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

• URL: http://arxiv.org/abs/2201.11967v3
• Date: Mon, 4 Mar 2024 11:12:38 GMT
• Title: Pseudo-Differential Neural Operator: Generalized Fourier Neural Operator for Learning Solution Operators of Partial Differential Equations
• Authors: Jin Young Shin, Jae Yong Lee, Hyung Ju Hwang
• Abstract summary: We propose a novel textitpseudo-differential integral operator (PDIO) to analyze and generalize the Fourier integral operator in FNO. We experimentally validate the effectiveness of the proposed model by utilizing Darcy flow and the Navier-Stokes equation.
• Score: 14.43135909469058
• License: http://creativecommons.org/licenses/by-nc-sa/4.0/
• 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.

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