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

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 Tc^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.