# Solving Statistical Mechanics Using Variational Autoregressive Networks

arXiv: https://arxiv.org/abs/1809.10606 · HTML: https://mlsanity.com/paper/1809.10606/

## Key facts

- Cited by 100 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 27 Sep 2018.

## Built on by (most cited first, 30 of 100)

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- [Diffusion Generative Flow Samplers: Improving learning signals through partial trajectory optimization](https://mlsanity.com/paper/2310.02679/) (2023)
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- [Aspects of scaling and scalability for flow-based sampling of lattice QCD](https://mlsanity.com/paper/2211.07541/) (2022)
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- [Variational Annealing on Graphs for Combinatorial Optimization](https://mlsanity.com/paper/2311.14156/) (2023)
- [Unbiased Monte Carlo Cluster Updates with Autoregressive Neural Networks](https://mlsanity.com/paper/2105.05650/) (2021)
- [Ab-initio study of interacting fermions at finite temperature with neural canonical transformation](https://mlsanity.com/paper/2105.08644/) (2021)
- [Efficient Bayesian Sampling Using Normalizing Flows to Assist Markov Chain Monte Carlo Methods](https://mlsanity.com/paper/2107.08001/) (2021)
- [Controlling Continuous Relaxation for Combinatorial Optimization](https://mlsanity.com/paper/2309.16965/) (2023)
- [Applications of Machine Learning to Lattice Quantum Field Theory](https://mlsanity.com/paper/2202.05838/) (2022)
- [Precision Machine Learning](https://mlsanity.com/paper/2210.13447/) (2022)
- [A theoretical perspective on mode collapse in variational inference](https://mlsanity.com/paper/2410.13300/) (2024)
- [Reinforcement Fine-Tuning for Materials Design](https://mlsanity.com/paper/2504.02367/) (2025)
- [Discrete Neural Flow Samplers with Locally Equivariant Transformer](https://mlsanity.com/paper/2505.17741/) (2025)
- [Hierarchical autoregressive neural networks for statistical systems](https://mlsanity.com/paper/2203.10989/) (2022)
- [Path-Gradient Estimators for Continuous Normalizing Flows](https://mlsanity.com/paper/2206.09016/) (2022)
- [Analysis of autocorrelation times in Neural Markov Chain Monte Carlo simulations](https://mlsanity.com/paper/2111.10189/) (2021)
- [Machine Learning of Thermodynamic Observables in the Presence of Mode Collapse](https://mlsanity.com/paper/2111.11303/) (2021)
- [Artificial intelligence for representing and characterizing quantum systems](https://mlsanity.com/paper/2509.04923/) (2025)
- [qecGPT: decoding Quantum Error-correcting Codes with Generative Pre-trained Transformers](https://mlsanity.com/paper/2307.09025/) (2023)
- [Neural-network solutions to stochastic reaction networks](https://mlsanity.com/paper/2210.01169/) (2022)
- [The autoregressive neural network architecture of the Boltzmann distribution of pairwise interacting spins systems](https://mlsanity.com/paper/2302.08347/) (2023)
- [m^ast of two-dimensional electron gas: a neural canonical transformation study](https://mlsanity.com/paper/2201.03156/) (2022)

## Abstract (arXiv)

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.