{"paper":{"title":"A Unified Framework for Structured Flow Modeling: From Continuous Fields to Data-Driven Representations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Structured flows in dynamical systems can be modeled uniformly by connecting continuous Helmholtz-Hodge decompositions to discrete graph representations.","cross_cats":["cs.LG"],"primary_cat":"physics.data-an","authors_text":"Diego Casadei","submitted_at":"2026-05-18T11:52:09Z","abstract_excerpt":"Many dynamical systems can be described in terms of structured flows combining source/sink behavior, cyclic dynamics, and topology-constrained transport. These features arise across a wide range of domains, including physical, engineered, and data-driven systems. This work provides a unified perspective on such systems by connecting continuous formulations based on the Helmholtz-Hodge decomposition with discrete and data-driven representations.\n  We review the recently proposed Graph Vector Field (GVF) framework, which enables a decomposition of complex dynamics into gradient, curl, and harmon"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"This work provides a unified perspective on such systems by connecting continuous formulations based on the Helmholtz-Hodge decomposition with discrete and data-driven representations, supported by a cross-domain validation strategy that leverages datasets from well-understood physical systems.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"That the recently proposed Graph Vector Field framework and the introduced hierarchy of models (parametric conditional, linear graph dynamical systems, reduced Hodge representations) can be systematically evaluated for trade-offs in complexity, interpretability, and performance using cross-domain physical datasets to verify correctness independently of the target application.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Proposes a unified framework linking Helmholtz-Hodge decompositions to Graph Vector Field models and reduced variants, with cross-domain validation on physical datasets to balance expressivity and tractability.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Structured flows in dynamical systems can be modeled uniformly by connecting continuous Helmholtz-Hodge decompositions to discrete graph representations.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"c3bfb5c03d28acb9d3172cc3dda63579021fa90cad7bb3bd6ad1d3094b826de9"},"source":{"id":"2605.18250","kind":"arxiv","version":1},"verdict":{"id":"a48d5a5b-a5a9-4e6b-844d-5221a97eaab8","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-19T23:48:44.948117Z","strongest_claim":"This work provides a unified perspective on such systems by connecting continuous formulations based on the Helmholtz-Hodge decomposition with discrete and data-driven representations, supported by a cross-domain validation strategy that leverages datasets from well-understood physical systems.","one_line_summary":"Proposes a unified framework linking Helmholtz-Hodge decompositions to Graph Vector Field models and reduced variants, with cross-domain validation on physical datasets to balance expressivity and tractability.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"That the recently proposed Graph Vector Field framework and the introduced hierarchy of models (parametric conditional, linear graph dynamical systems, reduced Hodge representations) can be systematically evaluated for trade-offs in complexity, interpretability, and performance using cross-domain physical datasets to verify correctness independently of the target application.","pith_extraction_headline":"Structured flows in dynamical systems can be modeled uniformly by connecting continuous Helmholtz-Hodge decompositions to discrete graph representations."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.18250/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"doi_compliance","ran_at":"2026-05-20T00:02:59.036445Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_title_agreement","ran_at":"2026-05-20T00:01:20.513568Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"ai_meta_artifact","ran_at":"2026-05-19T23:33:35.269820Z","status":"skipped","version":"1.0.0","findings_count":0},{"name":"claim_evidence","ran_at":"2026-05-19T23:21:58.994359Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"2f37032feceac502dd8b45538215de6fd4a4c9d83d54bbf5c2808d0dc2e33fb4"},"references":{"count":14,"sample":[{"doi":"","year":2026,"title":"Graph vector field: A unified framework for multimodal health risk assessment from heterogeneous wearable and environ- mental data streams, 2026","work_id":"d1f38e12-7f6f-492b-95c0-2441e1e98208","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2002,"title":"Cambridge University Press, 2002","work_id":"3ef1517f-ae63-42e5-9389-268b8f853d95","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1987,"title":"Thomas J. R. Hughes.The Finite Element Method: Linear Static and Dynamic Finite Element Analysis. Prentice Hall, 1987","work_id":"9c94b492-ff66-4722-835e-1b43ee0cbdf3","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2014,"title":"Klaus-Jürgen Bathe.Finite Element Procedures, 2nd Ed.Pearson Education, 2014","work_id":"8d4a6124-b401-4299-97e1-aa503965bc6a","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"10.1137/18m1223101","year":2020,"title":"Hodge laplacians on graphs.SIAM Review, 62(3):685–715, 2020","work_id":"8cb5f451-281c-4714-83bc-536172a51362","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":14,"snapshot_sha256":"45c78284a7ecc6800ac651a7138bf56b4e2073bd8dfaea779bf861e4d012d2ee","internal_anchors":0},"formal_canon":{"evidence_count":2,"snapshot_sha256":"192bd88a238764173d5636da3bc962dec3a699c5bde6fcce75bc61e2b933117e"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}