Geometric Laplace Neural Operator

TL;DR

The paper introduces GLNO, a geometric neural operator framework that extends pole-residue learning to arbitrary Riemannian manifolds by embedding the Laplace spectral representation into the Laplace-Beltrami eigenbasis. It replaces the traditional periodic Fourier basis with a generalized Laplace basis that includes exponential decay, enabling accurate modeling of non-periodic and transient dynamics on irregular geometries. A grid-invariant network (GLNONet) operationalizes GLNO, leveraging intrinsic geometric features and LBO spectra to learn mappings between functions on manifolds with strong performance gains over state-of-the-art operators, including on unstructured meshes and real-world geometric data. The approach offers superior expressiveness, parameter efficiency, and robust generalization, demonstrated across ODE/PDE benchmarks and diverse geometric tasks, with plans to release code for reproducibility.

Abstract

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.

Figures (5)

• Figure 1: GLNONet & GLNO block architectures. Encoder and Decoder are both multi-layer perceptron (MLP). In GLNO block, geometric features include coordinate, curvature and distance. GLNO achieves spectral learning of transient response through pole-residue calculation in manifolds and learnable frequency domain transform ($\sigma_i$). LBO: Laplace-Beltrami operator.
• Figure 2: Driven Gravity Pendulum Result: (a) The test Error of different models in $c=1.0$ pendulum task; (b) The prediction of different models in $c=1.0$ pendulum task; (c) The Relative Error$L^2$-$c$ curve for different models.
• Figure 3: Poisson Equation results from different models and unstructured mesh structure used in this task
• Figure 4: Example results on Shape-Net Car and human body segmentation. Local results of Shape-Net Car is displayed in prediction Error. For local results of human body segmentation, green areas: GLNO outperforms benchmark model; red areas: GLNO underperforms benchmark model.
• Figure 5: Result comparison on RNA classification task. Different colors represent different labels. Green points in the middle figure represent cases where our model outperforms the Diffusion Net.