Learning Rogue Waves of Nonlinear Schrödinger Equation with Enhanced Physics-Informed Neural Networks Simulator

Partial differential equations (PDEs) are fundamental tools for studying complex systems. However, the methods for solving PDEs are typically complex. Currently, numerical methods are commonly used to simulate the evolution of complex systems governed by PDEs. Nevertheless, these numerical methods often require significant computational resources and are time-consuming. In recent years, deep learning techniques, specifically Physics Informed Neural Networks (PINNs), have gained attention in the scientific community. PINNs have shown promising results in solving PDEs and simulating physical fields. In this study, the improved PINNs incorporate the constraints of PDEs and gradient information, enhancing the interpretability of the neural network model. Furthermore, an adaptive learning method is employed to update the weight coefficients of the loss function and dynamically adjust the ratio of each constraint term, resulting in accelerated training speed. Rogue waves are of significant research value in the field of nonlinear science and have found applications in various domains. In the experiment, we apply the improved PINNs to numerically simulate rogue wave solutions of PDEs. We evaluate the accuracy of the rogue wave simulation by conducting the error analysis of the results. The experimental results demonstrate that the improved PINNs outperform traditional PINNs in terms of training efficiency and prediction accuracy for rogue wave simulations.