{
  "abstract": "Parametrized PDEs with density-valued solutions are often difficult to approximate with classical linear reduced-order models, especially in transport-dominated regimes. We introduce an optimal-transport-based reduced-order modeling that represents each density by the Kantorovich potential transporting a fixed reference density to the target density, and then maps these potentials to transport signatures using a weighted Laplacian associated with the reference measure. This embeds the density-valued solution map in a Hilbert space while preserving control of the induced transport maps and Wasserstein error. We treat the signature map as a continuous matrix indexed by parameters and space, construct a low-rank skeleton decomposition using a maximal-volume criterion, and learn the parameter-to-coefficient map with a neural network for efficient non-intrusive online evaluation. The reconstructed solution is obtained by pushing forward the reference density, so mass and positivity preservation is built into the method. We prove a mean-squared Wasserstein error bound separating low-rank approximation, discretization, sampling, and learning errors, and demonstrate the method on density-valued continuity equation and Burgers equation, where transport signatures yield substantially lower-rank structure than the original density snapshots.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.01696",
  "author_total": 6,
  "authors": [
    {
      "name": "Jiajia Yu"
    },
    {
      "name": "Jingwei Hu"
    },
    {
      "name": "Fengyan Li"
    },
    {
      "name": "Shanyin Tong"
    },
    {
      "name": "Yunan Yang"
    },
    {
      "name": "Zhaiming Shen"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/esrzdr8nx4bka.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.01696",
  "fulltext_url": null,
  "id": "esrzdr8nx4bka",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.01696",
  "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": "W7167249233",
  "pdf_url": "https://arxiv.org/pdf/2607.01696",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-17",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/esrzdr8nx4bka.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:57Z",
      "native_id": "2607.01696",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.01696"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    }
  ],
  "title": "Structure-Preserving Reduced-Order Modeling via Low-Rank Transport Signatures",
  "title_text": "Structure-Preserving Reduced-Order Modeling via Low-Rank Transport Signatures",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:44Z",
  "url": "https://research.draconitic.com/paper/esrzdr8nx4bka",
  "venue": "arXiv",
  "versions": []
}