Prediction of phase transition and time-varying dynamics of the (2+1) -dimensional Boussinesq equation by parameter-integrated physics-informed neural networks with phase domain decomposition

A meaningful topic that needs to be explored in the field of nonlinear waves is whether a neural network can reveal the phase transition of different types of waves and novel dynamical properties. In this paper, a physics-informed neural network (PINN) with parameters is used to explore the phase transition and time-varying dynamics of nonlinear waves of the $(2+1)$-dimensional Boussinesq equation describing the propagation of gravity waves on the surface of water. We embed the physical parameters into the neural network for this purpose. Via such algorithm, we find the exact boundary of the phase transition that distinguishes the periodic lump chain and transformed wave, and the inexact boundaries of the phase transition for various transformed waves are detected through PINNs with phase domain decomposition. In particular, based only on the simple soliton solution, we discover types of nonlinear waves as well as their interesting time-varying properties for the $(2+1)$-dimensional Boussinesq equation. We further investigate the stability by adding noise to the initial data. Finally, we perform the parameters discovery of the equation in the case of data with and without noise, respectively. Our paper introduces deep learning into the study of the phase transition of nonlinear waves and paves the way for intelligent explorations of the unknown properties of waves by means of the PINN technique with a simple solution and small data set.