Diffeomorphism neural operator for various domains and parameters of partial differential equations

Abstract

Solving partial differential equations (PDEs) across varying geometric domains and parameters represents a significant challenge in fields such as materials science, engineering, design and medical imaging, primarily due to the high computing cost associated with recomputing the solution for every change in geometry or parameters. This paper presents a neural operator learning framework for solving PDEs with various domains and parameters, named diffeomorphism neural operator (DNO). The framework transforms the problem of operator learning on varying domains into learning on a generic domain through a diffeomorphic mapping. The efficiency and effectiveness of DNO are validated in experiments that rapidly provide solutions to various PDEs across different domains and parameters. Our method obtains solutions multiple orders of magnitude faster when adapted to changes in shape and size. DNO offers advangates for a broad spectrum of scientific and engineering applications that require dynamic domain and parameter handling. The authors introduce the Diffeomorphism Neural Operator (DNO) to solve partial differential equations across varying domains and parameters by diffeomorphism mapping various physical domains to a generic domain. Experiments demonstrate its efficiency and accuracy, with strong generalization across different shapes and sizes.