XFNO: Extended Fourier neural operator for solving parametric partial differential equations on general geometries

Hailong Sheng, Yulin Xie, Chao Yang

Abstract: In this paper, we propose an extended Fourier neural operator (XFNO) method which can efficiently solve parametric partial differential equations by extending the application range of the existing Fourier neural operator (FNO) method to irregular domains. The XFNO model is established according to the knowledge of physics equations rather than observational data and therefore is free of the labeled data. Inspired from the idea of the classical cut cell method, the boundary conditions are absorbed into the control equation within the loss function of XFNO, thus avoiding the computation on the irregular boundaries and enabling the application of Fast Fourier Transformation for accelerating the convolution calculation. Furthermore, the shape of computational domain is parameterized and handled along with other problem parameters, expanding the application of XFNO to a wide range of problems with variable domain shapes, boundary conditions and equation parameters. Through a series of numerical experiments of parametric problems on general geometries, we show that XFNO can effectively avoid data dependency and has a stronger generalization capability compared to the data-driven surrogate models, and has a broader scope of applicability, higher solution accuracy and computational efficiency than the other existing physics-informed algorithms.