Analysis of the toy model for the neural network applied to the classical Ising model

Abstract

Recently, neural network(NN) has been applied to a phase classification of many body systems. In the Ising model[Nature Physics 13, 431-434(2017)], outputs of the NN follows a size scaling behavior with critical exponent. We introduce a toy model to investigate an analytical relation between a system size and an outputs of the NN. At optimal points, we derive that parameters in the toy model follow the power law with exponent Ising critical exponent for the system size. To support numerical evidences, we implement the stochastic gradient descent method with the square and the triangular lattice to obtain optimal parameters in the toy model. We find that parameters show the power law and size scaling behavior with satisfying the Ising universality class