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

TL;DR

This paper introduces a pseudo-differential neural operator (PDNO) that generalizes Fourier neural operators by incorporating pseudo-differential operators, enabling more effective learning of PDE solution operators, validated on Darcy flow and Navier-Stokes equations.

Contribution

The paper proposes a novel pseudo-differential neural operator (PDNO) that extends Fourier neural operators using pseudo-differential integral operators, improving PDE solution learning.

Findings

PDNO outperforms existing neural operators in experiments.

The model effectively learns nonlinear PDE solution operators.

Theoretical analysis confirms PDIO as a bounded linear operator.

Abstract

Learning the mapping between two function spaces has garnered considerable research attention. However, learning the solution operator of partial differential equations (PDEs) remains a challenge in scientific computing. Fourier neural operator (FNO) was recently proposed to learn solution operators, and it achieved an excellent performance. In this study, we propose a novel \textit{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 generalized differential operator characterized by a certain symbol. We parameterize this symbol using a neural network and demonstrate that the neural network-based symbol is contained in a smooth symbol class. Subsequently, we verify 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 \textit{pseudo-differential neural operator} (PDNO) and learn the nonlinear solution operator of PDEs. We experimentally validate the effectiveness of the proposed model by utilizing Darcy flow and the Navier-Stokes equation. The obtained results indicate that the proposed PDNO outperforms the existing neural operator approaches in most experiments.