Solving nonlinear soliton equations using improved physics-informed neural networks with adaptive mechanisms

Abstract

Partial differential equation (PDE) is an important tool for scientific research and are widely used in various fields. However, it is usually very difficult to obtain accurate analytical solutions of partial differential equations, and numerical methods to solve PDEs are often computationally intensive and very time-consuming. In recent years, Physics Informed Neural Networks (PINNs) have been successfully applied to find numerical solutions of partial differential equations (PDEs) and have shown great potential. All the while, solitary waves have been of great interest to researchers in the field of nonlinear science. In this paper, we perform numerical simulations of solitary wave solutions of several partial differential equations using improved PINNs. The improved PINNs not only incorporate constraints on the control equations to ensure the interpretability of the prediction results, which is important for physical field simulations. In addition, an adaptive activation function is introduced. By introducing hyperparameters in the activation function to change the slope of the activation function to avoid the disappearance of the gradient, thereby saving computing time and speeding up training. In this paper, the mKdV equation, the improved Boussinesq equation, the CDGSK equation and the p-gBKP equation are selected for study, and the errors of the simulation results are analyzed to assess the accuracy of the predicted solitary wave solution. The experimental results show that the improved PINNs are significantly better than the traditional PINNs with shorter training time but more accurate prediction results. The improved PINNs improve the training speed by more than 1.5 times compared with the traditional PINNs, while maintaining the prediction error less than $10^{-2}$ in this order of magnitude.

MetaPINNs: Predicting soliton and rogue wave of nonlinear PDEs via the improved physics-informed neural networks based on meta-learned optimization

“…[8,9] Another exciting area of advancement is the utilization of deep neural networks for solving various types of partial differential equations (PDEs) and modeling complex natural physical phenomena. [10][11][12][13] In this context, Raissi et al introduced the concept of physicsinformed neural networks (PINNs), which represents an improvement over traditional neural network-based methods for solving forward and inverse problems of partial differential equations. Numerous studies have shown that PINNs can effectively integrate the intrinsic constraint information in the physical equations.…”