aiqbench analysis
This research introduces the Dynamic Kuramoto-Hodge Operator (DKHO), a novel neural architecture designed to solve partial differential equations on complex geometries by explicitly integrating topological constraints with adaptive dynamics. By leveraging Hodge theory to manage information flow across vertices, edges, and faces, the method achieves significant error reduction and parameter efficiency compared to existing baselines for tasks like porous-medium flow and magnetostatics. The work represents a meaningful advancement in scientific machine learning, demonstrating how topological inductive biases can enhance the accuracy of operator learning on irregular domains.
Dynamic Kuramoto-Hodge Operators for PDEs on Complex Geometries and Topologies
arXiv:2609.33693v1 Announce Type: new Abstract: 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.
Abstract: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.
Machine Learning (cs.LG); Artificial Intelligence (cs.AI)
arXiv:2609.33693 [cs.LG]
(or arXiv:2609.33693v1 [cs.LG] for this version)
https://doi.org/10.48550/arXiv.2609.33693 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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Full text under the publisher's open licence · 260 words
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