Many 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 a probability distribution of energy configuration or finding the ground states of a disordered, fully connected Ising model is essential for statistical mechanics and NP-hard problems. Despite tremendous efforts, a neural network architecture with the ability to high-accurately solve these fully connected and extremely intractable problems on larger systems is still lacking. Here we propose a variational autoregressive architecture with a message passing mechanism, which can effectively utilize the interactions between spin variables. The new network trained under an annealing framework outperforms existing methods in solving several prototypical Ising spin Hamiltonians, especially for larger spin systems at low temperatures. The advantages also come from the great mitigation of mode collapse during the training process of deep neural networks. Considering these extremely difficult problems to be solved, our method extends the current computational limits of unsupervised neural networks to solve combinatorial optimization problems.

翻译：许多深度神经网络已被用于求解伊辛模型，包括自回归神经网络、卷积神经网络、循环神经网络和图神经网络。学习能量构型的概率分布或寻找无序全连接伊辛模型的基态，对于统计力学和NP难题至关重要。尽管已有大量研究，但仍缺乏能够在更大系统中高精度求解这些全连接且极其难解问题的神经网络架构。为此，我们提出了一种融合消息传递机制的变分自回归架构，该架构能有效利用自旋变量间的相互作用。在退火框架下训练的新网络，在求解多个典型伊辛自旋哈密顿量时（尤其在低温大自旋系统中）性能优于现有方法。其优势还体现在深度神经网络训练过程中模式坍缩现象的显著缓解。针对这些极难求解的问题，我们的方法拓展了无监督神经网络求解组合优化问题的当前计算极限。