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Tang Y, Liu J, Zhang J, Zhang P. Learning nonequilibrium statistical mechanics and dynamical phase transitions. Nat Commun 2024; 15:1117. [PMID: 38321012 PMCID: PMC10847122 DOI: 10.1038/s41467-024-45172-8] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/31/2023] [Accepted: 01/15/2024] [Indexed: 02/08/2024] Open
Abstract
Nonequilibrium statistical mechanics exhibit a variety of complex phenomena far from equilibrium. It inherits challenges of equilibrium, including accurately describing the joint distribution of a large number of configurations, and also poses new challenges as the distribution evolves over time. Characterizing dynamical phase transitions as an emergent behavior further requires tracking nonequilibrium systems under a control parameter. While a number of methods have been proposed, such as tensor networks for one-dimensional lattices, we lack a method for arbitrary time beyond the steady state and for higher dimensions. Here, we develop a general computational framework to study the time evolution of nonequilibrium systems in statistical mechanics by leveraging variational autoregressive networks, which offer an efficient computation on the dynamical partition function, a central quantity for discovering the phase transition. We apply the approach to prototype models of nonequilibrium statistical mechanics, including the kinetically constrained models of structural glasses up to three dimensions. The approach uncovers the active-inactive phase transition of spin flips, the dynamical phase diagram, as well as new scaling relations. The result highlights the potential of machine learning dynamical phase transitions in nonequilibrium systems.
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Affiliation(s)
- Ying Tang
- Institute of Fundamental and Frontier Sciences, University of Electronic Sciences and Technology of China, Chengdu, 611731, China.
- International Academic Center of Complex Systems, Beijing Normal University, Zhuhai, 519087, China.
| | - Jing Liu
- CAS Key Laboratory for Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing, 100190, China
- School of Systems Science, Beijing Normal University, Beijing, 100875, China
| | - Jiang Zhang
- School of Systems Science, Beijing Normal University, Beijing, 100875, China
- Swarma Research, Beijing, 102308, China
| | - Pan Zhang
- CAS Key Laboratory for Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing, 100190, China.
- School of Fundamental Physics and Mathematical Sciences, Hangzhou Institute for Advanced Study, UCAS, Hangzhou, 310024, China.
- Hefei National Laboratory, Hefei, 230088, China.
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Braunstein A, Budzynski L, Mariani M. Statistical mechanics of inference in epidemic spreading. Phys Rev E 2023; 108:064302. [PMID: 38243547 DOI: 10.1103/physreve.108.064302] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/25/2023] [Accepted: 11/21/2023] [Indexed: 01/21/2024]
Abstract
We investigate the information-theoretical limits of inference tasks in epidemic spreading on graphs in the thermodynamic limit. The typical inference tasks consist in computing observables of the posterior distribution of the epidemic model given observations taken from a ground-truth (sometimes called planted) random trajectory. We can identify two main sources of quenched disorder: the graph ensemble and the planted trajectory. The epidemic dynamics however induces nontrivial long-range correlations among individuals' states on the latter. This results in nonlocal correlated quenched disorder which unfortunately is typically hard to handle. To overcome this difficulty, we divide the dynamical process into two sets of variables: a set of stochastic independent variables (representing transmission delays), plus a set of correlated variables (the infection times) that depend deterministically on the first. Treating the former as quenched variables and the latter as dynamic ones, computing disorder average becomes feasible by means of the replica-symmetric cavity method. We give theoretical predictions on the posterior probability distribution of the trajectory of each individual, conditioned to observations on the state of individuals at given times, focusing on the susceptible infectious (SI) model. In the Bayes-optimal condition, i.e., when true dynamic parameters are known, the inference task is expected to fall in the replica-symmetric regime. We indeed provide predictions for the information theoretic limits of various inference tasks, in form of phase diagrams. We also identify a region, in the Bayes-optimal setting, with strong hints of replica-symmetry breaking. When true parameters are unknown, we show how a maximum-likelihood procedure is able to recover them with mostly unaffected performance.
