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 <a href=\"https://github.com/ContinuumCoder/Hodge-Spectral-Duality\" rel=\"nofollow\">https://github.com/ContinuumCoder/Hodge-Spectral-Duality</a>.</p>\n","updatedAt":"2026-05-15T02:16:28.906Z","author":{"_id":"69b322027c89a6c6ac5e2474","avatarUrl":"/avatars/811a981d2c59e28f67c3afe6b5c2961e.svg","fullname":"Tao Zhong","name":"n3il666","type":"user","isPro":false,"isHf":false,"isHfAdmin":false,"isMod":false,"isUserFollowing":false}},"numEdits":0,"identifiedLanguage":{"language":"en","probability":0.8732939958572388},"editors":["n3il666"],"editorAvatarUrls":["/avatars/811a981d2c59e28f67c3afe6b5c2961e.svg"],"reactions":[],"isReport":false}}],"primaryEmailConfirmed":false,"paper":{"id":"2605.13834","authors":[{"_id":"6a053cb8b1a8cbabc9f087d2","name":"Dongzhe Zheng","hidden":false},{"_id":"6a053cb8b1a8cbabc9f087d3","name":"Tao Zhong","hidden":false},{"_id":"6a053cb8b1a8cbabc9f087d4","name":"Christine Allen-Blanchette","hidden":false}],"publishedAt":"2026-05-13T00:00:00.000Z","submittedOnDailyAt":"2026-05-15T00:00:00.000Z","title":"Topology-Preserving Neural Operator Learning via Hodge Decomposition","submittedOnDailyBy":{"_id":"69b322027c89a6c6ac5e2474","avatarUrl":"/avatars/811a981d2c59e28f67c3afe6b5c2961e.svg","isPro":false,"fullname":"Tao Zhong","user":"n3il666","type":"user","name":"n3il666"},"summary":"In this paper, we study solution operators of physical field equations on geometric meshes from a function-space perspective. 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Topology-Preserving Neural Operator Learning via Hodge Decomposition
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.
AI-generated summary
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
Community
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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