Prediction of optical solitons using an improved physics-informed neural network method with the conservation law constraint

In this work, based on the original physics-informed neural networks, we propose an improved physics-informed neural network method by combining the conservation laws. As one of the important integrable properties of nonlinear physical models, the conservation law can bring strong constraining force for the neural network to solve nonlinear physical models. Using this method, we study the standard nonlinear Schrödinger equation and predict various data-driven optical soliton solutions, including one-soliton, soliton molecules, two-soliton interaction, and rogue wave. In addition, from various exact solutions, we use the improved physics-informed neural network method to predict the dispersion and nonlinearity coefficients of the standard nonlinear Schrödinger equation based on the conservation law constraint. It turns out that the proposed method gives rise to the better results compared with the traditional physics-informed neural network method, and thus this method paves a way to simulate other physical models. • Strong restraint. The physical law increases the constrained effect of the neural network. • Wide range of application. It has good training effects for various different optical solitons. • The small training error. The training error of this improved method is smaller.