{
  "abstract": "Nonlinear least-squares objectives form the foundation of scientific machine-learning tasks, yet even curvature-aware optimizers remain geometrically inconsistent at finite step sizes: Levenberg-Marquardt (LM) derives its direction from local Riemannian metrics but realizes it as a straight parameter update. Here we introduce RNC-LM, which carries the LM direction along a locally constructed curved trajectory in Riemann normal coordinates. A recursive reformulation of the geodesic equation generates arbitrary finite-order corrections while reusing the same damped Gauss-Newton matrix factorization, and curve length is controlled separately from damping. On a reaction-diffusion physics-informed neural network benchmark, RNC-LM reduces relative $L^2$ errors below $8\\times10^{-3}$, whereas L-BFGS, LM and LM with geodesic acceleration remain near one. On a large-scale machine-learning potential fitting task with 985,160 configurations, fourth-order RNC-LM reaches a fixed training-error target with a $34\\times$ wall-clock speedup over LM. These results establish finite-step geometric realization as a distinct optimizer design principle.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.07623",
  "author_total": 2,
  "authors": [
    {
      "name": "Jianing Liu"
    },
    {
      "name": "Dong H. Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9y04v43ez3tqg.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.07623",
  "fulltext_url": null,
  "id": "9y04v43ez3tqg",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.07623",
  "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.LG"
    ],
    [
      "arxiv",
      "cs.NA"
    ],
    [
      "arxiv",
      "math.NA"
    ],
    [
      "arxiv",
      "physics.chem-ph"
    ],
    [
      "arxiv",
      "physics.comp-ph"
    ],
    [
      "arxiv",
      "physics.data-an"
    ],
    [
      "openalex",
      "1702|Stochastic Gradient Optimization Techniques"
    ]
  ],
  "openalex": "W7167852237",
  "pdf_url": "https://arxiv.org/pdf/2607.07623",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9y04v43ez3tqg.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:12Z",
      "native_id": "2607.07623",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.07623"
    }
  ],
  "subjects": [
    {
      "field": "chemistry",
      "origin": "arxiv:physics.chem-ph",
      "subfield": "physical"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    },
    {
      "field": "physics",
      "origin": "arxiv:physics.comp-ph",
      "subfield": "computational"
    }
  ],
  "title": "Higher-Order Geometric Updates for Levenberg-Marquardt Method via Riemann Normal Coordinates",
  "title_text": "Higher-Order Geometric Updates for Levenberg-Marquardt Method via Riemann Normal Coordinates",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:17Z",
  "url": "https://research.draconitic.com/paper/9y04v43ez3tqg",
  "venue": "arXiv",
  "versions": []
}