U-shaped factorized Fourier neural operator for solving partial differential equations

Abstract

In this study, we proposed a U-shaped Factorized Fourier neural operator (U-FFNO) by introducing the U-shaped architecture idea and skip connection method of U-Net and improving the F-FNO operator layer using a Gaussian low-pass filter. The Factorized Fourier neural operator (F-FNO) introduces a dimensional decomposition method to learn the nonlinear mapping from parameter space to solution space to solve a series of partial differential equations (PDEs). However, due to the existence of truncation coefficients, some high-frequency information will be lost in the process of learning the nonlinear mapping, resulting in an increase in the error when learning the solution space. The proposed U-FFNO can learn this part of the information before the high-frequency information is lost, and enhances the learning ability of low-frequency information. We conduct experiments on several challenging partial differential equations in regular grids and structured grids to demonstrate the excellent accuracy of U-FFNO. U-FFNO is a learning-based method for simulating partial differential equations. As a neural operator, it also has the characteristics of discretization invariance and still performs well in super-resolution prediction tasks.