Diffeomorphism Neural Operator for various domains and parameters of
  partial differential equations
Zhiwei Zhao Changqing Liu Yingguang Li Zhibin Chen Xu Liu
Many science and engineering applications demand partial differential
equations (PDE) evaluations that are traditionally computed with
resource-intensive numerical solvers. Neural operator models provide an
efficient alternative by learning the governing physical laws directly from
data in a class of PDEs with different parameters, but constrained in a fixed
boundary (domain). Many applications, such as design and manufacturing, would
benefit from neural operators with flexible domains when studied at scale. Here
we present a diffeomorphism neural operator learning framework towards
developing domain-flexible models for physical systems with various and complex
domains. Specifically, a neural operator trained in a shared domain mapped from
various domains of fields by diffeomorphism is proposed, which transformed the
problem of learning function mappings in varying domains (spaces) into the
problem of learning operators on a shared diffeomorphic domain. Meanwhile, an
index is provided to evaluate the generalization of diffeomorphism neural
operators in different domains by the domain diffeomorphism similarity.
Experiments on statics scenarios (Darcy flow, mechanics) and dynamic scenarios
(pipe flow, airfoil flow) demonstrate the advantages of our approach for neural
operator learning under various domains, where harmonic and volume
parameterization are used as the diffeomorphism for 2D and 3D domains. Our
diffeomorphism neural operator approach enables strong learning capability and
robust generalization across varying domains and parameters.