{
  "abstract": "We introduce a unified framework for learning stochastic dynamics from distributional observations and discretely observed trajectories. The key contribution is to combine soft law constraints, which regularize fitting to empirical distributions, with path-dependent supervision, which captures temporal structure that marginal distributions alone cannot identify. We formulate the problem as a discrete path-dependent McKean-Vlasov control problem, penalizing endpoint, intermediate marginal, and joint sampled-path laws within a single objective. Its optimality system takes the form of a McKean-Vlasov forward-backward stochastic differential equation (FBSDE), in which the marginal terms act continuously in time while the path-law term instead induces conditional adjoint jumps at the observation dates. We derive the reduced FBSDE for quadratic control cost and constant diffusion, establish local well-posedness for a Maximum Mean Discrepancy (MMD) path-law formulation, and develop a sample-based neural solver with causal representations. Experiments on synthetic transport, latent face transport, and human motion validate the framework. In particular, a two-state AMASS experiment under path-law supervision captures both low-to-high and high-to-low transitions, which marginal supervision alone cannot distinguish.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2605.08928",
  "author_total": 5,
  "authors": [
    {
      "name": "Samer El Boustany"
    },
    {
      "name": "Samy Mekkaoui"
    },
    {
      "name": "Yadh Hafsi"
    },
    {
      "name": "Alexandre Alouadi"
    },
    {
      "name": "Huyên Pham"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2xsy1gx5dtbqq.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2605.08928",
  "fulltext_url": null,
  "id": "2xsy1gx5dtbqq",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2605.08928",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.OC"
    ],
    [
      "arxiv",
      "stat.ML"
    ],
    [
      "openalex",
      "2003|Stochastic processes and financial applications"
    ]
  ],
  "openalex": "W7160893631",
  "pdf_url": "https://arxiv.org/pdf/2605.08928",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-19",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2xsy1gx5dtbqq.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:08Z",
      "native_id": "2605.08928",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2605.08928"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:stat.ML",
      "subfield": "statistics"
    }
  ],
  "title": "Learning Generative Dynamics with Soft Path-Law Constraints: A McKean-Vlasov FBSDE Approach",
  "title_text": "Learning Generative Dynamics with Soft Path-Law Constraints: A McKean-Vlasov FBSDE Approach",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:13Z",
  "url": "https://research.draconitic.com/paper/2xsy1gx5dtbqq",
  "venue": "arXiv",
  "versions": []
}