Learning Neural Operators on Riemannian Manifolds

TL;DR

The paper tackles the challenge of learning operators between functions defined on non-Euclidean domains by introducing Neural Operator on Riemannian Manifolds (NORM). NORM projects inputs/outputs onto a finite-dimensional Laplace–Beltrami eigenbasis, enabling a discretisation-free, universal-approximation-capable operator learner that works across complex 2D/3D geometries. Through five diverse case studies, including PDE operators and engineering problems like CFRP deformation and blood flow, NORM consistently outperforms established baselines (DeepONet, POD-DeepONet, FNO, GNN) in accuracy and convergence. The work highlights the theoretical universal approximation property and practical advantages of spectral encoding on manifolds, while outlining avenues for extending to non-Riemannian data and physics-informed integration.

Abstract

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.

Figures (8)

• Figure 1: The illustration of Neural Operator on Riemannian Manifolds (NORM).a, Operators defined on Riemannian manifolds, where the input function and output function can be defined on the same or different Riemannian manifolds. The example for this illustration is the operator learning problem of the composite curing case, where the input temperature function and the output deformation function are both defined on the same manifold, the composite part. b, The framework of NORM, consists of two feature mapping layers (P and Q) and multiple L-layers. c, The structure of L-layer, consists of the encoder-approximator-decoder block, the linear transformation, and the non-linear activation function. d, Laplace-Beltrami Operator (LBO) eigenfunctions for the geometric domain (the composite part).
• Figure 1: Experimental results of the Darcy problem.a, The mesh for the irregular geometric domain. b, c, The input and output fields for a representative sample. d-g, The prediction results of different methods. h-k, The prediction errors of different methods.
• Figure 2: Illustration of three toy case studies. a, Darcy problem (Case 1): the operator learning problem is the mapping from the diffusion coefficient field to the pressure field. b, Pipe turbulence (Case 2): the operator learning problem is the mapping from the current velocity field to the future velocity field. c, Heat transfer (Case 3): the operator learning problem is the mapping from the 2D boundary condition to the 3D temperature field of the part.
• Figure 2: Experimental results of the pipe turbulence.a, The mesh for the complex pipe shape. b, c, The input and output fields for a representative sample. d-g, The prediction results of different methods. h-k, The prediction errors of different methods.
• Figure 3: Composite workpiece deformation prediction case (Case 4).a, Illustration of the air-intake workpiece and the composite curing. b, The input and output of the operator learning problem, the predicted deformation of NORM, and the prediction error of comparison methods. c, The distribution of deformation prediction error over all nodes of all test samples. d, The maximum prediction errors of all test cases for the three methods.

Theorems & Definitions (2)

• Theorem
• proof : Proof of Theorem