Solving Statistical Mechanics Using Variational Autoregressive Networks

TL;DR

This paper introduces a neural network-based variational method for solving finite-size statistical mechanics problems, enabling direct sampling, free energy computation, and physical quantity estimation with improved accuracy.

Contribution

It extends variational mean-field methods with autoregressive neural networks, allowing exact probability calculations and efficient sampling for complex systems.

Findings

Successfully applied to 2D Ising, Hopfield, Sherrington-Kirkpatrick, and inverse Ising models.

Demonstrates advantages over traditional variational mean-field approaches.

Provides a deep learning framework for solving classical statistical physics problems.

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.