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

Jin Young Shin, Jae Yong Lee, Hyung Ju Hwang

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)

Previous TMLR Submission Url: https://openreview.net/forum?id=805jKZ0Gqf

Changes Since Last Submission: I have addressed the content corresponding to your feedback by making the necessary modifications. All reviewers either accept or are leaning towards acceptance, despite noting the incremental nature and addressing any misprints.

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License: Creative Commons Attribution 4.0 International (CC BY 4.0)

Submission Number: 1089