mlsanityData to 14 Oct 2026

Topological Neural Operators

arXiv 2606.09806 ↗First posted 8 Jun 2026cs.LG cs.AIcited by 4

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Abstract (arXiv)

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

Key facts. Cited by 4 papers in the map (data through 14 Oct 2026). Builds on 47 of its 92 references that are in the map. First posted on arXiv 8 Jun 2026.

Builds on

47 of 92 references.

References · 47

  1. 2017Attention Is All You Needcited by 60,361
  2. 2019PyTorch: An Imperative Style, High-Performance Deep Learning Librarycited by 20,305
  3. 2016PointNet: Deep Learning on Point Sets for 3D Classification and Segmentationcited by 5,248
  4. 2020Fourier Neural Operator for Parametric Partial Differential Equationscited by 2,908
  5. 2019DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operatorscited by 2,018
  6. 2020Learning Mesh-Based Simulation with Graph Networkscited by 868
  7. 2020Neural Operator: Graph Kernel Network for Partial Differential Equationscited by 734
  8. 2021Physics-Informed Neural Operator for Learning Partial Differential Equationscited by 527
  9. 2022Fourier Neural Operator with Learned Deformations for PDEs on General Geometriescited by 427
  10. 2023GNOT: A General Neural Operator Transformer for Operator Learningcited by 336
  11. 2024Transolver: A Fast Transformer Solver for PDEs on General Geometriescited by 326
  12. 2023Geometry-Informed Neural Operator for Large-Scale 3D PDEscited by 303
35 more
  1. 2022Transformer for Partial Differential Equations' Operator Learningcited by 283
  2. 2020Model Reduction and Neural Networks for Parametric PDEscited by 258
  3. 2021Factorized Fourier Neural Operatorscited by 249
  4. 2021Weisfeiler and Lehman Go Topological: Message Passing Simplicial Networkscited by 221
  5. 2023Neural Operators for Accelerating Scientific Simulations and Designcited by 221
  6. 2021Weisfeiler and Lehman Go Cellular: CW Networkscited by 218
  7. 2022Neural Sheaf Diffusion: A Topological Perspective on Heterophily and Oversmoothing in GNNscited by 214
  8. 2023Convolutional Neural Operators for robust and accurate learning of PDEscited by 214
  9. 2024Poseidon: Efficient Foundation Models for PDEscited by 200
  10. 2022Physics-Informed Deep Neural Operator Networkscited by 113
  11. 2022Topological Deep Learning: Going Beyond Graph Datacited by 97
  12. 2018Random Walks on Simplicial Complexes and the normalized Hodge 1-Laplaciancited by 86
  13. 2024Operator Learning: Algorithms and Analysiscited by 85
  14. 2020Sheaf Neural Networkscited by 84
  15. 2023A Mathematical Guide to Operator Learningcited by 76
  16. 2024Position: Topological Deep Learning is the New Frontier for Relational Learningcited by 66
  17. 2025Geometry Aware Operator Transformer as an Efficient and Accurate Neural Surrogate for PDEs on Arbitrary Domainscited by 62
  18. 2025AB-UPT: Scaling Neural CFD Surrogates for High-Fidelity Automotive Aerodynamics Simulations via Anchored-Branched Universal Physics Transformerscited by 53
  19. 2024Continuum Attention for Neural Operatorscited by 40
  20. 2023Domain Agnostic Fourier Neural Operatorscited by 38
  21. 2021One-shot learning for solution operators of partial differential equationscited by 30
  22. 2024HAMLET: Graph Transformer Neural Operator for Partial Differential Equationscited by 24
  23. 2025RIGNO: A Graph-based framework for robust and accurate operator learning for PDEs on arbitrary domainscited by 23
  24. 2023Beyond Regular Grids: Fourier-Based Neural Operators on Arbitrary Domainscited by 21
  25. 2024Physics-Informed Geometry-Aware Neural Operatorcited by 19
  26. 2023Combinatorial Complexes: Bridging the Gap Between Cell Complexes and Hypergraphscited by 15
  27. 2025Copresheaf Topological Neural Networks: A Generalized Deep Learning Frameworkcited by 12
  28. 2021The topological Dirac equation of networks and simplicial complexescited by 11
  29. 2023Equivariant Neural Operator Learning with Graphon Convolutioncited by 11
  30. 2022Higher-order signal processing with the Dirac operatorcited by 8
  31. 2023Neural Operator: Is data all you need to model the world? An insight into the paradigm of data-driven scientific MLcited by 8
  32. 2025Neural Green's Functionscited by 6
  33. 2025Going with the Speed of Sound: Pushing Neural Surrogates into Highly-turbulent Transonic Regimescited by 6
  34. 2024A simple and complete discrete exterior calculus on general polygonal meshescited by 1
  35. 2026Gauge-Equivariant Intrinsic Neural Operators for Geometry-Consistent Learning of Elliptic PDE Mapscited by 1

Built on by

Cited by year (2026 to 14 Oct)

Citing papers per year: 2026: 4.

4 papers.

  1. 2026Collapsed Effective Operators for Higher-order Structurescited by 0
  2. 2026Beyond Arbitrary Geometry: Topology Generalization In neural PDE Operatorscited by 0
  3. 2026Dynamic Kuramoto-Hodge Operators for PDEs on Complex Geometries and Topologiescited by 0
  4. 2026Learnable Hodge-Laplacian Driven Mixture of Experts for Steady-PDE Surrogates on Coarse Unstructured Meshescited by 0