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.

翻译：许多科学与工程应用需要求解偏微分方程，传统上依赖资源密集的数值求解器完成计算。神经网络算子模型通过直接从数据中学习物理定律，为具有不同参数但固定边界（域）的一类偏微分方程提供了高效替代方案。然而，在设计与制造等规模化应用中，具有灵活域结构的神经网络算子将更具优势。本文提出一种同胚神经网络算子学习框架，旨在为具有复杂多变域结构的物理系统开发域自适应模型。具体而言，我们提出一种在共享域上训练的神经网络算子——该共享域通过同胚映射从各类场域中变换获得，从而将不同域（空间）上的函数映射学习问题转化为共享同胚域上的算子学习问题。同时，我们构建了一个基于域同胚相似性的评价指标，用于评估同胚神经网络算子在异构域中的泛化能力。在静态场景（达西流、力学）与动态场景（管道流、翼型流）上的实验表明，本文方法在多种域结构下具有显著优势（采用调和参数化与体积参数化分别作为二维与三维域的同胚映射）。所提出的同胚神经网络算子方法在跨域与跨参数场景中展现出强大的学习能力与稳健的泛化性能。