Geometric Laplace Neural Operator
This work addresses the problem of operator learning on arbitrary Riemannian manifolds for researchers in computational science, offering a novel framework for non-periodic and decaying dynamics.
The paper tackles the challenge of learning mappings between function spaces for PDEs on irregular geometries and non-periodic excitations by proposing the Geometric Laplace Neural Operator (GLNO), which embeds Laplace spectral representation into the Laplace-Beltrami eigen-basis, achieving robust performance over state-of-the-art models in experiments.
Neural operators have emerged as powerful tools for learning mappings between function spaces, enabling efficient solutions to partial differential equations across varying inputs and domains. Despite the success, existing methods often struggle with non-periodic excitations, transient responses, and signals defined on irregular or non-Euclidean geometries. To address this, we propose a generalized operator learning framework based on a pole-residue decomposition enriched with exponential basis functions, enabling expressive modeling of aperiodic and decaying dynamics. Building on this formulation, we introduce the Geometric Laplace Neural Operator (GLNO), which embeds the Laplace spectral representation into the eigen-basis of the Laplace-Beltrami operator, extending operator learning to arbitrary Riemannian manifolds without requiring periodicity or uniform grids. We further design a grid-invariant network architecture (GLNONet) that realizes GLNO in practice. Extensive experiments on PDEs/ODEs and real-world datasets demonstrate our robust performance over other state-of-the-art models.