Long-time Integration of Nonlinear Wave Equations with Neural Operators
Abstract
Neural operators have shown promise in solving many types of Partial Differential Equations (PDEs). They are significantly faster compared to traditional numerical solvers once they have been trained with a certain amount of observed data. However, their numerical performance in solving time-dependent PDEs, particularly in long-time prediction of dynamic systems, still needs improvement. In this paper, we focus on solving the long-time integration of nonlinear wave equations via neural operators by replacing the initial condition with the prediction in a recurrent manner. Given limited observed temporal trajectory data, we utilize some intrinsic features of these nonlinear wave equations, such as conservation laws and well-posedness, to improve the algorithm design and reduce accumulated error. Our numerical experiments examine these improvements in the Korteweg-de Vries (KdV) equation, the sine-Gordon equation, and the Klein-Gordon wave equation on the irregular domain.
Keywords
Cite
@article{arxiv.2410.15617,
title = {Long-time Integration of Nonlinear Wave Equations with Neural Operators},
author = {Guanhang Lei and Zhen Lei and Lei Shi},
journal= {arXiv preprint arXiv:2410.15617},
year = {2025}
}