{"paper":{"title":"Topology-Preserving Neural Operator Learning via Hodge Decomposition","license":"http://creativecommons.org/licenses/by/4.0/","headline":"Hodge orthogonality isolates unlearnable topological degrees of freedom from learnable geometric dynamics in neural operators.","cross_cats":["cs.AI","cs.CG"],"primary_cat":"cs.LG","authors_text":"Christine Allen-Blanchette, Dongzhe Zheng, Tao Zhong","submitted_at":"2026-05-13T17:56:23Z","abstract_excerpt":"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 Dual"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"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.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"That the discrete Hodge decomposition cleanly isolates topological components from geometric dynamics in the operator without introducing discretization artifacts or requiring problem-specific tuning on geometric meshes.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Hodge Spectral Duality provides a topology-preserving neural operator by isolating unlearnable topological components via Hodge orthogonality and operator splitting.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Hodge orthogonality isolates unlearnable topological degrees of freedom from learnable geometric dynamics in neural operators.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"9aeb1ea525c5a48e180351dc5f8c31bd2ce9db424031d1ecb3f64d15fa746f79"},"source":{"id":"2605.13834","kind":"arxiv","version":1},"verdict":{"id":"a47b478d-39e3-4e0e-ae59-b9fc73cb4a9c","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-14T19:01:47.698354Z","strongest_claim":"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.","one_line_summary":"Hodge Spectral Duality provides a topology-preserving neural operator by isolating unlearnable topological components via Hodge orthogonality and operator splitting.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"That the discrete Hodge decomposition cleanly isolates topological components from geometric dynamics in the operator without introducing discretization artifacts or requiring problem-specific tuning on geometric meshes.","pith_extraction_headline":"Hodge orthogonality isolates unlearnable topological degrees of freedom from learnable geometric dynamics in neural operators."},"references":{"count":14,"sample":[{"doi":"","year":2006,"title":"On the bottleneck of graph neural networks and its practical implications.arXiv:2006.05205","work_id":"4dc88c40-c0d4-4c5d-a0f9-5c863c1fabdc","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"Splitting methods for differential equations.arXiv preprint arXiv:2401.01722,","work_id":"c2ced08f-09ee-4067-bce9-129da8f1d3f4","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2006,"title":"A note on over-smoothing for graph neural networks","work_id":"357166af-f327-48bd-8c0f-de9fc925688c","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2010,"title":"Sim- plicial neural networks.arXiv preprint arXiv:2010.03633","work_id":"d9db9dfd-e89f-44af-920b-dce5439a70a2","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"Dey, Soham Mukherjee, Shreyas N","work_id":"401a4ce8-6e1d-455b-b436-f2ba15c8053c","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":14,"snapshot_sha256":"9a9540772bd8442f458a6119e3b306ca2465c5f07341c876be62eea64cb72251","internal_anchors":2},"formal_canon":{"evidence_count":1,"snapshot_sha256":"1032eaadf41908bf54b41c51ca3551500871ad9672ed9b6fb3dc8121a60c7e11"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}