{
  "abstract": "We consider partial differential equation (PDE) inverse problems from sparse, noisy, and corrupted observations, with the aim of recovering unknown coefficient fields and associated state variables in a mesh-free setting. Under such sparse and corrupted observations, standard physics-informed and generative approaches typically treat all samples indiscriminately and therefore lack a principled mechanism for reconciling physical laws with contaminated data. We address this difficulty with a two-stage flow-matching framework. In the first stage, we develop physics-guided conditional flow matching (PG-CFM), which incorporates strong-form PDE information through residual regularization along the generative trajectories together with intermittent global collocation constraints. In the second stage, we introduce energy-regularized flow matching (ERFM), which fine-tunes the Stage-1 model by assigning each observation a physics--data energy score from a frozen teacher and reweighting the flow-matching objective to reduce the influence of high-energy, PDE-inconsistent samples. We show that the resulting Stage-2 objective is equivalent to flow matching under a teacher-induced reweighted data distribution, which gives a population-level interpretation of the robustness mechanism. Numerical experiments on several inverse benchmarks, including Poisson and Navier--Stokes problems, show that the proposed framework yields more accurate coefficient recovery than robust PINN variants and competing generative baselines.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.15536",
  "author_total": 4,
  "authors": [
    {
      "name": "Yongsheng Chen"
    },
    {
      "name": "Shuo Lu"
    },
    {
      "name": "Wei Guo"
    },
    {
      "name": "Xinghui Zhong"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6mpt96ze3hfhf.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.15536",
  "fulltext_url": null,
  "id": "6mpt96ze3hfhf",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.15536",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.NA"
    ],
    [
      "arxiv",
      "math.NA"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2609.15536",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-14",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6mpt96ze3hfhf.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:29Z",
      "native_id": "2609.15536",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.15536"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    }
  ],
  "title": "Physics-Guided Conditional Flow Matching with Energy Regularization for Robust PDE Inverse Problems",
  "title_text": "Physics-Guided Conditional Flow Matching with Energy Regularization for Robust PDE Inverse Problems",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:29Z",
  "url": "https://research.draconitic.com/paper/6mpt96ze3hfhf",
  "venue": "arXiv",
  "versions": []
}