mlsanityData to 14 Oct 2026

Beyond Arbitrary Geometry: Topology Generalization In neural PDE Operators

arXiv 2609.05860 ↗First posted 5 Sep 2026cs.LGcited by 0

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Abstract (arXiv)

Neural operators that accept arbitrary meshes are often treated as geometry-general, but unseen domain topology changes both the invariant and decaying subspaces of a PDE operator. We use Hodge heat flow as a controlled lens on this distinction and introduce TopoBox-3D, where tunnels and cavities vary Betti support while the exact Hodge decomposition separates the harmonic kernel from the positive spectrum. Across six architectures, models that infer topology implicitly suffer excess matched degradation in 37 of 45 model--task topology-OOD cells, yet cases that change harmonic dimension are not more strongly penalized on average. The dominant difficulty is instead spectral: the initial Rayleigh quotient is the most stable predictor of error, and spectral broadening adds information for edge and face cochains. Most strikingly, controlled probes show that explicit incidence and harmonic coordinates do not yield the best kernel-identity accuracy; nevertheless, TNO ranks first in mixed-input nonharmonic accuracy on all six tasks with nontrivial harmonic support. Together, these results establish topology as a distinct generalization axis beyond arbitrary-geometry compatibility and show that its influence extends across the Hodge spectrum rather than remaining confined to the harmonic kernel. More broadly, they suggest that global, low-frequency structural priors may help organize predictions in the faster-decaying complementary component, offering a new perspective on how neural operators may generalize across topology as well as geometry.

Key facts. No citing papers in the map yet (data through 14 Oct 2026). Builds on 32 of its 40 references that are in the map. First posted on arXiv 5 Sep 2026.

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32 of 40 references.

References · 32

  1. 2020Fourier Neural Operator for Parametric Partial Differential Equationscited by 2,908
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20 more
  1. 2022U-NO: U-shaped Neural Operatorscited by 230
  2. 2023Convolutional Neural Operators for robust and accurate learning of PDEscited by 214
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  4. 2025Transolver++: An Accurate Neural Solver for PDEs on Million-Scale Geometriescited by 97
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  7. 2021Principled Simplicial Neural Networks for Trajectory Predictioncited by 77
  8. 2025Geometry Aware Operator Transformer as an Efficient and Accurate Neural Surrogate for PDEs on Arbitrary Domainscited by 62
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  12. 2025RIGNO: A Graph-based framework for robust and accurate operator learning for PDEs on arbitrary domainscited by 23
  13. 2023Beyond Regular Grids: Fourier-Based Neural Operators on Arbitrary Domainscited by 21
  14. 2024Topological Neural Networks go Persistent, Equivariant, and Continuouscited by 17
  15. 2023NUNO: A General Framework for Learning Parametric PDEs with Non-Uniform Datacited by 15
  16. 2024Diffeomorphism Neural Operator for various domains and parameters of partial differential equationscited by 15
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  18. 2026Topological Neural Operatorscited by 4
  19. 2026Topology-Preserving Neural Operator Learning via Hodge Decompositioncited by 3
  20. 2025Learning to Solve PDEs on Neural Shape Representationscited by 2

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