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We investigate the connection between the supervised learning of the binaryphase classification in the ferromagnetic Ising model and the standardfinite-size-scaling theory of the second-order phase transition. Proposing aminimal one-free-parameter neural network model, we analytically formulate thesupervised learning problem for the canonical ensemble being used as a trainingdata set. We show that just one free parameter is capable enough to describethe data-driven emergence of the universal finite-size-scaling function in thenetwork output that is observed in a large neural network, theoreticallyvalidating its critical point prediction for unseen test data from differentunderlying lattices yet in the same universality class of the Isingcriticality. We also numerically demonstrate the interpretation with theproposed one-parameter model by providing an example of finding a criticalpoint with the learning of the Landau mean-field free energy being applied tothe real data set from the uncorrelated random scale-free graph with a largedegree exponent.

Emergence of a finite-size-scaling function in the supervised learning of the Ising phase transition