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

The machine learning (ML) of phase transitions (PTs) has gradually become an effective approach, which enables us to explore the nature of various PTs, more promptly, in both equilibrium and non-equilibrium systems. Unlike equilibrium systems, non-equilibrium systems display more complicated and diverse features, due to the extra dimension of time, which are not readily tractable, both theoretically and numerically. The combination of ML and the most renowned non-equilibrium model, the directed percolation (DP), has led to some significant findings. Here in this work, the ML technique will be applied to the (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 at the same system sizes. The dynamic exponent \;$z$\; and the spatial correlation length correlation exponent \;$\nu_{\perp}$\ are also measured and found to be consistent with the respective theoretical values. The unsupervised learning of (1+1)-d BAW, via autoencoder (AE), also gives rise to a transition point which is the same as the critical point. The output of AE, through a single neuron, can be regarded as the order parameter of the system, being re-scaled properly. We therefore have the reason to believe that ML has an exciting application prospect in such reaction-diffusion systems as the BAW and DP.