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

Published: 29 Feb 2024, Last Modified: 29 Feb 2024Accepted by TMLREveryoneRevisionsBibTeX
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.
Submission Length: Regular submission (no more than 12 pages of main content)
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Assigned Action Editor: ~Ivan_Oseledets1
License: Creative Commons Attribution 4.0 International (CC BY 4.0)
Submission Number: 1089