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.

翻译：在科学与工程应用中，求解不同参数与域上的偏微分方程通常依赖于资源密集型的数值方法。基于深度学习的神经算子通过直接从数据中学习物理规律，为偏微分方程求解提供了有前景的替代方案。然而，现有神经算子方法仅限于在固定域上求解偏微分方程。将神经算子扩展至多域求解在医学成像、工程设计与制造等领域具有重要应用前景，其中几何形态与参数变化至关重要。本文提出一种新颖的神经算子学习框架，用于求解物理系统中定义于多域多参数的偏微分方程，称为微分同胚神经算子。其核心思想是：神经算子在一个通用域上进行学习，该通用域通过微分同胚映射从表达相同偏微分方程的各种物理域转换而来。通过这种方式，多域上的算子学习挑战被转化为通用域上的算子学习问题。我们提出一种评估方法，通过计算新域与训练数据集各域在微分同胚映射后的几何相似度，来评估DNO在不同域上的泛化性能。我们在达西流、管流、翼型流及力学问题上进行了实验，其中二维与三维域分别采用调和参数化与体积参数化作为微分同胚映射。实验表明，DNO框架在不同域与参数上展现出稳健的学习能力与强大的泛化性能。