{
  "abstract": "Optimal control problems in mathematical biology are commonly solved by the forward-backward sweep on the Pontryagin's Maximum Principle necessary conditions. The sweep has known stability issues at constraint-saturated optima. We frame the same problem within the universal differential equation framework. A neural network represents the control. The cost functional serves as the training loss. We prove that any stationary point of the training loss satisfies the continuous first-order necessary conditions to a discretisation-controlled accuracy. On a grid of bifurcation-parameter and initial-condition scenarios drawn from a tumor-immune dynamics model, the standard sweep fails on most scenarios in the nominal regime. Neural network training reaches a stable optimum on every scenario, with a lower final cost than the converged damped sweep iterate. The pipeline accommodates cost functional variants by changing only the loss function. A Sobol sensitivity analysis confirms robustness of the trained schedule to perturbations of the cost weights.",
  "abstract_shown": true,
  "affiliations": [
    "Department of Mathematics and Statistics, Texas Tech University, Lubbock, TX, USA. wenjing.zhang@ttu.edu.",
    "Department of Mathematical Sciences, Middle Tennessee State University, Murfreesboro, TN, USA.",
    "Department of Mathematics and Statistics, York University, Toronto, ON, Canada."
  ],
  "article_type": "research article",
  "arxiv": null,
  "author_total": 3,
  "authors": [
    {
      "affs": [
        0
      ],
      "name": "Wenjing Zhang",
      "orcid": "0000-0001-8779-5106"
    },
    {
      "affs": [
        1
      ],
      "name": "Wandi Ding"
    },
    {
      "affs": [
        2
      ],
      "name": "Huaiping Zhu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/001yr4ht3krwd.bib",
  "cited_by": 0,
  "doi": "10.1007/s00285-026-02475-2",
  "fulltext_url": "https://europepmc.org/articles/PMC13719735",
  "id": "001yr4ht3krwd",
  "issns": [
    "0303-6812",
    "1432-1416"
  ],
  "landing_url": "https://europepmc.org/article/MED/42955091",
  "language": "eng",
  "licence": "CC-BY",
  "licence_label": "CC BY",
  "licence_source": "article",
  "licence_url": "https://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": null,
  "native_subjects": [
    [
      "heading",
      "Article"
    ],
    [
      "openalex",
      "2611|Mathematical Biology Tumor Growth"
    ]
  ],
  "openalex": "W7224414965",
  "pdf_url": "https://europepmc.org/articles/PMC13719735?pdf=render",
  "pmcid": "PMC13719735",
  "pmid": "42955091",
  "published": "2026-10-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/001yr4ht3krwd.ris",
  "sources": [
    {
      "first_seen": "2026-11-01T00:20:36Z",
      "native_id": "MED:42955091",
      "source": "europepmc",
      "url": "https://europepmc.org/article/MED/42955091"
    },
    {
      "first_seen": "2026-11-02T12:23:39Z",
      "native_id": "PMC13719735",
      "source": "pmc",
      "url": "https://doi.org/10.1007/s00285-026-02475-2"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "openalex:2611|Mathematical Biology Tumor Growth",
      "subfield": "general"
    }
  ],
  "title": "From forward-backward sweep to backpropagation for optimal control in tumor-immune dynamics.",
  "title_text": "From forward-backward sweep to backpropagation for optimal control in tumor-immune dynamics.",
  "type": "peer-reviewed",
  "updated_at": "2026-11-03T12:20:45Z",
  "url": "https://research.draconitic.com/paper/001yr4ht3krwd",
  "venue": "Journal of mathematical biology",
  "versions": []
}