  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.
