基於霍奇分解的拓撲保持神經算子學習

摘要

本文從函數空間的視角，探討幾何網格上物理場方程的解算子。我們揭示出，霍奇正交性透過將不可學習的拓撲自由度從可學習的幾何動力學中分離，從根本上解決了譜干擾問題，從而實現局限於結構保持子空間的加性近似。基於霍奇理論與算子分裂，我們推導出一個基於原則的算子層級分解。其成果是一種混合歐拉-拉格朗日架構，內含我們稱之為霍奇譜對偶（HSD）的代數層級歸納偏置。在我們的框架中，我們利用離散微分形式捕捉拓撲主導的分量，並透過一個正交的輔助環境空間來表示複雜的局部動力學。我們的方法在幾何圖形上實現了卓越的準確性與效率，同時對物理不變量具有更高的忠實度。我們的程式碼可於 https://github.com/ContinuumCoder/Hodge-Spectral-Duality 取得。

English

In this paper, we study solution operators of physical field equations on geometric meshes from a function-space perspective. We reveal that Hodge orthogonality fundamentally resolves spectral interference by isolating unlearnable topological degrees of freedom from learnable geometric dynamics, enabling an additive approximation confined to structure-preserving subspaces. Building on Hodge theory and operator splitting, we derive a principled operator-level decomposition. The result is a Hybrid Eulerian-Lagrangian architecture with an algebraic-level inductive bias we call Hodge Spectral Duality (HSD). In our framework, we use discrete differential forms to capture topology-dominated components and an orthogonal auxiliary ambient space to represent complex local dynamics. Our method achieves superior accuracy and efficiency on geometric graphs with enhanced fidelity to physical invariants. Our code is available at https://github.com/ContinuumCoder/Hodge-Spectral-Duality