Data-driven vector degenerate and nondegenerate solitons of coupled nonlocal nonlinear Schrödinger equation via improved PINN algorithm

In recent years, the Physics-Informed Neural Networks have demonstrated significant potential in solving nonlinear evolution equations, and exhibited high stability and applicability. However, it does not fully adapt to nonlocal nonlinear evolution equations. In this paper, we improve the traditional Physics-Informed Neural Network by incorporating prior information as a supplementary term in the loss function to effectively capture the amplitude distribution at the target location, thereby enhancing the predictive accuracy of the neural network. Additionally, we address the problem of multiple competing objectives in the loss function through stepwise training, leveraging adaptive weights and adaptive activation functions to optimize predictions. We apply these improved strategies of physical information neural networks to predict soliton solution of the coupled nonlocal nonlinear Schrödinger equation, including two kinds of nondegenerate one-soliton, and two kinds of degenerate double-soliton. Moreover, we also discuss the impact of Gaussian noise on data-driven parameter discovery of the coupled nonlocal nonlinear Schrödinger equation.