Stochastic dynamics generation of directed percolation using a two-channel convolutional neural network

Abstract

Neural networks provide a promising approach to learning and generating the stochastic dynamics of nonequilibrium systems. In this study, we propose a stochastic dynamics generative method based on a two-channel convolutional neural network, using the (1+1)-dimensional directed bond percolation model to investigate its ability to learn local stochastic transition rules and reproduce stochastic dynamics across different system sizes. The proposed method takes the current particle configuration and the control parameter as inputs, learns the conditional activation probabilities at the next time step through local convolutions, and generates stochastic temporal evolution through iterative Bernoulli sampling. The results show that the network-generated dynamics agree well with Monte Carlo simulations across different control parameters. Furthermore, applying the trained network to a substantially larger system with $L=10000$ yields an estimated critical probability of $p_c=0.643$ and an effective density decay exponent of $δ\approx0.1592$, close to the established theoretical values. These findings demonstrate that the proposed method can reproduce the stochastic dynamics of directed percolation and capture its macroscopic dynamical and statistical properties across different spatial sizes.