In Artificial Intelligence (AI) and computational science, learning the mappings between functions (called operators) defined on complex computational domains is a common theoretical challenge. Recently, Neural Operator emerged as a promising framework with a discretisation-independent model structure to break the fixed-dimension limitation of classical deep learning models. However, existing operator learning methods mainly focus on regular computational domains, and many components of these methods rely on Euclidean structural data. In real-life applications, many operator learning problems are related to complex computational domains such as complex surfaces and solids, which are non-Euclidean and widely referred to as Riemannian manifolds. Here, we report a new concept, Neural Operator on Riemannian Manifolds (NORM), which generalises Neural Operator from being limited to Euclidean spaces to being applicable to Riemannian manifolds, and can learn the mapping between functions defined on any real-life complex geometries, while preserving the discretisation-independent model structure. NORM shifts the function-to-function mapping to finite-dimensional mapping in the Laplacian eigenfunctions' subspace of geometry, and holds universal approximation property in learning operators on Riemannian manifolds even with only one fundamental block. The theoretical and experimental analysis prove that NORM is a significant step forward in operator learning and has the potential to solve complex problems in many fields of applications sharing the same nature and theoretical principle.

翻译：在人工智能与计算科学领域，学习定义在复杂计算域上的函数间映射（即算子）是普遍存在的理论挑战。近年来，神经算子作为一种具有离散化无关模型结构的创新框架应运而生，突破了经典深度学习模型的固定维度限制。然而，现有算子学习方法主要聚焦于规则计算域，其诸多组件依赖欧几里得结构数据。实际应用中，大量算子学习问题涉及复杂曲面与实体等非欧几里得计算域（统称为黎曼流形）。本文提出黎曼流形上的神经算子（NORM）新概念，将神经算子从欧几里得空间拓展至黎曼流形，使其能够学习定义于任意实际复杂几何构型上的函数映射，同时保持离散化无关模型结构。NORM将函数空间映射转化为几何拉普拉斯算子本征函数子空间中的有限维映射，即使仅采用单一基础模块，也能在黎曼流形上实现算子学习的通用逼近性质。理论与实验分析证明，NORM是算子学习领域的重要突破，可解决众多具有相同本质与理论原理的应用领域中的复杂问题。