NS-FUO: Fourier U-type operator based on nested structure

Abstract

For partial differential equations (PDE), neural operators can learn the mapping of input and output functions in infinite dimensional spaces by introducing kernel functions into linear transformations. Fourier neural operator (FNO) is a very representative neural operator, which filters out the high-frequency noise in PDE mainly through low frequency dominated Fourier space truncation, and can solve PDE with high precision and high efficiency. However, for some complex high-dimensional PDE, FNO and other algorithms usually have the problem of incomplete filtering out high-frequency noise, which will affect the solution accuracy. To filter out high-frequency noise more thoroughly and further improve the precision, we propose NS-FUO: Fourier U-type Operator Based on Nested Structure. Firstly, NS-FUO adds MLP to each Fourier layer to extract the nonlinear features of PDE in depth. Then, NS-FUO adds UNet to each Fourier layer to extract the multi-layer condition features of PDE in depth. Finally, NS-FUO adds nested UNet after the last Fourier layer to fuses the original input features of PDE with the filtered output features. The experimental results show that compared with 15 PDE intelligent methods such as FNO, U-FNO, LSM, etc, NS-FUO has the highest accuracy for solving three solid PDEs and four fluid PDEs, and achieves an average accuracy improvement of 11.9% compared with the previous best method LSM.