Topology-Preserving Neural Operator Learning via Hodge Decomposition
2026-05-13 • Machine Learning
Machine LearningArtificial IntelligenceComputational Geometry
AI summaryⓘ
The authors study how to solve physical field equations on complex geometric meshes by looking at them through the lens of function spaces. They use a mathematical idea called Hodge orthogonality to separate parts of the problem that are hard to learn (topological factors) from those that are easier to model (geometric factors). This leads to a new method combining Eulerian and Lagrangian approaches, which better preserves important physical properties. Their approach uses discrete differential forms to handle topology and an extra space to capture detailed local behavior. They show this method works more accurately and efficiently on geometric graphs.
Hodge orthogonalityspectral interferencetopological degrees of freedomEulerian-Lagrangian methodsoperator splittingdiscrete differential formsgeometric meshesfunction spacesphysical invariants
Authors
Dongzhe Zheng, Tao Zhong, Christine Allen-Blanchette
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