Pith Number

pith:RCTLJG2E

pith:2026:RCTLJG2EU4OWPQDHAAH5ZFWDTA

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Lagrangian Flow Matching: A Least-Action Framework for Principled Path Design

Junzhe Zhang, Shukai Du, Yiming Li

Minimizing a general Lagrangian's action under the continuity equation produces a family of probability paths for flow matching.

arxiv:2605.15419 v1 · 2026-05-14 · cs.LG

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Claims

C1strongest 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.

C2weakest 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.

C3one 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.

References

31 extracted · 31 resolved · 0 Pith anchors

[1] 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 2025

[2] M. S. Albergo and E. Vanden-Eijnden. Building normalizing flows with stochastic interpolants. InThe Eleventh International Conference on Learning Representations (ICLR), 2023 2023

[3] L. Ambrosio, N. Gigli, and G. Savaré.Gradient Flows: In Metric Spaces and in the Space of Probability Measures. Birkhäuser, Basel, 2005 2005

[4] 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 1989

[5] arXiv preprint arXiv:2504.10612 , year= 2025

Formal links

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Receipt and verification

First computed	2026-05-20T00:00:57.622873Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

88a6b49b44a71d67c067000fdc96c39829c8e2ddc70fb18ce352065a9fec1414

Aliases

arxiv: 2605.15419 · arxiv_version: 2605.15419v1 · doi: 10.48550/arxiv.2605.15419 · pith_short_12: RCTLJG2EU4OW · pith_short_16: RCTLJG2EU4OWPQDH · pith_short_8: RCTLJG2E

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/RCTLJG2EU4OWPQDHAAH5ZFWDTA \
  | jq -c '.canonical_record' \
  | python3 -c "import sys,json,hashlib; b=json.dumps(json.loads(sys.stdin.read()), sort_keys=True, separators=(',',':'), ensure_ascii=False).encode(); print(hashlib.sha256(b).hexdigest())"
# expect: 88a6b49b44a71d67c067000fdc96c39829c8e2ddc70fb18ce352065a9fec1414

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "f228cb44b3931ee5f81fcba193e146d068ee4a180fa2b59ee54e2d31ac0b64a2",
    "cross_cats_sorted": [],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "cs.LG",
    "submitted_at": "2026-05-14T21:05:03Z",
    "title_canon_sha256": "8f94b1d4a152b12dd6e95b734af9f9742444f422deed75309cdc13e4e4506dd1"
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  "source": {
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    "kind": "arxiv",
    "version": 1
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}