Solving Statistical Mechanics Using Variational Autoregressive Networks

Abstract

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.Consider a statistical physics model such as the celebrated Ising model, the joint probability of spins s ∈ {±1} N follows the Boltzmann distributionwhere β = 1/T is the inverse temperature and Z is the partition function. Given a problem instance, statistical mechanics problems concern about how to estimate the free energy F = − 1 β ln Z of the instance, how to compute macroscopic properties of the system such as magnetizations and correlations, and how to sample from the Boltzmann distribution efficiently. Solving these problems are not only relevant to physics, but also find broad applications in fields like Bayesian inference where the Boltzmann distribution naturally acts as posterior distribution, and in combinatorial optimizations where the task is equivalent to study zero temperature phase of a spin-glass model.When the system has finite size, computing exactly the free energy belongs to the class of #P-hard problems, hence is in general intractable. Therefore, usually one employs approximate algorithms such as variational approaches. The variational approach adopts an ansatz for the joint distribution q θ (s) parametrized by variational parameters θ, and adjusts them so that q θ (s) is as close as possible to the Boltzmann distribution p(s). The closeness between two distributions is measured bywhereis the variational free energy corresponding to distribution q θ (s). Since the KL divergence is non-negative, minimizing the KL divergence is equivalent to minimizing the variational free energy F q , an upper bound to the true free energy F.One of the most popular variational approaches, namely the variational mean-field method, assumes a factorized vari-is the marginal probability of the ith spin. In such parametrization, the variational free energy F q can be expressed as an analytical function of parameters q i (s i ), as well as its derivative with respect to q i (s i ). By setting the derivatives to zero, one obtains a set of iterative equations, known as the naïve mean-field (NMF) equations. Despite its simplicity, NMF has been used in various applica...

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