Topology-Preserving Neural Operator Learning via Hodge Decomposition
Summary
This paper proposes a topology-preserving neural operator learning method using Hodge decomposition to separate topological and geometric components, improving accuracy and efficiency on geometric meshes.
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Paper page - Topology-Preserving Neural Operator Learning via Hodge Decomposition
Source: https://huggingface.co/papers/2605.13834
Abstract
Physical field equations on geometric meshes are analyzed through Hodge theory to develop a hybrid Eulerian-Lagrangian architecture that improves accuracy and efficiency by separating topological and geometric components.
In this paper, we study solution operators of physical field equations on geometric meshes from a function-space perspective. We reveal thatHodge orthogonalityfundamentally resolvesspectral interferenceby isolating unlearnabletopological degrees of freedomfrom learnablegeometric dynamics, enabling an additive approximation confined to structure-preserving subspaces. Building on Hodge theory and operator splitting, we derive a principledoperator-level decomposition. The result is a Hybrid Eulerian-Lagrangian architecture with an algebraic-level inductive bias we callHodge Spectral Duality(HSD). In our framework, we usediscrete differential formsto capture topology-dominated components and anorthogonal auxiliary ambient spaceto represent complex local dynamics. Our method achieves superior accuracy and efficiency ongeometric graphswith enhanced fidelity tophysical invariants. Our code is available at https://github.com/ContinuumCoder/Hodge-Spectral-Duality
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