# Beyond Arbitrary Geometry: Topology Generalization In neural PDE Operators

arXiv: https://arxiv.org/abs/2609.05860 · HTML: https://mlsanity.com/paper/2609.05860/

## 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.

## Builds on (most cited first, 30 of 32)

- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges](https://mlsanity.com/paper/2104.13478/) (2021)
- [Learning Mesh-Based Simulation with Graph Networks](https://mlsanity.com/paper/2010.03409/) (2020)
- [Fourier Neural Operator with Learned Deformations for PDEs on General Geometries](https://mlsanity.com/paper/2207.05209/) (2022)
- [Message Passing Neural PDE Solvers](https://mlsanity.com/paper/2202.03376/) (2022)
- [Choose a Transformer: Fourier or Galerkin](https://mlsanity.com/paper/2105.14995/) (2021)
- [GNOT: A General Neural Operator Transformer for Operator Learning](https://mlsanity.com/paper/2302.14376/) (2023)
- [Transolver: A Fast Transformer Solver for PDEs on General Geometries](https://mlsanity.com/paper/2402.02366/) (2024)
- [Geometry-Informed Neural Operator for Large-Scale 3D PDEs](https://mlsanity.com/paper/2309.00583/) (2023)
- [Transformer for Partial Differential Equations' Operator Learning](https://mlsanity.com/paper/2205.13671/) (2022)
- [Multiwavelet-based Operator Learning for Differential Equations](https://mlsanity.com/paper/2109.13459/) (2021)
- [U-NO: U-shaped Neural Operators](https://mlsanity.com/paper/2204.11127/) (2022)
- [Convolutional Neural Operators for robust and accurate learning of PDEs](https://mlsanity.com/paper/2302.01178/) (2023)
- [Networks beyond pairwise interactions: structure and dynamics](https://mlsanity.com/paper/2006.01764/) (2020)
- [Transolver++: An Accurate Neural Solver for PDEs on Million-Scale Geometries](https://mlsanity.com/paper/2502.02414/) (2025)
- [Random Walks on Simplicial Complexes and the normalized Hodge 1-Laplacian](https://mlsanity.com/paper/1807.05044/) (2018)
- [Operator Learning with Neural Fields: Tackling PDEs on General Geometries](https://mlsanity.com/paper/2306.07266/) (2023)
- [Principled Simplicial Neural Networks for Trajectory Prediction](https://mlsanity.com/paper/2102.10058/) (2021)
- [Geometry Aware Operator Transformer as an Efficient and Accurate Neural Surrogate for PDEs on Arbitrary Domains](https://mlsanity.com/paper/2505.18781/) (2025)
- [Pretraining Codomain Attention Neural Operators for Solving Multiphysics PDEs](https://mlsanity.com/paper/2403.12553/) (2024)
- [Domain Agnostic Fourier Neural Operators](https://mlsanity.com/paper/2305.00478/) (2023)
- [E(n) Equivariant Message Passing Simplicial Networks](https://mlsanity.com/paper/2305.07100/) (2023)
- [RIGNO: A Graph-based framework for robust and accurate operator learning for PDEs on arbitrary domains](https://mlsanity.com/paper/2501.19205/) (2025)
- [Beyond Regular Grids: Fourier-Based Neural Operators on Arbitrary Domains](https://mlsanity.com/paper/2305.19663/) (2023)
- [Topological Neural Networks go Persistent, Equivariant, and Continuous](https://mlsanity.com/paper/2406.03164/) (2024)
- [NUNO: A General Framework for Learning Parametric PDEs with Non-Uniform Data](https://mlsanity.com/paper/2305.18694/) (2023)
- [Diffeomorphism Neural Operator for various domains and parameters of partial differential equations](https://mlsanity.com/paper/2402.12475/) (2024)
- [Operator Learning with Domain Decomposition for Geometry Generalization in PDE Solving](https://mlsanity.com/paper/2504.00510/) (2025)
- [Topological Neural Operators](https://mlsanity.com/paper/2606.09806/) (2026)

## 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.