Data driven solutions and parameter discovery of the nonlocal mKdV equation via deep learning method

TL;DR

This paper explores data-driven solutions and parameter discovery for the nonlocal mKdV equation using deep learning, including neural network methods for various soliton solutions and inverse problem solving.

Contribution

It introduces a systematic approach combining neural networks and integrability analysis to find solutions and parameters of the nonlocal mKdV equation.

Findings

Neural network methods successfully generate various soliton solutions.

Dynamic behaviors of solutions are validated through image simulations.

Parameters of the nonlinear terms are accurately identified via physics-informed neural networks.

Abstract

In this paper, we systematically study the integrability and data-driven solutions of the nonlocal mKdV equation. The infinite conservation laws of the nonlocal mKdV equation and the corresponding infinite conservation quantities are given through Riccti equation. The data driven solutions of the zero boundary for the nonlocal mKdV equation are studied by using the multi-layer physical information neural network algorithm, which including kink soliton, complex soliton, bright-bright soliton and the interaction between soliton and kink-type. For the data-driven solutions with non-zero boundary, we study kink, dark, anti-dark and rational solution. By means of image simulation, the relevant dynamic behavior and error analysis of these solutions are given. In addition, we discuss the inverse problem of the integrable nonlocal mKdV equation by applying the physics-informed neural network algorithm to discover the parameters of the nonlinear terms of the equation.