Topological Neural Operators

• URL: http://arxiv.org/abs/2606.09806v1
• Date: Mon, 08 Jun 2026 17:54:33 GMT
• Title: Topological Neural Operators
• Abstract summary: We introduce Topological Neural Operators (TNOs), a principled framework for operator learning on cell complexes.<n>TNOs represent data as features defined on cells of varying dimension and model their interactions through Discrete Exterior Calculus.<n>We propose Hierarchical TNOs (HTNOs), which incorporate learned coarse complexes to propagate long-range and topology-dependent information.
• Score: 22.267195753988418
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We introduce Topological Neural Operators (TNOs), a principled framework for operator learning on cell complexes that lifts neural operators (NOs) from functions on points and/or edges to topological domains. TNOs represent data as features defined on cells of varying dimension and model their interactions through Discrete Exterior Calculus, enabling explicit cross-dimensional coupling via gradient-, curl-, and divergence-type operators. The key design principle is to decouple where information flows, as governed by fixed topological operators, from how it is transformed (which is learned), yielding models that respect the geometric support of physical quantities and expose conservation and compatibility structure. We further propose Hierarchical TNOs (HTNOs), which incorporate learned coarse complexes to propagate long-range and topology-dependent information. Our framework subsumes existing NOs as a special case, providing a unified perspective on operator learning across discretizations. Across a range of PDE benchmarks, including irregular-geometry flow problems, TNOs and HTNOs improve accuracy; controlled studies further isolate the benefits of native higher-rank and topological structure. Project page: https://circle-group.github.io/research/TNO

Related papers

• Neural Operators as Efficient Function Interpolators [0.764671395172401]
  Neural operators (NOs) are designed to learn maps between infinite-dimensional function spaces.<n>We show that NOs can match or outperform standard multilayer perceptrons and Kolmogorov-Arnold Networks in accuracy.<n>A TFNO ensemble reaches a held-out root-mean-square error of 198.2 keV, placing it among the best recent neural-network approaches.
  arXiv  Detail & Related papers  (2026-05-08T14:30:12Z)
• An Algebraic Representation Theorem for Linear GENEOs in Geometric Machine Learning [1.3425748364842416]
  Group Equivariant Non-Expansive Operators (GENEOs) have emerged as a powerful class of operators for encoding symmetries.<n>We introduce a new representation theorem for linear GENEOs acting between different perception pairs.<n>We also prove the compactness and convexity of the space of linear GENEOs.
  arXiv  Detail & Related papers  (2026-01-07T13:21:44Z)
• Geometric Laplace Neural Operator [12.869633759181417]
  We propose a generalized operator learning framework based on a pole-residue decomposition enriched with exponential basis functions.<n>We introduce the Geometric Laplace Neural Operator (GLNO), which embeds the Laplace spectral representation into the eigen-basis of the Laplace-Beltrami operator.<n>We further design a grid-invariant network architecture (GLNONet) that realizes GLNO in practice.
  arXiv  Detail & Related papers  (2025-12-18T11:07:41Z)
• Neural Operators with Localized Integral and Differential Kernels [77.76991758980003]
  We present a principled approach to operator learning that can capture local features under two frameworks. We prove that we obtain differential operators under an appropriate scaling of the kernel values of CNNs. To obtain local integral operators, we utilize suitable basis representations for the kernels based on discrete-continuous convolutions.
  arXiv  Detail & Related papers  (2024-02-26T18:59:31Z)
• Resolution-Invariant Image Classification based on Fourier Neural Operators [1.3190581566723918]
  We investigate the use of generalization Neural Operators (FNOs) for image classification in comparison to standard Convolutional Neural Networks (CNNs) We derive the FNO architecture as an example for continuous and Fr'echet-differentiable neural operators on Lebesgue spaces.
  arXiv  Detail & Related papers  (2023-04-02T10:23:36Z)
• Designing Universal Causal Deep Learning Models: The Case of Infinite-Dimensional Dynamical Systems from Stochastic Analysis [7.373617024876726]
  Several non-linear operators in analysis depend on a temporal structure which is not leveraged by contemporary neural operators.<n>This paper introduces a deep learning model-design framework that takes suitable infinite-dimensional linear metric spaces.<n>We show that our framework can uniformly approximate on compact sets and across arbitrarily finite-time horizons H" or smooth trace class operators.
  arXiv  Detail & Related papers  (2022-10-24T14:43:03Z)
• Deep Architecture Connectivity Matters for Its Convergence: A Fine-Grained Analysis [94.64007376939735]
  We theoretically characterize the impact of connectivity patterns on the convergence of deep neural networks (DNNs) under gradient descent training. We show that by a simple filtration on "unpromising" connectivity patterns, we can trim down the number of models to evaluate.
  arXiv  Detail & Related papers  (2022-05-11T17:43:54Z)
• Neural Operator: Learning Maps Between Function Spaces [75.93843876663128]
  We propose a generalization of neural networks to learn operators, termed neural operators, that map between infinite dimensional function spaces. We prove a universal approximation theorem for our proposed neural operator, showing that it can approximate any given nonlinear continuous operator. An important application for neural operators is learning surrogate maps for the solution operators of partial differential equations.
  arXiv  Detail & Related papers  (2021-08-19T03:56:49Z)
• Stability of Algebraic Neural Networks to Small Perturbations [179.55535781816343]
  Algebraic neural networks (AlgNNs) are composed of a cascade of layers each one associated to and algebraic signal model. We show how any architecture that uses a formal notion of convolution can be stable beyond particular choices of the shift operator.
  arXiv  Detail & Related papers  (2020-10-22T09:10:16Z)
• Provably Efficient Neural Estimation of Structural Equation Model: An Adversarial Approach [144.21892195917758]
  We study estimation in a class of generalized Structural equation models (SEMs) We formulate the linear operator equation as a min-max game, where both players are parameterized by neural networks (NNs), and learn the parameters of these neural networks using a gradient descent. For the first time we provide a tractable estimation procedure for SEMs based on NNs with provable convergence and without the need for sample splitting.
  arXiv  Detail & Related papers  (2020-07-02T17:55:47Z)
• Neural Operator: Graph Kernel Network for Partial Differential Equations [57.90284928158383]
  This work is to generalize neural networks so that they can learn mappings between infinite-dimensional spaces (operators) We formulate approximation of the infinite-dimensional mapping by composing nonlinear activation functions and a class of integral operators. Experiments confirm that the proposed graph kernel network does have the desired properties and show competitive performance compared to the state of the art solvers.
  arXiv  Detail & Related papers  (2020-03-07T01:56:20Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.