# Geometric Laplace Neural Operator

arXiv: https://arxiv.org/abs/2512.16409 · HTML: https://mlsanity.com/paper/2512.16409/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 31 of its 48 references that are in the map.
- First posted on arXiv 18 Dec 2025.

## Builds on (most cited first, 30 of 31)

- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [FourCastNet: A Global Data-driven High-resolution Weather Model using Adaptive Fourier Neural Operators](https://mlsanity.com/paper/2202.11214/) (2022)
- [Neural Operator: Graph Kernel Network for Partial Differential Equations](https://mlsanity.com/paper/2003.03485/) (2020)
- [Fourier Neural Operator with Learned Deformations for PDEs on General Geometries](https://mlsanity.com/paper/2207.05209/) (2022)
- [Multipole Graph Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2006.09535/) (2020)
- [GNOT: A General Neural Operator Transformer for Operator Learning](https://mlsanity.com/paper/2302.14376/) (2023)
- [Transolver: A Fast Transformer Solver for PDEs on General Geometries](https://mlsanity.com/paper/2402.02366/) (2024)
- [Geometry-Informed Neural Operator for Large-Scale 3D PDEs](https://mlsanity.com/paper/2309.00583/) (2023)
- [U-FNO -- An enhanced Fourier neural operator-based deep-learning model for multiphase flow](https://mlsanity.com/paper/2109.03697/) (2021)
- [Spherical Fourier Neural Operators: Learning Stable Dynamics on the Sphere](https://mlsanity.com/paper/2306.03838/) (2023)
- [Neural Operators for Accelerating Scientific Simulations and Design](https://mlsanity.com/paper/2309.15325/) (2023)
- [Convolutional Neural Operators for robust and accurate learning of PDEs](https://mlsanity.com/paper/2302.01178/) (2023)
- [DiffusionNet: Discretization Agnostic Learning on Surfaces](https://mlsanity.com/paper/2012.00888/) (2020)
- [Solving High-Dimensional PDEs with Latent Spectral Models](https://mlsanity.com/paper/2301.12664/) (2023)
- [Operator Learning: Algorithms and Analysis](https://mlsanity.com/paper/2402.15715/) (2024)
- [Geometry Aware Operator Transformer as an Efficient and Accurate Neural Surrogate for PDEs on Arbitrary Domains](https://mlsanity.com/paper/2505.18781/) (2025)
- [Effective Rotation-invariant Point CNN with Spherical Harmonics kernels](https://mlsanity.com/paper/1906.11555/) (2019)
- [Domain Agnostic Fourier Neural Operators](https://mlsanity.com/paper/2305.00478/) (2023)
- [Physics informed WNO](https://mlsanity.com/paper/2302.05925/) (2023)
- [Huge Ensembles Part I: Design of Ensemble Weather Forecasts using Spherical Fourier Neural Operators](https://mlsanity.com/paper/2408.03100/) (2024)
- [Harnessing Scale and Physics: A Multi-Graph Neural Operator Framework for PDEs on Arbitrary Geometries](https://mlsanity.com/paper/2411.15178/) (2024)
- [Huge Ensembles Part II: Properties of a Huge Ensemble of Hindcasts Generated with Spherical Fourier Neural Operators](https://mlsanity.com/paper/2408.01581/) (2024)
- [Mesh-Informed Neural Operator : A Transformer Generative Approach](https://mlsanity.com/paper/2506.16656/) (2025)
- [Geometry-Informed Neural Operator Transformer](https://mlsanity.com/paper/2504.19452/) (2025)
- [Spatio-spectral graph neural operator for solving computational mechanics problems on irregular domain and unstructured grid](https://mlsanity.com/paper/2409.00604/) (2024)
- [Physics- and geometry-aware spatio-spectral graph neural operator for time-independent and time-dependent PDEs](https://mlsanity.com/paper/2508.09627/) (2025)
- [GeoMaNO: Geometric Mamba Neural Operator for Partial Differential Equations](https://mlsanity.com/paper/2505.12020/) (2025)
- [GITO: Graph-Informed Transformer Operator for Learning Complex Partial Differential Equations](https://mlsanity.com/paper/2506.13906/) (2025)

## Built on by (most cited first, 2 of 2)

- [Neural Operators for Biomedical Spherical Heterogeneity](https://mlsanity.com/paper/2601.03561/) (2026)
- [Factorized Neural Operators Decompose Dynamic and Persistent Responses](https://mlsanity.com/paper/2606.16900/) (2026)

## Abstract (arXiv)

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.