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

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.