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.

Figure 1

Autoregressive networks with different architectures for variational free energy calculation. The spin configuration $\mathbf{s}$ is the input to the network, $\widehat{\mathbf{s}}$ is the output of the network, and $h$ denotes hidden layer. The loss function $F_q$ is given by Eq. (3) and Eq. (4). The colored sites denote the receptive field of a site in $\widehat{\mathbf{s}}$. (a) The network has only one layer, which is densely connected, while the autoregressive property hold. (b) The network has a hidden layer. (c) The network has masked convolution layers on 2D lattice. Only connections in a convolution kernel are shown for clarity.

Figure 2

(a) Free energy per site and its relative error of ferromagnetic Ising model on $16\times 16$ square lattice with periodic boundary condition. (b) Entropy per site of antiferromagnetic Ising model on triangular lattices of various sizes $L$ with periodic boundary condition. The exact result (dashed line) at $T=0$ and $L\to \infty$ is $S/N=0.323066$ [30, 31]. The curves for $L=8$, 14, 16 are almost overlapped.

Figure 3

Log probability of sampled configurations from VAN trained for a Hopfield model with $N=100$ spins, and $P=2$ orthogonal patterns. The sampled configurations are projected onto the two-dimensional space spanned by the two patterns. $X$ axis ($O_1$) and $Y$ axis ($O_2$) are the overlap (inner product, normalized to $[-1,1]$) between each sampled configuration and the two patterns, respectively. (a) $\beta =0.3$, and the system is in the paramagnetic phase. (b) $\beta =1.5$, and the system is in the retrieval phase. Note the different scales in the color bars.

Figure 4

(a) Free energy of SK model with $N=20$ spins. The inset shows relative errors to exact values in a larger $\beta$ regime. Bethe converges only when $\beta \le 1.5$. (b) The reconstruction error in the inverse Ising problem. The underlying model is an SK model with $N=20$ spins. VAN uses a network with two layers (a hidden layer and an output layer).