Solving Localized Wave Solutions of the Nonlinear PDEs Using Physics-Constraint Deep Learning Method

In the field of nonlinear science, localized waves hold significant research value, and their theories have found applications across various domains. Partial Differential Equations (PDEs) serve as crucial tools for studying localized waves in nonlinear systems, and numerical methods for PDEs have been widely employed in the numerical simulation of localized waves. However, the complexity of solving partial differential equations has impeded progress in the study of nonlinear localized waves. In recent years, with the rapid advancement of deep learning, Physics-Informed Neural Networks (PINNs) have emerged as powerful tools for solving PDEs and simulating multiphysical phenomena, attracting significant attention from researchers. In this study, we apply an improved PINNs approach to solve PDEs governing localized waves. The enhanced PINNs not only incorporates the constraints of the PDEs but also introduces gradient information constraints, further enriching the physical constraints within the neural network model. Additionally, we employ an adaptive learning method to update the weight coefficients of the loss function and dynamically adjust the relative importance of each constraint term in the entire loss function to expedite the training process. In the experimental section, we selected Boussinesq equation and nonlinear Schrödinger equation (NLSE) for the study and evaluated the accuracy of localized wave simulation results through error analysis. The experimental results indicate that the improved PINNs are significantly better than traditional PINNs, with shorter training time and more accurate prediction results.