{
  "abstract": "We compute self-similar singularity profiles of partial differential equations (PDEs) using adaptive numerical discretizations based on optimizing a discrete loss (ODIL). The fields are represented in overlapping local Chebyshev patches joined with a mapped asymptotic patch. After factoring out the leading power laws of solution ansatzes, their limiting amplitudes at infinity are determined by the optimization, eliminating domain truncation and prescribed far-field conditions. Adaptive refinement is guided by Chebyshev-coefficient decay, and the nonlinear least-squares problem is solved using automatically differentiated Gauss--Newton iterations with Levenberg--Marquardt damping. The method achieves O(10^{-14})-level accuracy for Burgers profiles, residuals of O(10^{-14}) to O(10^{-10}) for the generalized De Gregorio family, and of O(10^{-8}) in a local region for the two-dimensional Boussinesq equations. These results demonstrate that ODIL can identify singularity profiles for PDEs with high accuracy using adaptive numerical methods, complementing ongoing efforts that rely on artificial intelligence.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.30492",
  "author_total": 3,
  "authors": [
    {
      "name": "Allen Yang"
    },
    {
      "name": "Lucas Amoudruz"
    },
    {
      "name": "Petros Koumoutsakos"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/20y2vads6jn5f.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.30492",
  "fulltext_url": null,
  "id": "20y2vads6jn5f",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.30492",
  "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.AP"
    ],
    [
      "arxiv",
      "math.NA"
    ],
    [
      "openalex",
      "3109|Model Reduction and Neural Networks"
    ]
  ],
  "openalex": "W7223335781",
  "pdf_url": "https://arxiv.org/pdf/2610.30492",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-21",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/20y2vads6jn5f.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:38Z",
      "native_id": "2610.30492",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.30492"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    }
  ],
  "title": "Discovery of Singularities in Partial Differential Equations Using Adaptive Numerical Discretizations",
  "title_text": "Discovery of Singularities in Partial Differential Equations Using Adaptive Numerical Discretizations",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:38Z",
  "url": "https://research.draconitic.com/paper/20y2vads6jn5f",
  "venue": "arXiv",
  "versions": []
}