Uncovering Magnetic Phases with Synthetic Data and Physics-Informed Training

TL;DR

The paper tackles learning magnetic phase diagrams with limited real data by combining synthetic data and physics-informed training. It introduces a supervised Dense Neural Network trained on ideal spin configurations to classify phases and estimate transition temperatures in the diluted Ising model, and an unsupervised convolutional autoencoder trained only on ordered configurations to detect phase transitions via reconstruction error (anomaly detection). Physics priors are embedded through architectural biases that amplify symmetry-breaking features and by including symmetry-breaking training samples, enabling robust phase discrimination even without Monte Carlo data. The approach yields Tc estimates and a percolation threshold near ρ_c ≈ 0.60, with a Tc scaling form Tc(ρ)/Tc(1) = -K/ln(ρ - ρ_c) + A where K ≈ 0.77 and A ≈ 0.18, aligning with known results and demonstrating low-cost, scalable applicability to broader condensed-matter contexts.

Abstract

We investigate the efficient learning of magnetic phases using artificial neural networks trained on synthetic data, combining computational simplicity with physics-informed strategies. Focusing on the diluted Ising model, which lacks an exact analytical solution, we explore two complementary approaches: a supervised classification using simple dense neural networks, and an unsupervised detection of phase transitions using convolutional autoencoders trained solely on idealized spin configurations. To enhance model performance, we incorporate two key forms of physics-informed guidance. First, we exploit architectural biases which preferentially amplify features related to symmetry breaking. Second, we include training configurations that explicitly break $\mathbb{Z}_2$ symmetry, reinforcing the network's ability to detect ordered phases. These mechanisms, acting in tandem, increase the network's sensitivity to phase structure even in the absence of explicit labels. We validate the machine learning predictions through comparison with direct numerical estimates of critical temperatures and percolation thresholds. Our results show that synthetic, structured, and computationally efficient training schemes can reveal physically meaningful phase boundaries, even in complex systems. This framework offers a low-cost and robust alternative to conventional methods, with potential applications in broader condensed matter and statistical physics contexts.

Figures (14)

• Figure 1: Schematic representation of the DNN used in this study. The input layer has a fixed size of 1600 neurons ($40\times 40$ square spin lattice). The hidden layer size was varied to analyze performance with a reduced number of neurons. The output layer size corresponds to the number of magnetic phases used during training.
• Figure 2: Probability distribution of each configuration as a function of temperature for the pure Ising model on a $40 \times 40$ square spin lattice, computed using a DNN with a hidden layer of 1000 neurons (main panel) and 8 neurons (inset).
• Figure 3: Probability distribution of each configuration as a function of temperature for the pure Ising model on a $40 \times 40$ square spin lattice, computed using a DNN with a hidden layer containing up to 400 neurons (main panel) and between 400 and 1000 neurons (inset). Note the marked improvement in performance when using 5 or more hidden neurons (main panel).
• Figure 4: Critical temperature predicted by the DNN as a function of the number of neurons in the hidden layer, for the pure Ising model on a $40 \times 40$ square spin lattice. Although a marked improvement in performance is observed for $n \geq 5$ (see Fig. \ref{['fig:ferro']}), a stable prediction of the critical temperature is not achieved until the number of hidden neurons exceeds 400. The red dashed line indicates the exact critical temperature in the thermodynamic limit.
• Figure 5: Probability of ferromagnetic configurations as a function of temperature for various spin densities in the diluted Ising model. The DNN is trained with ferromagnetic and paramagnetic configurations. For $\rho = 0.55$, the DNN does not find an ordered state for any temperature and predicts $0.55<\rho_c<0.6$.