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

翻译：本文从函数空间角度研究几何网格上物理场方程的解算子。我们揭示霍奇正交性通过将不可学习的拓扑自由度与可学习的几何动力学分离，从根本上解决了频谱干扰问题，从而实现了局限于结构保持子空间的加性逼近。基于霍奇理论和算子分裂，我们推导出一个原理性的算子级分解。最终形成一种混合欧拉-拉格朗日架构，其代数级归纳偏置被称为霍奇频谱对偶性（HSD）。在我们的框架中，使用离散微分形式捕捉拓扑主导分量，并通过正交辅助环境空间表示复杂的局部动力学。该方法在几何网格上实现了卓越的准确性和效率，并增强了对物理不变量的保真度。我们的代码可于 https://github.com/ContinuumCoder/Hodge-Spectral-Duality 获取。