Phase classification and finite-size analysis with supervised machine learning

We analyze the problem of supervised learning of ferromagnetic phase transitions from a statistical physics perspective [1]. We consider two systems in two universality classes, a two-dimensional Ising model and a two-dimensional Baxter-Wu model, and perform a thorough finite-dimensional analysis of the supervised learning phase results of each model. We found that the variance of the neural network (NN) output function (VOF) as a function of temperature peaks in the critical region. Qualitatively, VOF is related to the degree of NN classification. We find that the VOF peak width displays a finite size scaling determined by the correlation length metric of the model's universality class. We test this conclusion using several neural network architectures—a fully connected neural network, a convolutional neural network, and several members of the ResNet family—and discuss the accuracy of the extracted critical metrics.

We used this approach to investigate how the anisotropy of the model being tested affects the accuracy of the critical temperature and critical index estimates by conducting a training and validation test run on an isotropic model. In other words, we would like to understand how stable the predictions are on a test set with unknown isotropic properties. We simulate a two-dimensional Ising model with two types of anisotropy using precise knowledge of the orthogonal anisotropy of the Onsager solution and the Huthappel solution of the diagonal anisotropy. The result is interesting in that the universal property of the Ising model, expressed in the correlation length exponent, is well restored as a result of supervised machine learning, while the non-universal property - the critical temperature value - is shifted in some regular way. This is true even for the antiferromagnetic sector of the Hautapell model.

[1] V. Chertenkov, E. Burovski, and L. Shchur, PRE 108, L032102 (2023)