Machine Learning Domain Adaptation in Spin Models with Continuous Phase Transitions

TL;DR

The paper investigates whether neural networks trained to distinguish FM from PM spin configurations in one model can predict critical properties of another model from a different universality class. It combines supervised learning with finite-size scaling of the NN output, $V(T;L)$, to estimate the critical temperature $T_c$ and correlation length exponent $\nu$ using both spin- and energy-based data representations across Ising, Baxter-Wu, and four-state Potts models. Results show that cross-domain transfer is feasible in select cases (notably Ising–Baxter-Wu) and can be enhanced by energy-based representations, while many cross-pairs remain unreliable, especially with spin data alone. The work highlights the importance of data representation for domain adaptation in statistical physics and provides a path toward estimating critical properties of models with unknown behavior.

Abstract

The main question raised in the article is whether a neural network trained on a spin lattice model in one universality class can be used to test a model in another universality class. The quantities of interest are the critical phase transition temperature and the correlation length exponent. In other words, the question of transfer learning is how ``universal'' the trained network is and under what conditions. For this purpose, we applied a supervised learning procedure to three two-dimensional models for which critical properties are precisely known: the Ising model, the four-state Potts model, and the Baxter-Wu model. We consider two datasets: one with spins configurations and one with binding energy configurations. We find that estimates of the critical temperature agree well with the known results for both datasets, but not with the results of cross-testing using the energy datasets of the two models: the four-state Potts model and the Ising model. Estimates of the critical length exponent are less regular, and appear to be more accurate for energy datasets. A good example is the cross-testing using the energy dataset between Ising model and Baxter-Wu model in both training and testing directions.12

Figures (9)

• Figure 1: An example of the spin configuration of the four-state Potts model. Left: the initial configuration at temperature $T=0.9107$ in the critical region, followed by simple (S), ranked (R), and majority (M) representations in the input matrix. See the text for details.
• Figure 2: Probability $P(T;L)$ of the ferromagnetic phase (left) and its variation $V(T;L)$ (right), estimated for the Potts model using the dataset 4P-S and CNN.
• Figure 3: Probability $P(T)$ and variation $V(T)$ of the ferromagnetic phase estimated for the Baxter-Wu model using NN trained with the Ising model with spin dataset and FCNN model, i.e., the transfer learning BW@IS-FCNN.
• Figure 4: Probability $P(T)$ and variation $V(T)$ of the ferromagnetic phase estimated for the Ising model using NN trained with the Baxter-Wu model with spin dataset and FCNN model, i.e., the transfer learning IS@BW-FCNN. Demonstration of independent fits of the left and right wings of the variation $V(T)$.
• Figure 5: Final scaling of the maximum position $T^*(L)$ of the function $V(T;L)$ relative to $L^{-1/\nu}$, estimated for 4P-R@IS with FCNN. The dotted line represents an approximation of the data with the results given in the first row of Table \ref{['table6']}.