Finite-size analysis in neural network classification of critical phenomena

TL;DR

This paper investigates how neural networks can identify phase transitions in statistical physics models, revealing finite-size effects and extracting critical exponents through analysis of the neural network output variance.

Contribution

It introduces a finite-size scaling analysis of neural network outputs to determine critical phenomena and exponents in physical models, validated across multiple neural network architectures.

Findings

Neural network output variance peaks near critical points.

Peak width scales with system size according to the correlation length exponent.

Multiple neural network architectures yield consistent critical exponent estimates.

Abstract

We analyze the problem of supervised learning of ferromagnetic phase transitions from the statistical physics perspective. We consider two systems in two universality classes, the two-dimensional Ising model and two-dimensional Baxter-Wu model, and perform careful finite-size analysis of the results of the supervised learning of the phases of each model. We find that the variance of the neural network (NN) output function (VOF) as a function of temperature has a peak in the critical region. Qualitatively, the VOF is related to the classification rate of the NN. We find that the width of the VOF peak displays the finite-size scaling governed by the correlation length exponent, $\nu$, of the universality class of the model. We check this conclusion using several NN architectures -- a fully connected NN, a convolutional NN and several members of the ResNet family -- and discuss the accuracy of the extracted critical exponents $\nu$.