We investigate the connection between the supervised learning of the binary phase classification in the ferromagnetic Ising model and the standard finite-size-scaling theory of the second-order phase transition. Proposing a minimal one-free-parameter neural network model, we analytically formulate the supervised learning problem for the canonical ensemble being used as a training data set. We show that just one free parameter is capable enough to describe the data-driven emergence of the universal finite-size-scaling function in the network output that is observed in a large neural network, theoretically validating its critical point prediction for unseen test data from different underlying lattices yet in the same universality class of the Ising criticality. We also numerically demonstrate the interpretation with the proposed one-parameter model by providing an example of finding a critical point with the learning of the Landau mean-field free energy being applied to the real data set from the uncorrelated random scale-free graph with a large degree exponent.

我们研究了铁磁Ising模型中二元相分类的有监督学习与二阶相变的标准有限尺寸缩放理论之间的联系。提出一个最小的单参数神经网络模型，我们分析地规范了被用作训练数据集的合奏的监督学习问题。我们证明，只有一个自由参数足以描述在大型神经网络中观察到的网络输出中通用有限大小缩放函数的数据驱动出现，从理论上验证了其临界点预测可用于从中看不见的测试数据在Ising临界度的同一通用性类别中，存在不同的底层晶格。