{
  "abstract": "Physics-informed machine learning (PIML) integrates mechanistic knowledge, typically through partial differential equations (PDEs), into data-driven models. Despite strong empirical performance, its statistical generalisation properties remain poorly understood, especially for regression with unbounded losses. We develop a PAC-Bayesian framework for PIML that provides high-probability generalisation guarantees under potentially unbounded losses. Exploiting the structure of physics-informed objectives, we derive component-wise bounds whose complexity scales with the input-gradient energy of each loss, establishing a direct link between physical regularity and generalisation. We further introduce a PAC-Bayesian calibration procedure that yields computable gradient-based complexities while controlling rare large-gradient events through a residual-tail correction. Adopting a multi-task view of data fidelity, PDE residuals, initial conditions, and boundary conditions, we obtain a refined certificate with a single PAC-Bayesian complexity penalty, avoiding the looseness of independently bounding each component. Building on this certificate, we propose a bound-aware learning procedure that promotes empirical accuracy, proximity to a physics-informed prior, and low input-gradient complexity. Experiments on six PDE benchmarks yield substantially tighter certificates than bounded-loss and sub-Gaussian alternatives, while ablations quantify the effects of posterior-training data, calibration data, gradient envelopes, and hypothesis localisation.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2605.26341",
  "author_total": 3,
  "authors": [
    {
      "name": "Thien V. Nguyen"
    },
    {
      "name": "Amaury Habrard"
    },
    {
      "name": "Benjamin Guedj"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/d1gw8kpsb59wd.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2605.26341",
  "fulltext_url": null,
  "id": "d1gw8kpsb59wd",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2605.26341",
  "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.LG"
    ],
    [
      "arxiv",
      "stat.ML"
    ],
    [
      "openalex",
      "3109|Model Reduction and Neural Networks"
    ]
  ],
  "openalex": "W7162544223",
  "pdf_url": "https://arxiv.org/pdf/2605.26341",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/d1gw8kpsb59wd.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:16Z",
      "native_id": "2605.26341",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2605.26341"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:stat.ML",
      "subfield": "statistics"
    }
  ],
  "title": "A PAC-Bayesian View of Generalisation for Physics-Informed Machine Learning",
  "title_text": "A PAC-Bayesian View of Generalisation for Physics-Informed Machine Learning",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:16Z",
  "url": "https://research.draconitic.com/paper/d1gw8kpsb59wd",
  "venue": "arXiv",
  "versions": []
}