Physics

Inverse problem

Mathematics

Principle of maximum entropy

Entropy (arrow of time)

Statistical physics

Statistics

Autoregressive model

Mathematical analysis

Artificial intelligence

Computer science

Ising model

Quantum mechanics

Applied mathematics

Variational principle

Artificial neural network

Statistical mechanics

Physical Review Letters

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.

Solving Statistical Mechanics Using Variational Autoregressive Networks