For Ising models with complex energy landscapes, whether the ground state can
be found by neural networks depends heavily on the Hamming distance between the
training datasets and the ground state. Despite the fact that various recently
proposed generative models have shown good performance in solving Ising models,
there is no adequate discussion on how to quantify their generalization
capabilities. Here we design a Hamming distance regularizer in the framework of
a class of generative models, variational autoregressive networks (VAN), to
quantify the generalization capabilities of various network architectures
combined with VAN. The regularizer can control the size of the overlaps between
the ground state and the training datasets generated by networks, which,
together with the success rates of finding the ground state, form a
quantitative metric to quantify their generalization capabilities. We conduct
numerical experiments on several prototypical network architectures combined
with VAN, including feed-forward neural networks, recurrent neural networks,
and graph neural networks, to quantify their generalization capabilities when
solving Ising models. Moreover, considering the fact that the quantification of
the generalization capabilities of networks on small-scale problems can be used
to predict their relative performance on large-scale problems, our method is of
great significance for assisting in the Neural Architecture Search field of
searching for the optimal network architectures when solving large-scale Ising
models.

通过设计一个 Hamming 距离约束器来量化生成模型的泛化能力，本研究对几种常见的神经网络架构（前馈神经网络、递归神经网络和图神经网络）结合变分自回归网络（VAN）的泛化能力进行了数值实验，并发现网络在小规模问题上的泛化能力可以预测其在大规模问题上的性能，这对于神经架构搜索中搜索大规模 Ising 模型的最佳网络架构具有重要意义。