Deep Learning of the Biswas-Chatterjee-Sen Model

TL;DR

This work tackles disorder-induced continuous phase transitions in the Biswas-Chatterjee-Sen model with continuous spins by generating kinetic Monte Carlo data on square and triangular lattices and applying both supervised and unsupervised deep learning. A dense neural network classifies ferromagnetic versus paramagnetic configurations, achieving accurate identification of the critical point and showing finite-size scaling with Ising-like exponent $\nu=1$. PCA uncovers phase-transition signatures via cluster evolution and universal eigenvalue ratios, while a variational autoencoder reveals a latent representation and a universal correlation function with losses that scale according to Ising critical exponents, all consistent across lattice geometries. The results demonstrate the effectiveness of DL methods in identifying and characterizing nonequilibrium phase transitions with continuous degrees of freedom and varying topology, providing a data-driven route to extract universal critical behavior. The study thereby strengthens the connection between machine learning techniques and critical phenomena in statistical physics.

Abstract

We investigate the critical properties of kinetic continuous opinion dynamics using deep learning techniques. The system consists of $N$ continuous spin variables in the interval $[-1,1]$. Dense neural networks are trained on spin configuration data generated via kinetic Monte Carlo simulations, accurately identifying the critical point on both square and triangular lattices. Classical unsupervised learning with principal component analysis reproduces the magnetization and allows estimation of critical exponents. Additionally, variational autoencoders are implemented to study the phase transition through the loss function, which behaves as an order parameter. A correlation function between real and reconstructed data is defined and found to be universal at the critical point.

Figures (8)

• Figure S1: Neural network outputs for the BChS model on the square lattice. For each $L$, two curves are shown: $\rho_1$ and $\rho_2$. The score of ferromagnetic phase $\rho_1$ is close to $1$ at low noise values and decreases at high noise values, while the score of paramagnetic phase $\rho_2$ behaves oppositely. The crossing of $\rho_1$ and $\rho_2$ marks the point of maximum confusion, corresponding to the transition threshold. In panel (a), the crossing point $\rho_1=\rho_2=0.5$ closely matches the critical noise $q^s_c$, indicated by the dashed vertical line. In panel (b), the outputs collapse according to Equation (\ref{['classification-fss']}) with the critical exponent $\nu=1$ for the square lattice; $q^\prime_c$ denotes the crossing abscissas.
• Figure S2: Neural network outputs $\rho_1$ and $\rho_2$ for the BChS model on the triangular lattice, trained with square lattice data. The curves have the same interpretation as in Figure \ref{['confidence-mv-square']}. In panel (a), the crossing points $q^\prime_c$ ($\rho_1=\rho_2=0.5$) are used to estimate the critical noise via the process in Figure \ref{['regression-mv-triangular']}. The critical noise $q^t_c$ is indicated by the dashed vertical line. In panel (b), the outputs scale according to Equation \ref{['classification-fss']} with critical exponent $\nu=1$.
• Figure S3: Linear regression of the crossing points $q^{\prime}_c$ of the neural network outputs $\rho_1$ and $\rho_2$ for BChS model configurations on the triangular lattice. Extrapolation according to Equation \ref{['regression-threshold']} yields an estimate for the critical noise, $q_c \approx 0.2397 \pm 0.0002$.
• Figure S4: Observables for the BChS model on the triangular lattice obtained from standard Monte Carlo simulations. Panel (a): Binder cumulant $U$ as a function of noise for different lattice sizes $L$. The curves intersect at the critical noise $q^t_c \approx 0.240$, indicated by the dashed vertical line. Panel (b): scaling transformation allows estimation of the critical exponent $\nu=1$. Panel (c): order parameter $M$ as a function of noise, which scales with $L^{\beta/\nu}$ where $\beta/\nu=1/8$, as shown in panel (d). Panel (e): susceptibility $\chi$, whose maximum increases with $L^{\gamma/\nu}$ at the critical point where $\gamma/\nu=7/4$, as shown in panel (f).
• Figure S5: Projection of BChS model training data with $L=40$ onto the first two principal components as a function of the noise. PCA was performed separately for each noise value in the training set; $\hat{q}$ denotes normalized noise values from $0.5q^s_c$ to $1.5q^s_c$. For low noise, two clusters appear at $(0,-L)$ and $(0,L)$; for high noise, a single cluster emerges at $(0,0)$. The clustering illustrates the phase transition.