{
  "abstract": "Score-learning methods generally lack pointwise supervision, which affects the training accuracy, particularly in regions where SDE trajectories are rare. To address this challenge, we introduce a novel method called \\emph{BEL anchors}, which uses Monte Carlo score estimates at user-selected space-time query points. By combining a Feynman--Kac representation of the Fokker--Planck equation with the Bismut--Elworthy--Li formula, our estimator requires neither closed-form transition densities, spatial discretization, nor derivatives of the initial density. We derive its asymptotic variance and use the anchors as sparse supervised targets for existing score-learning objectives. Experiments show that BEL anchors improve implicit score matching in poorly sampled regions, prevent score-PINN collapse, and enable accurate sample regeneration in several numerical examples.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.03915",
  "author_total": 5,
  "authors": [
    {
      "name": "Jiatong Sun"
    },
    {
      "name": "Long Nguyen"
    },
    {
      "name": "Ning Jiang"
    },
    {
      "name": "Weiqi Chu"
    },
    {
      "name": "Yao Li"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/cezeaxjvgy20f.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.03915",
  "fulltext_url": null,
  "id": "cezeaxjvgy20f",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.03915",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.NA"
    ],
    [
      "arxiv",
      "math.NA"
    ],
    [
      "arxiv",
      "math.PR"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.03915",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/cezeaxjvgy20f.ris",
  "sources": [
    {
      "first_seen": "2026-11-04T15:21:29Z",
      "native_id": "2611.03915",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.03915"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "A probabilistic score estimation using the Bismut--Elworthy--Li formula",
  "title_text": "A probabilistic score estimation using the Bismut--Elworthy--Li formula",
  "type": "preprint",
  "updated_at": "2026-11-06T18:21:01Z",
  "url": "https://research.draconitic.com/paper/cezeaxjvgy20f",
  "venue": "arXiv",
  "versions": []
}