{
  "abstract": "We consider system identification for discovering parameterized operators in governing partial differential equations (PDEs) from noisy spatiotemporal data. Building on variational system identification (VSI), which identifies PDEs through Galerkin weak-form residuals, we develop a Bayesian VSI (B-VSI) framework for operator selection, parameter estimation, and uncertainty quantification. The central idea is to define the likelihood directly in weak-form residual space by propagating observation uncertainty through the weak-form residual map. The resulting likelihood captures heteroscedastic and correlated residual errors while avoiding repeated forward PDE solves during inference. For efficient computation, we use lagged-covariance updates that yield generalized least-squares estimates and conjugate posterior approximations when applicable, together with gradient-based and particle-based methods for more general priors and posterior structures. Model-form uncertainty is handled through sequential operator elimination guided by a residual-space Bayesian information criterion. We demonstrate the framework on state-linear and nonlinear PDEs, including the Fokker--Planck equation and a two-field Cahn--Hilliard equation. The results show that B-VSI accurately recovers active operators and coefficients from noisy data, improves robustness relative to classical VSI, and provides posterior uncertainty estimates for coefficients and derived physical quantities.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2606.14942",
  "author_total": 4,
  "authors": [
    {
      "name": "Chengyang Huang"
    },
    {
      "name": "Siddhartha Srivastava"
    },
    {
      "name": "Krishna Garikipati"
    },
    {
      "name": "Xun Huan"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/82br221xh0ewq.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2606.14942",
  "fulltext_url": null,
  "id": "82br221xh0ewq",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2606.14942",
  "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.CE"
    ],
    [
      "arxiv",
      "math-ph"
    ],
    [
      "arxiv",
      "math.MP"
    ],
    [
      "openalex",
      "3109|Model Reduction and Neural Networks"
    ]
  ],
  "openalex": "W7164890031",
  "pdf_url": "https://arxiv.org/pdf/2606.14942",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-23",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/82br221xh0ewq.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:35Z",
      "native_id": "2606.14942",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2606.14942"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    }
  ],
  "title": "Bayesian Variational System Identification with Weak-Form Residual Likelihoods",
  "title_text": "Bayesian Variational System Identification with Weak-Form Residual Likelihoods",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:40Z",
  "url": "https://research.draconitic.com/paper/82br221xh0ewq",
  "venue": "arXiv",
  "versions": []
}