The tricritical point of tricritical directed percolation based on neural network

In recent years, neural networks have been increasingly applied to the identification of critical points in phase transitions. In the tricritical directed percolation model, the dynamics are governed by two probabilities: p, the particle creation probability, and q, the probability associated with higher-order reactions. Steady-state configurations in this model encompass both first-order and second-order phase transitions. The presence of crossover effects makes the precise identification of phase transition boundaries challenging. In this study, Monte Carlo simulations are employed to generate steady-state configurations for various (p,q) values. By analyzing the increments in the average particle density, we identify first-order transitions, second-order transitions, and regions where signatures of both coexist. These Monte Carlo-generated configurations serve as the input for constructing and training a convolutional neural network (CNN), from which the critical points pc are obtained for different q. In addition, by learning the steady-state configurations associated with the superheated point, we determine the tricritical point as (pt,qt)=(0.3512,0.893). A three-output CNN model is further employed to extract the phase transition boundaries and to indicate the extent of the crossover region.