{
  "abstract": "We develop and analyze neural stochastic residual methods for fully nonlinear parabolic terminal problems under exact diffusion sampling. A quadratic stochastic residual supports global training and backward regression with scalar networks and losses involving only spatial derivatives. For global training, the correction removes a Hessian penalty from the limiting first-order loss. Under a uniform output regularity budget, we bound the continuous residual norm by the square roots of the global loss or normalized backward-regression excess loss, an $O(h^{1/2})$ discretization term for time step $h$, and terminal error. Residual stability then gives value, gradient and Hessian error bounds. Under suitable solution and terminal assumptions, we construct admissible networks with $O(h^{1/2})$ bounds on these approximation errors and the square roots of both loss criteria. In ten-dimensional backward-regression tests using a shared schedule tuned for the quadratic residual, mean relative Hessian errors are $5$--$6\\%$ with the quadratic residual and about $15\\%$ with the first-order residual. Tuned backward regression achieves Hessian accuracy comparable to physics-informed neural networks across the tested nonlinear coefficients. Translation tests show that the ratio of Hessian error to residual can grow sharply under a fixed sampling law.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.16730",
  "author_total": 2,
  "authors": [
    {
      "name": "Zhenhua Zhao"
    },
    {
      "name": "Jihao Long"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dms0ekahz9eyt.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.16730",
  "fulltext_url": null,
  "id": "dms0ekahz9eyt",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.16730",
  "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.OC"
    ],
    [
      "arxiv",
      "stat.ML"
    ],
    [
      "openalex",
      "3109|Model Reduction and Neural Networks"
    ]
  ],
  "openalex": "W7169883617",
  "pdf_url": "https://arxiv.org/pdf/2607.16730",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-19",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dms0ekahz9eyt.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:57Z",
      "native_id": "2607.16730",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.16730"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:stat.ML",
      "subfield": "statistics"
    }
  ],
  "title": "Deep Stochastic Residual Methods for Fully Nonlinear Parabolic Equations",
  "title_text": "Deep Stochastic Residual Methods for Fully Nonlinear Parabolic Equations",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:54Z",
  "url": "https://research.draconitic.com/paper/dms0ekahz9eyt",
  "venue": "arXiv",
  "versions": []
}