Geometric Laplace Neural Operator
Hao Tang, Jiongyu Zhu, Zimeng Feng, Hao Li, Chao Li
TL;DR
The paper introduces the Geometric Laplace Neural Operator, a novel framework that extends neural operator learning to arbitrary Riemannian manifolds, effectively handling non-periodic, transient, and irregular signals with superior performance.
Contribution
It proposes a generalized operator learning framework using pole-residue decomposition and introduces GLNO, embedding Laplace spectral representation on manifolds, with a grid-invariant architecture for broad applicability.
Findings
Outperforms state-of-the-art models on PDEs and ODEs.
Effectively models aperiodic and decaying dynamics.
Extends operator learning to arbitrary Riemannian manifolds.
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…
Peer Reviews
Decision·Submitted to ICLR 2026
1. The paper addresses an important problem related to Fourier transform-based neural operators, and the authors propose a very principled approach to tackle it. 2. The technique's ability to generalize to different geometries is a strong point. 3. The authors go beyond simply solving partial differential equations (PDEs) and demonstrate the performance on real-world data.
### Write Up I believe the write-up could be better organized. Some sections need consolidation, while others require more elaboration. For instance, Section 4.1 can be omitted, allowing the authors to directly introduce the generalized geometric Laplace basis. Additionally, the authors should expand upon the sections between lines 225 and 252, as I currently do not understand the message being conveyed in those line. ### Baselines Two important baselines were overlooked. The following works ar
The paper is well written and it also illustrates the algorithm on real world dataset.
Despite the strength, the contribution is somewhat incremental (using generalized laplace transform). It would strengthen the paper if theoretical analysis can also be included, which right now is missing. The bechamarking is not exhaustive. For example, I was expecting comprison with respect to Geo-FNO, GNO, Sp2GNO, and GINO at the very least as all of these handle non-Euclidean domain. The fact that graph based approaches are slower is well taken, but it does has its advantages and hence, com
The paper contains a detailed mathematical formulation of the proposed method, and the authors performed experiments across a diverse set of problems, which I appreciate. I also appreciate that the authors provided timing and parametric complexity metrics to support the downsides of transformer and graph-based approaches that they argue in Section 2.
The authors write that they want to address limitations of existing spectral operators (page 3). However, there have recently been many neural operator architectures that are not explicitly spectral, such as attention-based and convolution-based neural operators. The authors should clarify the benefits of spectral operators compared to these other types that have been explored in the literature. I would also recommend that the authors add some more challenging experimental problems, particularl
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsModel Reduction and Neural Networks · Generative Adversarial Networks and Image Synthesis · Gaussian Processes and Bayesian Inference