RAR-PINN algorithm for the data-driven vector-soliton solutions and parameter discovery of coupled nonlinear equations

Abstract

This work aims to provide an effective deep learning framework to predict the vectorsoliton solutions of the coupled nonlinear equations and their interactions. The method we propose here is a physics-informed neural network (PINN) combining with the residual-based adaptive refinement (RAR-PINN) algorithm. Different from the traditional PINN algorithm which takes points randomly, the RAR-PINN algorithm uses an adaptive point-fetching approach to improve the training efficiency for the solutions with steep gradients. A series of experiment comparisons between the RAR-PINN and traditional PINN algorithms are implemented to a coupled generalized nonlinear Schrödinger (CGNLS) equation as an example. The results indicate that the RAR-PINN algorithm has faster convergence rate and better approximation ability, especially in modeling the shape-changing vector-soliton interactions in the coupled systems. Finally, the RAR-PINN method is applied to perform the data-driven discovery of the CGNLS equation, which shows the dispersion and nonlinear coefficients can be well approximated.

A Deep Double Ritz Method (D2rm) for Solving Partial Differential Equations Using Neural Networks

“…Alternatively, there exist approaches to directly represent the PDE solutions via NNs. To mention a few: [9,10] minimize the strong form of the residual via collocation methods; [11] proposes a Deep Ritz Method (DRM) for symmetric and positive definite problems; [12] introduces a Deep Fourier Method; and [13][14][15][16][17] propose (Petrov-)Galerkin frameworks in the context of trial NNs with test functions belonging to linear spaces. All these NN-based methods exhibit multiple features but also present several limitations (see, e.g., [18][19][20][21]).…”