{"paper":{"title":"Lagrangian Flow Matching: A Least-Action Framework for Principled Path Design","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Minimizing a general Lagrangian's action under the continuity equation produces a family of probability paths for flow matching.","cross_cats":[],"primary_cat":"cs.LG","authors_text":"Junzhe Zhang, Shukai Du, Yiming Li","submitted_at":"2026-05-14T21:05:03Z","abstract_excerpt":"Flow matching trains a neural velocity field by regression against a target velocity associated with a prescribed probability path connecting a simple initial distribution to the data distribution. A central design choice is the path itself. Existing constructions, including rectified and optimal-transport-based paths, transport samples along straight lines between coupled endpoints and thus cover only a narrow class of dynamics. We observe that this corresponds to the simplest case of the least-action principle in classical mechanics, in which the kinetic Lagrangian yields free-particle strai"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We show that this dynamic problem admits an equivalent static optimal transport (OT) formulation, yielding a family of simulation-free training objectives that recover OT-based flow matching as the kinetic special case and the trigonometric variance-preserving diffusion path as the harmonic-oscillator case.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The assumption that minimizing the action of an arbitrary Lagrangian subject to the continuity equation and fixed marginal endpoints produces a valid probability path whose induced velocity field can be regressed by a neural network without introducing inconsistencies or requiring simulation.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Lagrangian flow matching minimizes action of a general Lagrangian subject to continuity equation and endpoints to generate new probability paths and simulation-free training objectives that recover prior flow matching methods as special cases.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Minimizing a general Lagrangian's action under the continuity equation produces a family of probability paths for flow matching.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"7e6f8dabf95526acfeefd9cdbffc147c0be5de4d4416ceb3f3e2647e9571d26c"},"source":{"id":"2605.15419","kind":"arxiv","version":1},"verdict":{"id":"f5bd1302-09a9-41f3-bc4e-09ce28b782cc","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-19T15:49:51.168263Z","strongest_claim":"We show that this dynamic problem admits an equivalent static optimal transport (OT) formulation, yielding a family of simulation-free training objectives that recover OT-based flow matching as the kinetic special case and the trigonometric variance-preserving diffusion path as the harmonic-oscillator case.","one_line_summary":"Lagrangian flow matching minimizes action of a general Lagrangian subject to continuity equation and endpoints to generate new probability paths and simulation-free training objectives that recover prior flow matching methods as special cases.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The assumption that minimizing the action of an arbitrary Lagrangian subject to the continuity equation and fixed marginal endpoints produces a valid probability path whose induced velocity field can be regressed by a neural network without introducing inconsistencies or requiring simulation.","pith_extraction_headline":"Minimizing a general Lagrangian's action under the continuity equation produces a family of probability paths for flow matching."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.15419/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"cited_work_retraction","ran_at":"2026-05-19T16:22:12.760587Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T16:04:39.999601Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_title_agreement","ran_at":"2026-05-19T16:01:17.991806Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"citation_quote_validity","ran_at":"2026-05-19T15:50:43.812513Z","status":"skipped","version":"0.1.0","findings_count":0},{"name":"claim_evidence","ran_at":"2026-05-19T14:21:54.143788Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"ai_meta_artifact","ran_at":"2026-05-19T13:33:22.705146Z","status":"skipped","version":"1.0.0","findings_count":0}],"snapshot_sha256":"52a0a8190e6e5461db9bb455ea7bf00ea0adbef34f1930688608c34fc45fefbc"},"references":{"count":31,"sample":[{"doi":"","year":2025,"title":"M. S. Albergo, N. M. Boffi, and E. Vanden-Eijnden. Stochastic interpolants: A unifying framework for flows and diffusions.Journal of Machine Learning Research, 26(209):1–80, 2025","work_id":"4a85a9da-2d60-428f-9fe7-d66bc074a919","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2023,"title":"M. S. Albergo and E. Vanden-Eijnden. Building normalizing flows with stochastic interpolants. InThe Eleventh International Conference on Learning Representations (ICLR), 2023","work_id":"215142f4-ac3b-4521-842a-627c67ca1904","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2005,"title":"L. Ambrosio, N. Gigli, and G. Savaré.Gradient Flows: In Metric Spaces and in the Space of Probability Measures. Birkhäuser, Basel, 2005","work_id":"cee8e776-92fa-402c-a7af-c27333127ca0","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1989,"title":"V . I. Arnold, K. V ogtmann, and A. Weinstein.Mathematical Methods of Classical Mechanics, volume 60 ofGraduate Texts in Mathematics. Springer, New York, 2 edition, 1989","work_id":"30cf9856-0dbc-4ad7-aeaf-e821ebfb6e59","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2025,"title":"arXiv preprint arXiv:2504.10612 , year=","work_id":"d0d065c8-17d7-404d-87f8-af8a3656897d","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":31,"snapshot_sha256":"f684f6838efa533d413cf95c7764a4c22458f6eb842227df82e2fe4e48aea795","internal_anchors":0},"formal_canon":{"evidence_count":2,"snapshot_sha256":"4089b2c0a19547d088830b91c924d1aafe9d3faedeea0f89d4dd0bdbafc2b08d"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}