Hodge分解为神经算子学习提供了严格的数学框架，解决了物理场模拟中拓扑与几何动力学的分离难题。做计算物理、几何深度学习或科学机器学习的团队，可以直接用开源的HSD架构提升模型对物理不变量的保真度。
本文从函数空间视角研究几何网格上物理场方程的解算子，揭示了Hodge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，从根本上解决频谱干扰问题，从而实现结构保持子空间内的加性逼近。基于Hodge理论和算子分裂，作者推导出原则性的算子级分解，提出一种混合欧拉-拉格朗日架构，并引入称为Hodge谱对偶（HSD）的代数级归纳偏置。该方法使用离散微分形式捕捉拓扑主导成分，并用正交辅助环境空间表示复杂的局部动力学。实验表明，该方法在几何图上实现了更高的精度和效率，并增强了对物理不变量的保真度。代码已开源。
Topology-Preserving Neural Operator Learning via Hodge Decomposition
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