A-WPINN algorithm for the data-driven vector-soliton solutions and parameter discovery of general coupled nonlinear equations

This work aims to provide an effective deep learning framework to predict the vector-soliton solutions of the coupled nonlinear equations and their collisions. The method we propose here is a weighted physics-informed neural network (WPINN) combining with the adaptive residual points distribution (A-WPINN) algorithm. Different from the traditional PINN algorithm which takes points randomly, the A-WPINN algorithm uses an adaptive point-fetching approach to improve the training efficiency for the solutions with steep gradients. Furthermore, the A-WPINN algorithm weights the training samples to achieve the goal of accelerating the learning progress. We implement series of experimental comparisons between the A-WPINN and traditional PINN algorithms with a generalized coupled nonlinear Schrödinger (GCNLS) equation as an example. The results indicate that the A-WPINN algorithm has faster convergence rate and better approximation ability. Finally, the A-WPINN method is applied to the data-driven parameters discovery of the equation, which shows the dispersion and nonlinear coefficients can be well approximated.