Topology-Preserving Neural Operator Learning via Hodge Decomposition
It addresses the challenge of learning solution operators for physical field equations on geometric meshes, providing a principled approach to preserve topological structure.
The paper introduces a topology-preserving neural operator learning method using Hodge decomposition to isolate topological degrees of freedom from geometric dynamics, achieving superior accuracy and efficiency on geometric graphs with enhanced fidelity to physical invariants.
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