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Journal of Experimental and Theoretical Physics Letters

Classification of Phase States of the Ising Model Using a Graph Neural Network

P. A. Ovchinnikov, Z. A. Moskovchenko, M. E. Tsuranova, Konstantin S Soldatov

Published Jul 1, 2026• 0 citations
Publisher

Abstract

An approach based on a graph neural network has been developed to recognize phase states and to estimate the critical temperature of finite-size systems within the Ising model. The advantage of the new approach is the explicit specification of the lattice geometry and bond structure using a graph. This allows a unified transfer of equilibrium spin configurations of lattices of different spatial dimensions and geometries to the input of the neural network. This method has been applied to seven different lattices: two-dimensional square, triangular, hexagonal, and two-dimensional Apamea lattice, as well as three-dimensional simple cubic, bcc, and fcc lattices. It has been shown that the critical temperature can be extracted by crossing the output of the neural network with the 1/2 level. Feeding a small sample of equilibrium configurations to the input of a graph neural network allows achieving the mean absolute error 9.7 × 10–3 relative to the reference value $$T_{c}^{{{\text{ref}}}}$$ . An additional scaling analysis has been performed for the Apamea lattice, which has shown that the neural network estimate of the critical temperature is consistent with the intersection point of the Binder cumulant curves for systems of different sizes.

DOI
10.1134/s0021364026601491

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An approach based on a graph neural network has been developed to recognize phase states and to estimate the critical temperature of finite-size systems within the Ising model. The advantage of the new approach is the explicit specification of the lattice geometry and bond structure using a graph. This allows a unified transfer of equilibrium spin configurations of lattices of different spatial dimensions and geometries to the input of the neural network. This method has been applied to seven different lattices: two-dimensional square, triangular, hexagonal, and two-dimensional Apamea lattice, as well as three-dimensional simple cubic, bcc, and fcc lattices. It has been shown that the critical temperature can be extracted by crossing the output of the neural network with the 1/2 level. Feeding a small sample of equilibrium configurations to the input of a graph neural network allows achieving the mean absolute error 9.7 × 10–3 relative to the reference value TcrefT_{c}^{{{\text{ref}}}}Tcref​ . An additional scaling analysis has been performed for the Apamea lattice, which has shown that the neural network estimate of the critical temperature is consistent with the intersection point of the Binder cumulant curves for systems of different sizes.
Abstract Recent advancements in deep neural networks have improved phase transition analysis. However, their performance depends heavily on the labeling method, and traditional approaches based on critical values for labeling lack physical interpretability and overlook the behaviors of different configurations under fluctuations. To address these issues, we propose a labeling strategy based on the average value of the observable at the critical point. In the percolation model, the threshold is determined by the average relative size of the largest connected cluster, while in the Ising model, it is based on the average absolute magnetization and energy. We validate the strategy on classical models, including site and bond percolation on 2D square and triangular lattices, bond percolation on Erdõs–Rényi (ER) random networks, and the Ising model on 2D square and triangular lattices. Convolutional neural networks are used for lattice data, and graph convolutional networks for ER network data. The results demonstrate that this observable-based labeling strategy can accurately identify critical points and exponents, offering universality and stability across various phase transition types.
The authors describe a method for determining the critical point of a second-order phase transitions using a convolutional neural network based on the Ising model on a square lattice. Data for training were obtained using Metropolis algorithm for different temperatures. The neural network was trained on the data corresponding to the low-temperature phase, that is a ferromagnetic one and high-temperature phase, that is a paramagnetic one, respectively. After training, the neural network analyzed input data from the entire temperature range: from 0.1 to 5.0 (in dimensionless units) and determined (the Curie temperature T_c). The accuracy of the obtained results was estimated relative to the Onsager solution for a flat lattice of Ising spins.
We present a graph-based deep learning framework for predicting the magnetic properties of quasi-one-dimensional Ising spin systems. The lattice geometry is encoded as a graph and processed by a graph neural network (GNN) followed by fully connected layers. The model is trained on Monte Carlo simulation data and accurately reproduces key features of the magnetization curve, including plateaus, critical transition points, and the effects of geometric frustration. It captures both local motifs and global symmetries, demonstrating that GNNs can infer magnetic behavior directly from structural connectivity. The proposed approach enables efficient prediction of magnetization without the need for additional Monte Carlo simulations.
In recent years Machine Learning has proved to be successful in many technological applications and scientific tasks, such as image and speech recognition, natural language understanding and more. In this work we apply both supervised and unsupervised Machine Learning to three key problems in statistical physics: design of Hamiltonian from data, phase recognition and study of critical properties of a system undergoing phase transition. As an example of model design, we use a dataset with spin configurations and corresponding energies randomly sampled from a one dimensional Ising model. We then make two guesses for the correct Hamiltonian: the former involving interactions of spins with an external local field, the latter involving two body interactions among spins. The coupling constants are determined with a linear penalized regression, comparing the effects of L1 and L2 penalization terms of the cost function. We pay specific attention to the problem of overfitting and to the validation process, which is critical for accepting or rejecting the proposed model. The phase recognition problem is faced with the two dimensional Ising and XY models as examples. After showing the limits of a simple softmax regression for this task, we build suitable neural networks to overcome these limits. In particular, a feed forward network is built and the learning process is investigated for the Ising model; while a more sophisticated convolutional network is proposed for the XY model in order to detect local topological structures. The last part of the work is dedicated to the unsupervised study of phase transitions, and the determination of critical properties (order parameter, critical temperature, critical exponents). The discussed techniques are Principal Component Analysis (PCA) for dimensional reduction and K-means clustering for organizing data into subsets with specific properties. PCA is applied to the two dimensional Ising, Potts and XY models, and it is used to find suitable order parameters. The study of the proposed order parameter as a function of temperature provides evidence of phase transition, and a finite size scaling allows to extrapolate both the critical temperature in the thermodynamic limit, and the ν\nuν critical exponent for the correlation length. K-means clustering is applied to equilibrium configurations of the two dimensional Ising model, before and after the dimensional reduction, and the critical temperature is estimated. Moreover, a clustering of relaxation curves of magnetization in a Monte Carlo dynamics is used to build a phase diagram on the parameter space of temperature and magnetic field. A synergy of C++, Python and Wolfram Mathematica 12.0 is used throughout this work in order to sample input datasets and to build and control customized neural networks and learning tools. The most relevant codes are provided in the appendix.
Learning with an artificial neural network encodes the system behavior in a feed-forward function with a number of parameters optimized by data-driven training. An open question is whether one can minimize the network complexity without loss of performance to reveal how and why it works. Here we investigate the learning of the phase transition in the Ising model and find that having two hidden neurons can be enough for an accurate prediction of critical temperature. We show that the networks learn the scaling dimension of the order parameter while being trained as a phase classifier, demonstrating how the machine learning exploits the Ising universality to work for different lattices of the same criticality within a single set of trainings in one lattice geometry.