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Journal of the Physical Society of Japan

Image Preparation for Machine-Learning Analysis of Multiple-Size Spin Systems

Akimichi Sakamoto, Kenta Shiina, 弘之 森

Published May 28, 2021• 0 citations
Publisher

Abstract

A new idea of preparing data is proposed for application of machine-learning methods to analysis of phase transitions of Ising model. Presenting spin configurations with different sizes in single-size images, we could test the different-size spin configurations by using a neural network trained with spin configurations with a fixed size. We also analyzed the Ising model on triangular and hexagonal lattices using the neural network trained for square lattice.

DOI
10.7566/jpsj.90.065001

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A new idea of preparing data is proposed for application of machine-learning methods to analysis of phase transitions of Ising model. Presenting spin configurations with different sizes in single-size images, we could test the different-size spin configurations by using a neural network trained with spin configurations with a fixed size. We also analyzed the Ising model on triangular and hexagonal lattices using the neural network trained for square lattice.
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.
We combine machine-learning (ML) 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 ννν. 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 yt=1/νy_t = 1/νyt​=1/ν and yhy_hyh​, 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) 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.
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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.
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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.