Solving Statistical Mechanics using Variational Autoregressive Networks

We propose a general framework for solving statistical mechanics of systems with finite size. The approach extends the celebrated variational mean-field approaches using autoregressive neural networks, which support direct sampling and exact calculation of normalized probability of configurations. It computes variational free energy, estimates physical quantities such as entropy, magnetizations and correlations, and generates uncorrelated samples all at once. Training of the network employs the policy gradient approach in reinforcement learning, which unbiasedly estimates the gradient of variational parameters. We apply our approach to several classic systems, including 2D Ising models, the Hopfield model, the Sherrington-Kirkpatrick model, and the inverse Ising model, for demonstrating its advantages over existing variational mean-field methods. Our approach sheds light on solving statistical physics problems using modern deep generative neural networks.

[1]  Haijun Zhou,et al.  Topologically invariant tensor renormalization group method for the Edwards-Anderson spin glasses model , 2014 .

[2]  R. Kikuchi.  A Theory of Cooperative Phenomena , 1951 .

[3]  R. Palmer,et al.  Solution of 'Solvable model of a spin glass' , 1977 .

[4]  Matthias Troyer,et al.  Solving the quantum many-body problem with artificial neural networks , 2016, Science.

[5]  H. J. Mclaughlin,et al.  Learn , 2002 .

[6]  Z. Y. Xie,et al.  Second renormalization of tensor-network states. , 2008, Physical review letters.

[7]  Sompolinsky,et al.  Storing infinite numbers of patterns in a spin-glass model of neural networks. , 1985, Physical review letters.

[8]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[9]  L. Onsager.  Crystal statistics. I. A two-dimensional model with an order-disorder transition , 1944 .

[10]  F. Ricci-Tersenghi.  The Bethe approximation for solving the inverse Ising problem: a comparison with other inference methods , 2011, 1112.4814.

[11]  J. Berg,et al.  Bethe–Peierls approximation and the inverse Ising problem , 2011, 1112.3501.

[12]  Christopher M. Bishop,et al.  Pattern Recognition and Machine Learning (Information Science and Statistics) , 2006 .

[13]  Kevin Barraclough,et al.  I and i , 2001, BMJ : British Medical Journal.

[14]  Ronald J. Williams,et al.  Simple Statistical Gradient-Following Algorithms for Connectionist Reinforcement Learning , 2004, Machine Learning.

[15]  W. Marsden.  I and J , 2012 .

[16]  Sompolinsky,et al.  Spin-glass models of neural networks. , 1985, Physical review. A, General physics.

[17]  Thierry Mora,et al.  Constraint satisfaction problems and neural networks: A statistical physics perspective , 2008, Journal of Physiology-Paris.

[18]  J J Hopfield,et al.  Neural networks and physical systems with emergent collective computational abilities. , 1982, Proceedings of the National Academy of Sciences of the United States of America.

[19]  Lei Wang,et al.  Neural Network Renormalization Group , 2018, Physical review letters.

[20]  M. Mézard,et al.  Spin Glass Theory And Beyond: An Introduction To The Replica Method And Its Applications , 1986 .

[21]  Brendan J. Frey,et al.  Graphical Models for Machine Learning and Digital Communication , 1998 .

[22]  Michael I. Jordan,et al.  An Introduction to Variational Methods for Graphical Models , 1999, Machine Learning.

[23]  L. Goddard.  Information Theory , 1962, Nature.

[24]  H. Bethe.  Statistical Theory of Superlattices , 1935 .

[25]  Pierre Vandergheynst,et al.  Geometric Deep Learning: Going beyond Euclidean data , 2016, IEEE Signal Process. Mag..

[26]  Florent Krzakala,et al.  Statistical physics of inference: thresholds and algorithms , 2015, ArXiv.

[27]  S. Kirkpatrick,et al.  Solvable Model of a Spin-Glass , 1975 .

[28]  J. Hertz,et al.  Ising model for neural data: model quality and approximate methods for extracting functional connectivity. , 2009, Physical review. E, Statistical, nonlinear, and soft matter physics.

[29]  Richard S. Sutton,et al.  Reinforcement Learning: An Introduction , 1998, IEEE Trans. Neural Networks.

[30]  G. Parisi.  The order parameter for spin glasses: a function on the interval 0-1 , 1980 .

[31]  Robert B. Ash,et al.  Information Theory , 2020, The SAGE International Encyclopedia of Mass Media and Society.

[32]  Hilbert J. Kappen,et al.  Efficient Learning in Boltzmann Machines Using Linear Response Theory , 1998, Neural Computation.

[33]  Michael Levin,et al.  Tensor renormalization group approach to two-dimensional classical lattice models. , 2006, Physical review letters.

[34]  R. Zecchina,et al.  Inverse statistical problems: from the inverse Ising problem to data science , 2017, 1702.01522.

[35]  R. Monasson,et al.  Small-correlation expansions for the inverse Ising problem , 2008, 0811.3574.

[36]  G. H. Wannier.  Antiferromagnetism. The Triangular Ising Net , 1973 .

[37]  Hugo Larochelle,et al.  Neural Autoregressive Distribution Estimation , 2016, J. Mach. Learn. Res..