Non-equispaced Fourier Neural Solvers for PDEs

Haitao Lin, Lirong Wu, Yongjie Xu, Yufei Huang, Siyuan Li, Guojiang Zhao, Stan Z. Li

Keywords: Neural PDE Solver, Neural Operatpor

Abstract:

Recently proposed neural resolution-invariant models,
despite their effectiveness and efficiency, usually require equispaced spatial points of data for solving partial differential equations. However, sampling in spatial domain is sometimes inevitably non-equispaced in real-world systems, limiting their applicability.
In this paper, we propose a Non-equispaced Fourier PDE Solver (\textsc{NFS}) with adaptive interpolation on resampled equispaced points and a variant of Fourier Neural Operators as its components.
Experimental results on complex PDEs demonstrate its advantages in accuracy and efficiency. Compared with the spatially-equispaced benchmark methods, it achieves superior performance with $42.85\%$ improvements on MAE, and is able to handle non-equispaced data with a tiny loss of accuracy.
Besides, \textsc{NFS} as a model with mesh invariant inference ability, can successfully model turbulent flows in non-equispaced scenarios, with a minor deviation of the error on unseen spatial points.