Supervised and unsupervised learning of (1+1)-dimensional even-offspring branching annihilating random walks

TL;DR

This paper demonstrates how machine learning techniques, both supervised and unsupervised, can accurately identify phase transition points and critical exponents in (1+1)-dimensional even-offspring branching annihilating random walks, a non-DP universality class.

Contribution

It introduces the application of CNN and autoencoder ML models to analyze non-DP-like systems, achieving precise critical point prediction and extracting order parameters.

Findings

CNN predicts critical points more accurately than Monte Carlo methods.

Measured critical exponents match theoretical values.

Autoencoder's latent layer acts as a scaled order parameter.

Abstract

Machine learning (ML) of phase transitions (PTs) has gradually become an effective approach that enables us to explore the nature of various PTs more promptly in equilibrium and nonequilibrium systems. Unlike equilibrium systems, non-equilibrium systems display more complicated and diverse features because of the extra dimension of time, which is not readily tractable, both theoretically and numerically. The combination of ML and most renowned nonequilibrium model, directed percolation (DP), led to some significant findings. In this study, ML is applied to (1+1)-d, even offspring branching annihilating random walks (BAW), whose universality class is not DP-like. The supervised learning of (1+1)-d BAW via convolutional neural networks (CNN) results in a more accurate prediction of the critical point than the Monte Carlo (MC) simulation for the same system sizes. The dynamic exponent \;$z$\; and spatial correlation length correlation exponent \;$\nu_{\perp}$\ were also measured and found to be consistent with their respective theoretical values. Furthermore, the unsupervised learning of (1+1)-d BAW via an autoencoder (AE) gives rise to a transition point, which is the same as the critical point. The latent layer of AE, through a single neuron, can be regarded as the order parameter of the system being properly re-scaled. Therefore, we believe that ML has exciting application prospects in reaction-diffusion systems such as BAW and DP.