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：本文从函数空间视角研究几何网格上物理场方程的求解算子。我们揭示了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相分离，从根本上解决了谱干扰问题，从而实现了限定于保结构子空间内的加性逼近。基于霍奇理论与算子分裂方法，我们推导出一个原则性的算子级分解。最终形成一种混合欧拉-拉格朗日架构，其中包含我们称为"霍奇谱对偶"（HSD）的代数级归纳偏置。在该框架中，我们利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征复杂的局部动力学。实验表明，我们的方法在几何图网络上展现出更优的精度与效率，同时增强了对物理不变量的保真度。代码开源地址：https://github.com/ContinuumCoder/Hodge-Spectral-Duality