approach; (c) the “learning with blanking” approach.
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[1] Melko R G, Carrasquilla J 2017 Nat. Phys. 13 431
Google Scholar
[2] van Nieuwenburg E P L, Liu Y H, Huber S D 2017 Nat. Phys. 13 435
Google Scholar
[3] Guo W C, Ai B Q, He L 2021 EPL 136 48002
Google Scholar
[4] Venderley J, Khemani V, Kim E A 2018 Phys. Rev. Lett. 120 257204
Google Scholar
[5] Beach M J S, Golubeva A, Melko R G 2018 Phys. Rev. B 97 045207
Google Scholar
[6] Lee S S, Kim B J 2019 Phys. Rev. E 99 043308
Google Scholar
[7] Ch’ng K, Carrasquilla J, Melko R G, Khatami E 2017 Phys. Rev. X 7 031038
[8] Broecker P, Carrasquilla J, Melko R G, Trebst S 2017 Sci. Rep. 7 1
Google Scholar
[9] Carrasquilla J, 2020 Adv. Phys. X 5 1797528
[10] Yu L W, Zhang S Y, Shen P X, Deng D L 2023 Fundamental Research (In Press
[11] Rem B S, Käming N, Tarnowski M, Asteria L, Fläschner N, Becker C, Sengstock K, Weitenberg C 2019 Nat. Phys. 15 917
Google Scholar
[12] Gökmen D E, Ringel Z, Huber S D, Koch-Janusz M 2021 Phys. Rev. Lett. 127 240603
Google Scholar
[13] Gökmen D E, Ringel Z, Huber S D, Koch-Janusz M 2021 Phys. Rev. E 104 064106
[14] Miles C, Bohrdt A, Wu R, Chiu C, Xu M, Ji G, Greiner M, Weinberger K Q, Demler E, Kim E A 2021 Nat. Commun. 12 3905
Google Scholar
[15] Nguyen H C, Zecchina R, Berg J 2017 Adv. Phys. 66 197
Google Scholar
[16] Udrescu S M, Tegmark M 2021 Phys. Rev. E 103 043307
[17] Liu Z, Tegmark M 2022 Phys. Rev. Lett. 128 180201
Google Scholar
[18] Liu Z, Tegmark M 2021 Phys. Rev. Lett. 126 180604
Google Scholar
[19] Udrescu S M, Tegmark M 2020 Sci. Adv. 6 eaay2631
Google Scholar
[20] Guo W C, He L 2023 New J. Phys. 25 083037
[21] Binder K 1987 Rep. Prog. Phys. 50 783
Google Scholar
[22] Falkovich G, Gawȩdzki K, Vergassola M 2001 Rev. Mod. Phys. 73 913
Google Scholar
[23] Jarzynski C 2015 Nat. Phys. 11 105
Google Scholar
[24] Vicsek T, Czirók A, Ben-Jacob E, Cohen I, Shochet O 1995 Phys. Rev. Lett. 75 1226
Google Scholar
[25] Toner J, Tu Y, Ramaswamy S 2005 Ann. Phys. 318 170
Google Scholar
[26] Grégoire G, Chaté H 2004 Phys. Rev. Lett. 92 025702
Google Scholar
[27] Chaté H, Ginelli F, Grégoire G, Raynaud F 2008 Phys. Rev. E 77 046113
Google Scholar
[28] He K, Zhang X, Ren S, Sun J 2016 Proceedings of 2016 IEEE Conference on Computer Vision and Pattern Recognition (CVPR) Las Vegas, USA, June 27–30, 2016 p770
[29] Nguyen H C, Berg J 2012 Phys. Rev. Lett. 109 050602
Google Scholar
[30] Jo J, Hoang D T, Periwal V 2020 Phys. Rev. E 101 032107
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