{
  "abstract": "Integrating physical conservation laws into deep neural architectures remains a central SciML challenge. Soft-penalty approaches, including many PINNs, may be sensitive to loss weighting and optimization; generic DEQ and implicit-layer solvers may require instance-dependent iteration counts. We study a norm-linearization scheme for networked systems with linear continuity constraints and nonlinear quadratic or power-law energy balances. We derive sharp universal and finite-$n$ bounds on relative variance and, via the weighted Kantorovich inequality, relate them to worst-case single-step squared-norm distortion when target and operating states have a bounded multiplicative ratio. Euler's theorem shows that the scheme is Newton's method for degree-one norm differences $u_i(\\mathbf{x})-v_i(\\mathbf{x})=0$. Recasting raw degree-$α$ balances as norm differences removes radial curvature, yields a degree-zero homogeneous Jacobian, eliminates the affine Newton term, and row-equilibrates the loop Jacobian at equilibrium. Under regularity and nonsingularity assumptions, this gives local quadratic convergence for Darcy-Weisbach ($α=2$) and Hazen-Williams ($α=1.852$) models. We implement dynamic sign partitioning in a FlowDEQ layer using a reduced incidence matrix and implicit-function adjoint differentiation.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.25506",
  "author_total": 2,
  "authors": [
    {
      "name": "A. A. Pak"
    },
    {
      "name": "A. S. Pak"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/be1kte0sy2gag.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.25506",
  "fulltext_url": null,
  "id": "be1kte0sy2gag",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.25506",
  "language": null,
  "licence": "CC-BY-NC-SA-4.0",
  "licence_label": "CC BY-NC-SA 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-sa/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.NA"
    ],
    [
      "arxiv",
      "math.NA"
    ],
    [
      "openalex",
      "3109|Model Reduction and Neural Networks"
    ]
  ],
  "openalex": "W7223005622",
  "pdf_url": "https://arxiv.org/pdf/2610.25506",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-18",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/be1kte0sy2gag.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:57Z",
      "native_id": "2610.25506",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.25506"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    }
  ],
  "title": "A Method for Linearization of Systems of Quadratic Equations",
  "title_text": "A Method for Linearization of Systems of Quadratic Equations",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:44Z",
  "url": "https://research.draconitic.com/paper/be1kte0sy2gag",
  "venue": "arXiv",
  "versions": []
}