Topology-Preserving Neural Operator Learning via Hodge Decomposition
Dongzhe Zheng, Tao Zhong, Christine Allen-Blanchette
TLDR
This paper introduces a topology-preserving neural operator learning method using Hodge decomposition to model physical field equations on geometric meshes.
Key contributions
- Reveals Hodge orthogonality isolates unlearnable topological degrees of freedom from learnable geometric dynamics.
- Derives a principled operator-level decomposition using Hodge theory and operator splitting.
- Introduces Hodge Spectral Duality (HSD), a Hybrid Eulerian-Lagrangian architecture with algebraic bias.
- Achieves superior accuracy, efficiency, and fidelity to physical invariants on geometric graphs.
Why it matters
Current methods struggle with spectral interference in physical field equations on geometric meshes. This paper offers a novel approach that resolves this by isolating topological and geometric components, leading to more accurate and efficient neural operators. It enhances fidelity to physical invariants.
Original Abstract
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
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