{"paper":{"title":"Exact Fixed-Point Constraints in Neural-ODEs with Provable Universality","license":"http://creativecommons.org/licenses/by/4.0/","headline":"Neural-ODEs can approximate arbitrary velocity fields while exactly forcing velocity to zero at any finite set of prescribed points.","cross_cats":["cs.LG","physics.bio-ph"],"primary_cat":"cond-mat.dis-nn","authors_text":"Diego Febbe, Duccio Fanelli, Feliciano Giuseppe Pacifico, Lorenzo Buffoni, Lorenzo Chicchi, Raffaele Marino","submitted_at":"2026-05-11T14:10:31Z","abstract_excerpt":"We introduce a technique that enables Neural-ODEs to approximate arbitrary velocity fields with a priori planted fixed-points. Specifically, a recipe is given to explicitly accommodate for a finite collection of points in the reference multi-dimensional space of the Neural-ODE where the velocity field is exactly equal to zero. In this way, the gradient-based training is rigorously constrained inside the prescribed hypothesis class while leaving the expressive power of the Neural-ODE unaltered. We rigorously prove the universality of the Neural-ODE under any local constraints in the velocity fi"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We rigorously prove the universality of the Neural-ODE under any local constraints in the velocity field and give a computationally convenient way of imposing the fixed points.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"That the explicit accommodation for fixed points can be introduced without reducing the expressive power of the underlying Neural-ODE architecture.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"A technique plants exact fixed points in Neural-ODE velocity fields with a rigorous proof that universality is preserved under local constraints.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Neural-ODEs can approximate arbitrary velocity fields while exactly forcing velocity to zero at any finite set of prescribed points.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"f516d737a038bd4a5a965319dcaadeb6d08e25633bdbcef27050fd26c22600bb"},"source":{"id":"2605.10613","kind":"arxiv","version":1},"verdict":{"id":"75c9c7d9-f6c2-43c4-918e-800a2aa54c5e","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-12T05:16:53.050561Z","strongest_claim":"We rigorously prove the universality of the Neural-ODE under any local constraints in the velocity field and give a computationally convenient way of imposing the fixed points.","one_line_summary":"A technique plants exact fixed points in Neural-ODE velocity fields with a rigorous proof that universality is preserved under local constraints.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"That the explicit accommodation for fixed points can be introduced without reducing the expressive power of the underlying Neural-ODE architecture.","pith_extraction_headline":"Neural-ODEs can approximate arbitrary velocity fields while exactly forcing velocity to zero at any finite set of prescribed points."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.10613/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"claim_evidence","ran_at":"2026-05-20T05:42:00.804160Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"ai_meta_artifact","ran_at":"2026-05-19T14:40:33.025652Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_title_agreement","ran_at":"2026-05-19T11:01:17.521371Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T09:06:16.142850Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"3ea4b50293f365bd1ab2c7a6d19fedbb086e6e7102f7d1033999049ab7d3a034"},"references":{"count":32,"sample":[{"doi":"","year":null,"title":"Advances in Neural Information Processing Systems (NeurIPS) , year =","work_id":"a4bda9df-6b80-4026-b380-b13ae65db551","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2024,"title":"Multiplicity of Time Scales in Complex Systems: Challenges for Sciences and Communication II , editor =","work_id":"9b9905ba-56cd-4e0a-8838-70329103f076","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2025,"title":"Daniel A. 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