Pseudo-Differential Integral Operator for Learning Solution Operators of Partial Differential Equations
- Title
- Pseudo-Differential Integral Operator for Learning Solution Operators of Partial Differential Equations
- Authors
- 신진영
- Date Issued
- 2022
- Publisher
- 포항공과대학교
- 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. Therefore, in this study, we propose a novel pseudo-differential integral operator (PDIO) inspired by a pseudo-differential operator, which is a generalization of a differential operator and characterized by a 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 Burgers’ equation, Darcy flow, and the Navier-Stokes equation. The results reveal that the proposed PDNO outperforms the existing neural operator approaches in most experiments.
- URI
- http://postech.dcollection.net/common/orgView/200000632196
https://oasis.postech.ac.kr/handle/2014.oak/117406
- Article Type
- Thesis
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