Analysis of neural network methods for obtaining soliton solutions of the nonlinear Schrödinger equation

Abstract

The paper addresses the practically significant problem of transmitting signals through nonlinear optical media by solving generalized nonlinear Schrödinger equations using various modifications of Physics-Informed Neural Networks (PINNs). The study provides numerical soliton solutions for Schrödinger equations of the order as high as four. To tackle this problem, the paper compares segmental modifications of PINNs, including BC-PINNs, FB-PINNs, and MoE-PINNs. Additionally, an adaptive option for selecting collocation points is proposed and explored. The efficiency of the numerical solutions is evaluated using three approaches: comparison with the precise analytical solutions, and two metrics based on conservation laws. The results show that the modified segmentation approach, combined with the developed adaptive selection of collocation points, greatly improves the accuracy and the convergence of PINNs compared to the initial version of the method. On such example problems as the interaction of a soliton with a Gaussian function, two solitons interaction, and the solution of a 4th-order equation, the proposed method demonstrates improved convergence of the numerical solution.