Deep neural networks for solving forward and inverse problems of (2+1)-dimensional nonlinear wave equations with rational solitons

TL;DR

This paper employs deep neural networks to solve forward and inverse problems related to (2+1)-dimensional nonlinear wave equations, specifically focusing on rational solitons in KP-I and spin-NLS equations, demonstrating data-driven solution approximation.

Contribution

It introduces a neural network-based framework for solving both forward and inverse problems of complex (2+1)D nonlinear wave equations with rational solitons, advancing data-driven PDE solutions.

Findings

Neural networks effectively approximate solutions to (2+1)D nonlinear wave equations.

The approach successfully addresses inverse problems for KP-I and spin-NLS equations.

The method demonstrates potential for solving complex nonlinear PDEs with deep learning.

Abstract

In this paper, we investigate the forward problems on the data-driven rational solitons for the (2+1)-dimensional KP-I equation and spin-nonlinear Schr\"odinger (spin-NLS) equation via the deep neural networks leaning. Moreover, the inverse problems of the (2+1)-dimensional KP-I equation and spin-NLS equation are studied via deep learning. The main idea of the data-driven forward and inverse problems is to use the deep neural networks with the activation function to approximate the solutions of the considered (2+1)-dimensional nonlinear wave equations by optimizing the chosen loss functions related to the considered nonlinear wave equations.