Pseudo-Differential Neural Operator: Generalize Fourier Neural Operator for Learning Solution Operators of Partial Differential Equations

Abstract

Learning mapping between two function spaces has attracted considerable research attention. However, learning the solution operator of partial differential equations (PDEs) remains a challenge in scientific computing. Fourier neural operator (FNO) is recently proposed to learn the solution operators with an excellent performance. In this study, we propose a novel pseudo-differential integral operator (PDIO) to analyze and generalize the Fourier integral operator in FNO. PDIO is inspired by a pseudo-differential operator, which is a generalization of a differential operator and characterized by a certain symbol. We parameterize the symbol by using a neural network and show that the neural-network-based symbol is contained in a smooth symbol class. Subsequently, we prove that the PDIO is a bounded linear operator, and thus is continuous in the Sobolev space. We combine the PDIO with the neural operator to develop a pseudo-differential neural operator (PDNO) to learn the nonlinear solution operator of PDEs. We experimentally validate the effectiveness of the proposed model by using Darcy flow and the Navier-Stokes equation. The results reveal that the proposed PDNO outperforms the existing neural operator approaches in most experiments.