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Affiliation(s)
- Alfredo Braunstein
- DISAT, Politecnico di Torino, Corso Duca Degli Abruzzi 24, 10129 Torino
- Collegio Carlo Alberto, P.za Arbarello 8, 10122 Torino, Italy
- Italian Institute for Genomic Medicine, IRCCS Candiolo, SP-142, I-10060 Candiolo (TO), Italy
| | - Louise Budzynski
- DISAT, Politecnico di Torino, Corso Duca Degli Abruzzi 24, 10129 Torino
- Italian Institute for Genomic Medicine, IRCCS Candiolo, SP-142, I-10060 Candiolo (TO), Italy
- Dipartimento di Fisica, Università "La Sapienza", P.le A. Moro 5, 00185 Rome, Italy
| | - Matteo Mariani
- DISAT, Politecnico di Torino, Corso Duca Degli Abruzzi 24, 10129 Torino
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Ghio D, Aragon ALM, Biazzo I, Zdeborová L. Bayes-optimal inference for spreading processes on random networks. Phys Rev E 2023; 108:044308. [PMID: 37978700 DOI: 10.1103/physreve.108.044308] [Citation(s) in RCA: 1] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/06/2023] [Accepted: 09/25/2023] [Indexed: 11/19/2023]
Abstract
We consider a class of spreading processes on networks, which generalize commonly used epidemic models such as the SIR model or the SIS model with a bounded number of reinfections. We analyze the related problem of inference of the dynamics based on its partial observations. We analyze these inference problems on random networks via a message-passing inference algorithm derived from the belief propagation (BP) equations. We investigate whether said algorithm solves the problems in a Bayes-optimal way, i.e., no other algorithm can reach a better performance. For this, we leverage the so-called Nishimori conditions that must be satisfied by a Bayes-optimal algorithm. We also probe for phase transitions by considering the convergence time and by initializing the algorithm in both a random and an informed way and comparing the resulting fixed points. We present the corresponding phase diagrams. We find large regions of parameters where even for moderate system sizes the BP algorithm converges and satisfies closely the Nishimori conditions, and the problem is thus conjectured to be solved optimally in those regions. In other limited areas of the space of parameters, the Nishimori conditions are no longer satisfied and the BP algorithm struggles to converge. No sign of a phase transition is detected, however, and we attribute this failure of optimality to finite-size effects. The article is accompanied by a Python implementation of the algorithm that is easy to use or adapt.
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Affiliation(s)
- Davide Ghio
- Ide PHICS Laboratory, Ecole Polytechnique Fédérale de Lausanne (EPFL), Rte Cantonale, 1015 Lausanne, Switzerland
| | - Antoine L M Aragon
- SPOC Laboratory, Ecole Polytechnique Fédérale de Lausanne (EPFL), Rte Cantonale, 1015 Lausanne, Switzerland
| | - Indaco Biazzo
- SPOC Laboratory, Ecole Polytechnique Fédérale de Lausanne (EPFL), Rte Cantonale, 1015 Lausanne, Switzerland
| | - Lenka Zdeborová
- SPOC Laboratory, Ecole Polytechnique Fédérale de Lausanne (EPFL), Rte Cantonale, 1015 Lausanne, Switzerland
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Braunstein A, Catania G, Dall'Asta L, Mariani M, Muntoni AP. Inference in conditioned dynamics through causality restoration. Sci Rep 2023; 13:7350. [PMID: 37147382 PMCID: PMC10163042 DOI: 10.1038/s41598-023-33770-3] [Citation(s) in RCA: 1] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/25/2023] [Accepted: 04/18/2023] [Indexed: 05/07/2023] Open
Abstract
Estimating observables from conditioned dynamics is typically computationally hard. While obtaining independent samples efficiently from unconditioned dynamics is usually feasible, most of them do not satisfy the imposed conditions and must be discarded. On the other hand, conditioning breaks the causal properties of the dynamics, which ultimately renders the sampling of the conditioned dynamics non-trivial and inefficient. In this work, a Causal Variational Approach is proposed, as an approximate method to generate independent samples from a conditioned distribution. The procedure relies on learning the parameters of a generalized dynamical model that optimally describes the conditioned distribution in a variational sense. The outcome is an effective and unconditioned dynamical model from which one can trivially obtain independent samples, effectively restoring the causality of the conditioned dynamics. The consequences are twofold: the method allows one to efficiently compute observables from the conditioned dynamics by averaging over independent samples; moreover, it provides an effective unconditioned distribution that is easy to interpret. This approximation can be applied virtually to any dynamics. The application of the method to epidemic inference is discussed in detail. The results of direct comparison with state-of-the-art inference methods, including the soft-margin approach and mean-field methods, are promising.
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Affiliation(s)
- Alfredo Braunstein
- DISAT, Politecnico di Torino, Corso Duca Degli Abruzzi 24, 10129, Turin, Italy
- INFN, Sezione di Torino, Turin, Italy
- Italian Institute for Genomic Medicine, IRCCS Candiolo, SP-142, 10060, Candiolo, TO, Italy
| | - Giovanni Catania
- Departamento de Física Téorica I, Universidad Complutense, 28040, Madrid, Spain
| | - Luca Dall'Asta
- DISAT, Politecnico di Torino, Corso Duca Degli Abruzzi 24, 10129, Turin, Italy
- INFN, Sezione di Torino, Turin, Italy
- Italian Institute for Genomic Medicine, IRCCS Candiolo, SP-142, 10060, Candiolo, TO, Italy
- Collegio Carlo Alberto, P.za Arbarello 8, 10122, Turin, Italy
| | - Matteo Mariani
- DISAT, Politecnico di Torino, Corso Duca Degli Abruzzi 24, 10129, Turin, Italy.
| | - Anna Paola Muntoni
- DISAT, Politecnico di Torino, Corso Duca Degli Abruzzi 24, 10129, Turin, Italy
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