Estimating free energy is a fundamental and difficult problem in statistical physics. In recent years, machine-learning methods, especially deep generative models, have been extensively applied to this problem. The variational autoregressive network (VAN) is one of the most successful methods. VAN has notable advantages, such as tractable probabilities from normalized distribution or the absolute value of the free energy. However, the sampling process of VAN is serial, which limits its application to high-dimensional and large-scale systems. In this work, we utilize renormalization group (RG) with VAN, namely RG-VAN, to improve the sampling efficiency of VAN. The method reduces the computational complexity from polynomial to logarithmic with respect to the degrees of freedom. Numerical experiments on the two-dimensional Ising model and Edwards-Anderson model show that RG-VAN can accelerate the training and inference process by an order of magnitude, with little loss of accuracy.

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• Spin glasses
• Square lattice
• Artificial neural networks
• Ising model
• Machine learning
• Monte Carlo methods
• Reinforcement learning