{
  "abstract": "This paper proposes gappy probabilistic manifold decomposition (Gappy PMD), a nonlinear method for reconstructing high-dimensional fields from extremely sparse measurements. Gappy PMD reconstructs the field on the nonlinear manifold learned by probabilistic manifold decomposition (PMD). We further propose a differentiable point selection method for reduced-order model (ROM)-based field reconstruction (DPS). Using differentiable meshless interpolation within the ROM-based reconstruction framework, DPS makes the full-field reconstruction error differentiable with respect to the sampling locations and directly optimizes these locations. In addition, a theoretical error analysis for Gappy PMD is also given. It splits the squared reconstruction error into two orthogonal parts: one normal to the reconstruction manifold and the other induced by sparse sampling and observation noise. Under a stability condition on the sampling operator, this error vanishes with the PMD approximation error and the noise. The Gappy PMD is evaluated on three numerical test cases: flow past a cylinder, lid-driven cavity flow, and backward-facing step flow. For the same reduced dimension and sampling points, Gappy PMD attains mean relative $L^2$ errors one to two orders of magnitude below Gappy POD. Optimizing the sampling points with DPS further improves reconstruction accuracy and robustness.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.29014",
  "author_total": 4,
  "authors": [
    {
      "name": "Qihan Feng"
    },
    {
      "name": "Jiaming Guo"
    },
    {
      "name": "Jiarun Meng"
    },
    {
      "name": "Dunhui Xiao"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bkrcsakn0ag95.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.29014",
  "fulltext_url": null,
  "id": "bkrcsakn0ag95",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.29014",
  "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"
    ],
    [
      "openalex",
      "3109|Model Reduction and Neural Networks"
    ]
  ],
  "openalex": "W7204861923",
  "pdf_url": "https://arxiv.org/pdf/2608.29014",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bkrcsakn0ag95.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:48Z",
      "native_id": "2608.29014",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.29014"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    }
  ],
  "title": "Gappy probabilistic manifold decomposition for nonlinear field reconstruction",
  "title_text": "Gappy probabilistic manifold decomposition for nonlinear field reconstruction",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:48Z",
  "url": "https://research.draconitic.com/paper/bkrcsakn0ag95",
  "venue": "arXiv",
  "versions": []
}