Diffeomorphism Neural Operator for various domains and parameters of partial differential equations

TL;DR

This paper introduces the diffeomorphism neural operator (DNO), a novel framework that enables neural operators to solve PDEs across various domains and parameters by leveraging diffeomorphic mappings from a generic domain.

Contribution

The paper proposes a new neural operator framework that generalizes across different physical domains using diffeomorphic mappings, expanding the applicability of neural operators to variable geometries.

Findings

DNO demonstrates robust learning across multiple PDEs and domains.

The method effectively evaluates geometric similarity between domains.

Experiments show strong generalization to unseen domains and parameters.

Abstract

In scientific and engineering applications, solving partial differential equations (PDEs) across various parameters and domains normally relies on resource-intensive numerical methods. Neural operators based on deep learning offered a promising alternative to PDEs solving by directly learning physical laws from data. However, the current neural operator methods were limited to solve PDEs on fixed domains. Expanding neural operators to solve PDEs on various domains hold significant promise in medical imaging, engineering design and manufacturing applications, where geometric and parameter changes are essential. This paper presents a novel neural operator learning framework for solving PDEs with various domains and parameters defined for physical systems, named diffeomorphism neural operator (DNO). The main idea is that a neural operator learns in a generic domain which is diffeomorphically mapped from various physics domains expressed by the same PDE. In this way, the challenge of operator learning on various domains is transformed into operator learning on the generic domain. The generalization performance of DNO on different domains can be assessed by a proposed method which evaluates the geometric similarity between a new domain and the domains of training dataset after diffeomorphism. Experiments on Darcy flow, pipe flow, airfoil flow and mechanics were carried out, where harmonic and volume parameterization were used as the diffeomorphism for 2D and 3D domains. The DNO framework demonstrated robust learning capabilities and strong generalization performance across various domains and parameters.