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The European Physical Journal Plus

A comprehensive neural network study of the non-equilibrium phase transition of the two-dimensional Ising model on the square lattice

Yuan-Heng Tseng, Shang-Wei Li, Fu-Jiun Jiang

Published Sep 1, 2024• 3 citations
Publisher
DOI
10.1140/epjp/s13360-024-05563-8

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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.
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 and analysis were obtained using Monte Carlo simulations. 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 J) and determined the Curie point Tc.