mlsanityData to 14 Oct 2026

Topology-Preserving Neural Operator Learning via Hodge Decomposition

arXiv 2605.13834 ↗First posted 13 May 2026cs.LG cs.AI cs.CGcited by 3

Not on the map. Why?

Abstract (arXiv)

In this paper, we study solution operators of physical field equations on geometric meshes from a function-space perspective. We reveal that Hodge orthogonality fundamentally resolves spectral interference by isolating unlearnable topological degrees of freedom from learnable geometric dynamics, enabling an additive approximation confined to structure-preserving subspaces. Building on Hodge theory and operator splitting, we derive a principled operator-level decomposition. The result is a Hybrid Eulerian-Lagrangian architecture with an algebraic-level inductive bias we call Hodge Spectral Duality (HSD). In our framework, we use discrete differential forms to capture topology-dominated components and an orthogonal auxiliary ambient space to represent complex local dynamics. Our method achieves superior accuracy and efficiency on geometric graphs with enhanced fidelity to physical invariants. Our code is available at https://github.com/ContinuumCoder/Hodge-Spectral-Duality

Key facts. Cited by 3 papers in the map (data through 14 Oct 2026). Builds on 24 of its 51 references that are in the map. First posted on arXiv 13 May 2026.

Builds on

24 of 51 references.

References · 24

  1. 2018How Powerful are Graph Neural Networks?cited by 4,393
  2. 2020Fourier Neural Operator for Parametric Partial Differential Equationscited by 2,908
  3. 2019DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operatorscited by 2,018
  4. 2018Deeper Insights into Graph Convolutional Networks for Semi-Supervised Learningcited by 1,102
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  9. 2023GNOT: A General Neural Operator Transformer for Operator Learningcited by 336
  10. 2019Gauge Equivariant Convolutional Networks and the Icosahedral CNNcited by 267
  11. 2020A Note on Over-Smoothing for Graph Neural Networkscited by 249
  12. 2020Simplicial Neural Networkscited by 110
12 more
  1. 2022Topological Deep Learning: Going Beyond Graph Datacited by 97
  2. 2023Operator Learning with Neural Fields: Tackling PDEs on General Geometriescited by 84
  3. 2024DrivAerNet++: A Large-Scale Multimodal Car Dataset with Computational Fluid Dynamics Simulations and Deep Learning Benchmarkscited by 75
  4. 2021Coordinate Independent Convolutional Networks -- Isometry and Gauge Equivariant Convolutions on Riemannian Manifoldscited by 73
  5. 2024Position: Topological Deep Learning is the New Frontier for Relational Learningcited by 66
  6. 2023Improved Operator Learning by Orthogonal Attentioncited by 61
  7. 2023Topological Deep Learning: A Review of an Emerging Paradigmcited by 34
  8. 2024TopoX: A Suite of Python Packages for Machine Learning on Topological Domainscited by 22
  9. 2021BScNets: Block Simplicial Complex Neural Networkscited by 21
  10. 2024Topological Signal Processing and Learning: Recent Advances and Future Challengescited by 12
  11. 2023Weighted Combinatorial Laplacian and its Application to Coverage Repair in Sensor Networkscited by 2
  12. 2025Resolving Oversmoothing with Opinion Dissensuscited by 2

Built on by

Cited by year (2026 to 14 Oct)

Citing papers per year: 2026: 3.

3 papers.

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