Dynamic kuramoto-hodge operators improve pde learning on complex shapes
Dynamic Kuramoto-Hodge Operators for PDEs on Complex Geometries and Topologies
Machine LearningArtificial Intelligence
Summary
Simulating physical processes described by partial differential equations (PDEs) on complicated shapes is challenging because interactions happen in different ways across points, edges, and surfaces, and the overall shape can affect behavior. The authors developed a new method called Dynamic Kuramoto-Hodge Operator that respects these shape constraints and learns how interactions change based on the problem. It uses concepts from topology to control where information flows and learns the details of how it flows for each case. Their approach made more accurate and efficient predictions for various physics problems on complex forms compared to existing methods.
What this means in practice
- •For simulation engineers: Improve accuracy and reduce model size when predicting physics phenomena on complex geometries like porous materials and magnetics setups.
- •For computational fluid dynamics teams: Enhance modeling of fluid and heat flow on complex curved surfaces including torus-like shapes with fewer computational resources.
Authors
Xiang Li, Yue Song
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.