# Dynamic Kuramoto-Hodge Operators for PDEs on Complex Geometries and Topologies

arXiv: https://arxiv.org/abs/2609.33693 · HTML: https://mlsanity.com/paper/2609.33693/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 13 of its 29 references that are in the map.
- First posted on arXiv 27 Sep 2026.

## Builds on (most cited first, 13 of 13)

- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [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)
- [Geometry-Informed Neural Operator for Large-Scale 3D PDEs](https://mlsanity.com/paper/2309.00583/) (2023)
- [Simplicial Neural Networks](https://mlsanity.com/paper/2010.03633/) (2020)
- [Topological Deep Learning: Going Beyond Graph Data](https://mlsanity.com/paper/2206.00606/) (2022)
- [Domain Agnostic Fourier Neural Operators](https://mlsanity.com/paper/2305.00478/) (2023)
- [Artificial Kuramoto Oscillatory Neurons](https://mlsanity.com/paper/2410.13821/) (2024)
- [Kuramoto Orientation Diffusion Models](https://mlsanity.com/paper/2509.15328/) (2025)
- [Topological Neural Operators](https://mlsanity.com/paper/2606.09806/) (2026)
- [Topology-Preserving Neural Operator Learning via Hodge Decomposition](https://mlsanity.com/paper/2605.13834/) (2026)
- [Winfree Oscillatory Neural Network](https://mlsanity.com/paper/2605.20922/) (2026)
- [The Kuramoto Neural Operator: Learning to Solve PDEs via Coupled Oscillator Dynamics](https://mlsanity.com/paper/2608.10234/) (2026)

## Abstract (arXiv)

Learning PDE operators on complex domains requires capturing interactions among fields on vertices, edges, and faces, alongside global responses shaped by topology. Existing neural operators accommodate irregular geometries but often overlook these distinct field supports or their condition-dependent coupling. We introduce the Dynamic Kuramoto--Hodge Operator (DKHO), which combines topology-constrained interactions with learned coordination. DKHO encodes conditions on their native cochain supports, evolves Kuramoto-inspired relation states through the boundary and coboundary operators that compose the Dirac operator, and decodes non-harmonic and harmonic responses in orthogonal Hodge subspaces. Topology thus determines where information can flow, while learned dynamics adapts how it is exchanged to each PDE instance. Across porous-medium Darcy flow, torus transport--diffusion, and cavity magnetostatics, DKHO-large reduces prediction error by approximately 61% on average over leading baselines, while DKHO-small remains competitive using only 11.5--24.3% as many parameters. These results suggest that coupling topological structure with adaptive dynamics provides an effective inductive bias for accurate and parameter-efficient PDE operator learning on complex geometries and topologies.