Investigation of forward and inverse problems in a nonlocal reverse-time nonlinear Schrödinger equation using physics-informed neural networks

Abstract In this paper, we investigate data-driven bright soliton solutions of the nonlocal reverse-time nonlinear Schrödinger (NLS) equation and the parameter identification using the physically informed neural networks (PINNs) algorithm. Accurate simulations and comparative analyses of relative and absolute errors are performed for two-soliton and four-soliton solutions including linear solitary waves and periodic waves. In the training process, the standard PINNs scheme is employed for linear solitary wave solutions, while the prior information is added at local sharp regions for periodic wave solutions due to the complicated collision behaviors. For the parameter identification, we accurately recognize the nonlinear coefficients of the nonlocal NLS equation from known solutions with different noises. These results reinforce the application of deep learning with the PINNs framework to successfully study nonlocal integrable systems.