Supervised and Unsupervised Deep Learning Applied to the Majority Vote Model

TL;DR

The paper addresses the challenge of characterizing a continuous phase transition in the nonequilibrium majority vote model by applying supervised learning, PCA, and variational autoencoders to spin configurations from kinetic Monte Carlo simulations on square and triangular lattices. It demonstrates that neural classifiers can accurately locate the critical point and exhibit finite-size scaling with $ u=1$, while PCA and VAEs provide unsupervised pathways to extract critical exponents and universal correlations, including cross-lattice transferability. The results establish that machine learning approaches can recover Ising-universal behavior in a nonequilibrium setting and offer general, data-driven tools for detecting phase transitions in complex systems. This work highlights the practical impact of ML methods for analyzing critical phenomena and suggests broad applicability to other models and domains where analytical solutions are challenging.

Abstract

We employ deep learning techniques to investigate the critical properties of the continuous phase transition in the majority vote model. In addition to deep learning, principal component analysis is utilized to analyze the transition. For supervised learning, dense neural networks are trained on spin configuration data generated via the kinetic Monte Carlo method. Using independently simulated configuration data, the neural network accurately identifies the critical point on both square and triangular lattices. Classical unsupervised learning with principal component analysis reproduces the magnetization and enables estimation of critical exponents, typically obtained via Monte Carlo importance sampling. Furthermore, deep unsupervised learning is performed using variational autoencoders, which reconstruct input spin configurations and generate artificial outputs. The autoencoders detect the phase transition through the loss function, quantifying the preservation of essential data features. We define a correlation function between the real and reconstructed data, and find that this correlation function is universal at the critical point. Variational autoencoders also serve as generative models, producing artificial spin configurations.

Figures (10)

• Figure 1: Panel (a) illustrates a square lattice of size $L=10$ with periodic boundary conditions. Panel (b) displays a triangular lattice with identical parameters. Lattice nodes are depicted as dots, while edges represent nearest-neighbor interactions. Dashed edges indicate periodic boundary conditions, connecting nodes on opposite sides of the lattice.
• Figure 2: Basic architecture of a neural network. Neurons, depicted as circles, are arranged in ordered layers from left to right. The input layer is on the left, the output layer on the right, and the intermediate layers are the hidden layers. Each neuron receives inputs from all neurons in the previous layer (dense connectivity), applies a linear transformation followed by a nonlinear activation function, and passes its output to the next layer. For spin configuration classification, the input layer consists of lattice spins $s_i = \pm 1$, and the output layer uses softmax activation to produce normalized scores.
• Figure 3: Neural network output for the MV model on the square lattice, using inference data distinct from the training set. For each system size, two curves are shown: $\rho_1$ and $\rho_2$. The output $\rho_1$ (ferromagnetic phase) is close to $1$ at low noise values and decreases at high noise values, while $\rho_2$ (paramagnetic phase) behaves oppositely. The crossing of $\rho_1$ and $\rho_2$ marks the point of maximum confusion, where the network cannot distinguish between ferromagnetic and paramagnetic configurations. In panel (a), the crossing point $\rho_1=\rho_2=0.5$ closely matches the critical noise $q^\square_c$, indicated by the dashed vertical line. In panel (b), the outputs collapse according to Eq. (\ref{['classification-fss']}) with the critical exponent $\nu=1$ for the square lattice; $q^\prime_c$ denotes the crossing abscissas.
• Figure 4: Neural network outputs $\rho_1$ and $\rho_2$ for the MV model on the triangular lattice, trained with square lattice data. The curves have the same interpretation as in Fig. (\ref{['confidence-mv-square']}). In panel (a), the crossing points $q^\prime_c$ ($\rho_1=\rho_2=0.5$) were used to estimate the critical noise via the process in Fig. (\ref{['regression-mv-triangular']}). The critical noise $q^\triangle_c$ is indicated by the dashed vertical line. In panel (b), the outputs scale according to Eq. (\ref{['classification-fss']}) with critical exponent $\nu=1$.
• Figure 5: Linear regression of the crossing points $q^{\prime}_c$ of the neural network outputs $\rho_1$ and $\rho_2$ for MV model configurations on the triangular lattice. Extrapolation according to Eq. (\ref{['regression-threshold']}) yields an estimate for the critical noise, $q_c \sim 0.1115 \pm 0.0007$, which is close to the critical noise $q^\triangle_c \sim 0.10910$ for the MV model on the triangular lattice Yu-2017, obtained via Monte Carlo simulations.