Physics-informed neural networks method in high-dimensional integrable systems

TL;DR

This paper applies physics-informed neural networks to high-dimensional integrable systems, successfully solving complex PDEs like the KP equation and reproducing various wave phenomena with high accuracy.

Contribution

It introduces a data-driven PINN approach for high-dimensional integrable systems, demonstrating effective solutions for complex PDEs and wave dynamics.

Findings

Error magnitude is significantly smaller than wave height

Classical solutions are accurately obtained

Reproduces solitons, breathers, lumps, and rogue waves

Abstract

In this paper, the physics-informed neural networks (PINN) is applied to high-dimensional system to solve the (N+1)-dimensional initial boundary value problem with 2N+1 hyperplane boundaries. This method is used to solve the most classic (2+1)-dimensional integrable Kadomtsev-Petviashvili (KP) equation and (3+1)-dimensional reduced KP equation. The dynamics of (2+1)-dimensional local waves such as solitons, breathers, lump and resonance rogue are reproduced. Numerical results display that the magnitude of the error is much smaller than the wave height itself, so it is considered that the classical solutions in these integrable systems are well obtained based on the data-driven mechanism.