Optimized Machine Learning Methods for Studying the Thermodynamic Behavior of Complex Spin Systems

TL;DR

The paper develops CNN-based approaches to study thermodynamic behavior in complex spin systems, tackling (i) regression of the temperature-dependent energy $\langle E\rangle_T$ from spatial exchange patterns in Edwards–Anderson spin glasses and (ii) a universal phase classifier for Ising models across multiple lattice geometries. It demonstrates that two-channel input encoding (horizontal and vertical couplings) and geometry-aware data layouts yield significant RMSE improvements over fully connected networks, with gradient-scale optimization further stabilizing training. A single CNN3 classifier achieves robust phase discrimination across square, triangular, hexagonal, and kagome lattices, using a fixed input representation and BFS-based embeddings to minimize critical-region blurring. These results provide a scalable, geometry-agnostic ML framework for analyzing thermodynamic properties of frustrated spin systems, with potential extensions to graph-based architectures and 3D systems.

Abstract

This paper presents a systematic study of the application of convolutional neural networks (CNNs) as an efficient and versatile tool for the analysis of critical and low-temperature phase states in spin system models. The problem of calculating the dependence of the average energy on the spatial distribution of exchange integrals for the Edwards-Anderson model on a square lattice with frustrated interactions is considered. We further construct a single convolutional classifier of phase states of the ferromagnetic Ising model on square, triangular, honeycomb, and kagome lattices, trained on configurations generated by the Swendsen-Wang cluster algorithm. Computed temperature profiles of the averaged posterior probability of the high-temperature phase form clear S-shaped curves that intersect in the vicinity of the theoretical critical temperatures and allow one to determine the critical temperature for the kagome lattice without additional retraining. It is shown that convolutional models substantially reduce the root-mean-square error (RMSE) compared with fully connected architectures and efficiently capture complex correlations between thermodynamic characteristics and the structure of magnetic correlated systems.

Figures (8)

• Figure 1: Предложенная архитектура CNN1.
• Figure 2: Предложенная архитектура CNN2.
• Figure 3: Сравнение зависимостей средних значений энергии от температуры для случайных конфигураций, полученных с помощью метода трансфер матриц (True Values) и нейронных сетей (CNN1 и CNN2) для систем разного размера: a) $6\times6$, CNN1; b) $6\times6$, CNN2; c) $10\times10$, CNN1; d) $10\times10$, CNN2.
• Figure 4: Предложенная архитектура CNN3.
• Figure 5: Квадратная и треугольная решётки. Усреднённые по конфигурациям кривые $P_{\mathrm{high}}(T)$ (красные) и $1-P_{\mathrm{high}}(T)$ (синие) для разных размеров. Чёрная вертикальная линия $T_c^{\text{ref}}$.