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Springer proceedings in physics

On the Validity of Transfer Learning for Temperature and Critical Exponent Extraction

Diana Sukhoverkhova, Щур Лев Николаевич

Published Jan 1, 2026• 0 citations
Publisher

Abstract

Abstract We analyze the Ising model with anisotropic interactions in two dimensions using deep learning neural networks (NN) trained to classify paramagnetic (PM) and ferromagnetic (FM) phases. The spin configurations used as training data are labeled as belonging to either PM state or the FM state using analytically known phase transition temperatures. We use the variation of the output function to determine the critical temperature and to estimate the correlation length critical exponent, $$\nu $$ ν , in a wide range of the anisotropy parameter. We find that on increasing the anisotropy, the critical exponent remains consistent with the expected value $$\nu {=}1$$ ν = 1 for the Ising universality class. At the same time, the NN prediction for the critical temperature, $$T_c$$ T c , is shifted from the analytically known value, with the magnitude of the shift increasing with anisotropy.

DOI
10.1007/978-981-92-0844-9_17

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Abstract We analyze the Ising model with anisotropic interactions in two dimensions using deep learning neural networks (NN) trained to classify paramagnetic (PM) and ferromagnetic (FM) phases. The spin configurations used as training data are labeled as belonging to either PM state or the FM state using analytically known phase transition temperatures. We use the variation of the output function to determine the critical temperature and to estimate the correlation length critical exponent, ν\nu ν ν , in a wide range of the anisotropy parameter. We find that on increasing the anisotropy, the critical exponent remains consistent with the expected value ν=1\nu {=}1ν=1 ν = 1 for the Ising universality class. At the same time, the NN prediction for the critical temperature, TcT_cTc​ T c , is shifted from the analytically known value, with the magnitude of the shift increasing with anisotropy.
Abstract We investigate the possibility of extracting features of second-order phase transitions using transfer machine learning. We have performed supervised machine learning for binary classification of snapshots of the spin distribution of the isotropic Ising model. The binary classification is performed in ferromagnetic and paramagnetic phases using a known critical temperature. The trained network is used to predict whether a snapshot obtained from model simulations with orthogonal anisotropy belongs to the paramagnetic phase. Using finite-dimensional prediction analysis, we estimate the critical temperature and the exponent of the correlation length. This gives us an estimate of the interval of the anisotropy parameter in which the neural network can make correct estimates.
In this paper, we applied a deep neural network to study the issue of knowledge transferability between statistical mechanics models. The following computer experiment was conducted. A convolutional neural network was trained to solve the problem of binary classification of snapshots of the Ising model’s spin configuration on a two-dimensional lattice. During testing, snapshots of the Ising model spins on a lattice with diagonal ferromagnetic and antiferromagnetic connections were fed to the input of the neural network. Estimates of the probability of samples belonging to the paramagnetic phase were obtained from the outputs of the tested network. The analysis of these probabilities allowed us to estimate the critical temperature and the critical correlation length exponent. It turned out that at weak anisotropy the neural network satisfactorily predicts the transition point and the value of the correlation length exponent. Strong anisotropy leads to a noticeable deviation of the predicted values from the precisely known ones. Qualitatively, strong anisotropy is associated with the presence of oscillations of the correlation function above the Stefenson disorder temperature and further approach to the point of the fully frustrated case.
Over the past several years, there have been many studies demonstrating the ability of deep neural networks to identify phase transitions in many physical systems, notably in classical statistical physics systems. One often finds that the prediction of deep learning methods trained on many ensembles below and above the critical temperature T_{c} behaves similarly to an order parameter, and this analogy has been successfully used to locate T_{c} and estimate universal critical exponents. In this work, we pay particular attention to the ability of a convolutional neural network to capture these critical parameters for the two-dimensional Ising model when the network is trained on configurations at T=0 and T=∞ only. We directly compare its output to the same network trained at multiple temperatures below and above T_{c} to gain understanding of how this extreme restriction of training data can impact a neural network's ability to classify phases. We find that the network trained on two temperatures is still able to identify T_{c} and ν, while the extraction of γ becomes more challenging.
The main question raised in the article is whether a neural network trained on a spin lattice model in one universality class can be used to test a model in another universality class. The quantities of interest are the critical phase transition temperature and the correlation length exponent. In other words, the question of transfer learning is how "universal" the trained network is and under what conditions. For this purpose, we applied a supervised learning procedure to three two-dimensional models for which critical properties are precisely known: The Ising model, the four-state Potts model, and the Baxter-Wu model. We consider two datasets: One with spin configurations and one with binding energy configurations. We find that estimates of the critical temperature agree well with the known results for both datasets, but not with the results of cross-testing using the energy datasets of the two models: The four-state Potts model and the Ising model. Estimates of the critical length exponent are less regular, and appear to be more accurate for energy datasets. A good example is the cross-testing using the energy dataset between the Ising model and the Baxter-Wu model in both training and testing directions.
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 ν 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 ν.
We combine machine-learning techniques with Monte Carlo (MC) simulations and finite-size scaling (FSS) to study continuous and first-order phase transitions in Ising, Blume-Capel, and Ising-metamagnet spin models. We go beyond earlier studies that had concentrated on obtaining the correlation-length exponent ν. In particular, we show (a) how to combine neural networks (NNs), trained with data from MC simulations of Ising-type spin models on finite lattices, with FSS to obtain both thermal magnetic exponents y_{t}=1/ν and y_{h}, respectively, at both critical and tricritical points, (b) how to obtain the NN counterpart of two-scale-factor universality at an Ising-type critical point, and (c) how to get FSS at a first-order transition. We also obtain the FSS forms for the output of our trained NNs as functions of both the temperature and the magnetic field.