Message passing variational autoregressive network for solving intractable Ising models

Deep neural networks have been used to solve Ising models, including autoregressive neural networks, convolutional neural networks, recurrent neural networks, and graph neural networks. Learning probability distributions of energy configuration or finding ground states of disordered, fully connected Ising models is essential for statistical mechanics and NP-hard problems. Despite tremendous efforts, neural network architectures with abilities to high-accurately solve these intractable problems on larger systems remain a challenge. Here we propose a variational autoregressive architecture with a message passing mechanism, which effectively utilizes the interactions between spin variables. The architecture trained under an annealing framework outperforms existing neural network-based methods in solving several prototypical Ising spin Hamiltonians, especially for larger systems at low temperatures. The advantages also come from the great mitigation of mode collapse during training process. Considering these difficult problems to be solved, our method extends computational limits of unsupervised neural networks to solve combinatorial optimization problems.