# Topology-Preserving Neural Operator Learning via Hodge Decomposition

arXiv: https://arxiv.org/abs/2605.13834 · HTML: https://mlsanity.com/paper/2605.13834/

## 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 (most cited first, 24 of 24)

- [How Powerful are Graph Neural Networks?](https://mlsanity.com/paper/1810.00826/) (2018)
- [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)
- [Deeper Insights into Graph Convolutional Networks for Semi-Supervised Learning](https://mlsanity.com/paper/1801.07606/) (2018)
- [Geometric deep learning: going beyond Euclidean data](https://mlsanity.com/paper/1611.08097/) (2016)
- [Learning Mesh-Based Simulation with Graph Networks](https://mlsanity.com/paper/2010.03409/) (2020)
- [Neural Operator: Graph Kernel Network for Partial Differential Equations](https://mlsanity.com/paper/2003.03485/) (2020)
- [On the Bottleneck of Graph Neural Networks and its Practical Implications](https://mlsanity.com/paper/2006.05205/) (2020)
- [GNOT: A General Neural Operator Transformer for Operator Learning](https://mlsanity.com/paper/2302.14376/) (2023)
- [Gauge Equivariant Convolutional Networks and the Icosahedral CNN](https://mlsanity.com/paper/1902.04615/) (2019)
- [A Note on Over-Smoothing for Graph Neural Networks](https://mlsanity.com/paper/2006.13318/) (2020)
- [Simplicial Neural Networks](https://mlsanity.com/paper/2010.03633/) (2020)
- [Topological Deep Learning: Going Beyond Graph Data](https://mlsanity.com/paper/2206.00606/) (2022)
- [Operator Learning with Neural Fields: Tackling PDEs on General Geometries](https://mlsanity.com/paper/2306.07266/) (2023)
- [DrivAerNet++: A Large-Scale Multimodal Car Dataset with Computational Fluid Dynamics Simulations and Deep Learning Benchmarks](https://mlsanity.com/paper/2406.09624/) (2024)
- [Coordinate Independent Convolutional Networks -- Isometry and Gauge Equivariant Convolutions on Riemannian Manifolds](https://mlsanity.com/paper/2106.06020/) (2021)
- [Position: Topological Deep Learning is the New Frontier for Relational Learning](https://mlsanity.com/paper/2402.08871/) (2024)
- [Improved Operator Learning by Orthogonal Attention](https://mlsanity.com/paper/2310.12487/) (2023)
- [Topological Deep Learning: A Review of an Emerging Paradigm](https://mlsanity.com/paper/2302.03836/) (2023)
- [TopoX: A Suite of Python Packages for Machine Learning on Topological Domains](https://mlsanity.com/paper/2402.02441/) (2024)
- [BScNets: Block Simplicial Complex Neural Networks](https://mlsanity.com/paper/2112.06826/) (2021)
- [Topological Signal Processing and Learning: Recent Advances and Future Challenges](https://mlsanity.com/paper/2412.01576/) (2024)
- [Weighted Combinatorial Laplacian and its Application to Coverage Repair in Sensor Networks](https://mlsanity.com/paper/2312.04825/) (2023)
- [Resolving Oversmoothing with Opinion Dissensus](https://mlsanity.com/paper/2501.19089/) (2025)

## Built on by (most cited first, 3 of 3)

- [Beyond Arbitrary Geometry: Topology Generalization In neural PDE Operators](https://mlsanity.com/paper/2609.05860/) (2026)
- [Dynamic Kuramoto-Hodge Operators for PDEs on Complex Geometries and Topologies](https://mlsanity.com/paper/2609.33693/) (2026)
- [Learnable Hodge-Laplacian Driven Mixture of Experts for Steady-PDE Surrogates on Coarse Unstructured Meshes](https://mlsanity.com/paper/2610.20585/) (2026)

## 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