{"paper":{"title":"SplineFlow: Flow Matching for Dynamical Systems with B-Spline Interpolants","license":"http://creativecommons.org/licenses/by/4.0/","headline":"SplineFlow uses B-spline interpolation to build stable conditional paths for flow matching in dynamical systems.","cross_cats":[],"primary_cat":"cs.LG","authors_text":"Pietro Li\\`o, Santanu Subhash Rathod, Xiao Zhang","submitted_at":"2026-01-30T15:19:48Z","abstract_excerpt":"Flow matching is a scalable generative framework for characterizing continuous normalizing flows with wide-range applications. However, current state-of-the-art methods are not well-suited for modeling dynamical systems, as they construct conditional paths using linear interpolants that may not capture the underlying state evolution, especially when learning higher-order dynamics from irregular sampled observations. Constructing unified paths that satisfy multi-marginal constraints across observations is challenging, since na\\\"ive higher-order polynomials tend to be unstable and oscillatory. W"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We introduce SplineFlow, a theoretically grounded flow matching algorithm that jointly models conditional paths across observations via B-spline interpolation. Specifically, SplineFlow exploits the smoothness and stability of B-spline bases to learn the complex underlying dynamics in a structured manner while ensuring the multi-marginal requirements are met.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"That B-spline bases of appropriate order can capture higher-order dynamics from irregular observations without introducing instability or violating the required multi-marginal constraints, an assumption stated in the abstract but not derived in detail here.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"SplineFlow uses B-spline interpolation inside flow matching to jointly construct stable conditional paths that satisfy multi-marginal constraints for dynamical systems with irregular observations.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"SplineFlow uses B-spline interpolation to build stable conditional paths for flow matching in dynamical systems.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"8dfe25fad14ee97feaed40ebce01da60eed9523b07007301dff5436d93155a0a"},"source":{"id":"2601.23072","kind":"arxiv","version":2},"verdict":{"id":"2f7edd4c-50b6-4194-aa10-ec9cd99e533a","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-16T09:24:44.198954Z","strongest_claim":"We introduce SplineFlow, a theoretically grounded flow matching algorithm that jointly models conditional paths across observations via B-spline interpolation. Specifically, SplineFlow exploits the smoothness and stability of B-spline bases to learn the complex underlying dynamics in a structured manner while ensuring the multi-marginal requirements are met.","one_line_summary":"SplineFlow uses B-spline interpolation inside flow matching to jointly construct stable conditional paths that satisfy multi-marginal constraints for dynamical systems with irregular observations.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"That B-spline bases of appropriate order can capture higher-order dynamics from irregular observations without introducing instability or violating the required multi-marginal constraints, an assumption stated in the abstract but not derived in detail here.","pith_extraction_headline":"SplineFlow uses B-spline interpolation to build stable conditional paths for flow matching in dynamical systems."},"references":{"count":19,"sample":[{"doi":"","year":null,"title":"Building Normalizing Flows with Stochastic Interpolants","work_id":"83654cbd-832b-4269-bec9-26faa87ececd","ref_index":1,"cited_arxiv_id":"2209.15571","is_internal_anchor":true},{"doi":"","year":1994,"title":"Random dynamical systems","work_id":"62e262ff-705e-4512-831c-347f5fe416c9","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1973,"title":"Good approximation by splines with variable knots","work_id":"b9a5a32c-87a1-49f5-9b76-9e6574d44018","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"A survey of the","work_id":"1406850e-f2ef-4676-b930-15f522dc644d","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"Flow matching meets biology and life science: A survey.arXiv preprint arXiv:2507.17731,","work_id":"886076b6-603b-48b7-af21-da9fa5631deb","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":19,"snapshot_sha256":"12953fd84d50fbabde0f05c84041623744503083260cce1fb10e30620d2ebdcc","internal_anchors":5},"formal_canon":{"evidence_count":1,"snapshot_sha256":"a91fa05bea7d062908315118ec5d9f82c675491f82810862910cef48708b8cb7"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}