Data-driven soliton solutions and model parameters of nonlinear wave models via the conservation-law constrained neural network method

TL;DR

This paper introduces a conservation-law constrained neural network approach that integrates integrable theory into deep learning to more accurately predict solutions and parameters of nonlinear wave models with less information.

Contribution

It proposes a novel neural network method that incorporates conservation laws, improving prediction accuracy for nonlinear wave models over traditional physics-informed neural networks.

Findings

More accurate solution and parameter predictions for nonlinear wave models.

Effective even with limited information, such as missing boundary conditions.

Applicable to various models like nonlinear Schrödinger and KdV equations.

Abstract

In the process of the deep learning, we integrate more integrable information of nonlinear wave models, such as the conservation law obtained from the integrable theory, into the neural network structure, and propose a conservation-law constrained neural network method with the flexible learning rate to predict solutions and parameters of nonlinear wave models. As some examples, we study real and complex typical nonlinear wave models, including nonlinear Schr\"odinger equation, Korteweg-de Vries and modified Korteweg-de Vries equations. Compared with the traditional physics-informed neural network method, this new method can more accurately predict solutions and parameters of some specific nonlinear wave models even when less information is needed, for example, in the absence of the boundary conditions. This provides a reference to further study solutions of nonlinear wave models by combining the deep learning and the integrable theory.