动态Kuramoto-Hodge算子用于复杂几何与拓扑上的PDE
Dynamic Kuramoto-Hodge Operators for PDEs on Complex Geometries and Topologies
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中文总结 AI 辅助
本文提出动态Kuramoto-Hodge算子（DKHO），将拓扑约束与学习动力学结合，在复杂几何上学习PDE算子，显著降低预测误差并减少参数需求。
中文摘要 AI 辅助
在复杂域上学习PDE算子需要捕捉顶点、边和面上的场之间的相互作用，以及由拓扑塑造的全局响应。现有的神经算子能够适应不规则几何，但往往忽视这些不同的场支撑或其条件依赖的耦合。我们引入了动态Kuramoto-Hodge算子（DKHO），它将拓扑约束的相互作用与学习到的协调相结合。DKHO在原始上链支撑上编码条件，通过构成Dirac算子的边界和上边界算子演化受Kuramoto启发的关联状态，并在正交Hodge子空间中解码非调和与调和响应。因此，拓扑决定了信息可以在何处流动，而学习到的动力学则适应每个PDE实例如何交换信息。在多孔介质Darcy流、环面输运-扩散和腔体静磁学中，DKHO-large相比领先基线平均减少约61%的预测误差，而DKHO-small仅使用11.5%-24.3%的参数数量仍保持竞争力。这些结果表明，将拓扑结构与自适应动力学耦合为在复杂几何和拓扑上进行准确且参数高效的PDE算子学习提供了有效的归纳偏置。
英文摘要
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.
发表机构
- Tsinghua University（清华大学）
- Beijing University of Chemical Technology（北京化工大学）
机构由 AI 辅助整理，请以论文原文为准。
