Title: Enhancing Neural Operator Learning with Topology Preservation through Hodge Decomposition

Abstract: This study investigates the solution operators governing physical field equations on geometric meshes, analyzed through the lens of function spaces. We demonstrate that Hodge orthogonality effectively mitigates spectral interference by separating unlearnable topological degrees of freedom from learnable geometric dynamics. This separation facilitates an additive approximation restricted to subspaces that preserve structure. Leveraging operator splitting and Hodge theory, we establish a rigorous operator-level decomposition. This approach yields a Hybrid Eulerian-Lagrangian architecture characterized by a specific algebraic inductive bias, which we term Hodge Spectral Duality (HSD). Within this framework, discrete differential forms are employed to model topology-dominated components, while an orthogonal auxiliary ambient space captures intricate local dynamics. Consequently, our method delivers enhanced accuracy and efficiency on geometric graphs, ensuring greater fidelity to physical invariants. The source code is accessible at https://github.com/ContinuumCoder/Hodge-Spectral-Duality

Source: arXiv Generated at: 2026-06-02 00:00:00 UTC