{
  "abstract": "Discovering governing partial differential equations (PDEs) from observed data is a central inverse problem in computational science and engineering: the goal is not only to predict trajectories, but to identify interpretable symbolic laws that can be analyzed and reused. Existing PDE discovery methods often depend on hand-designed libraries, accurate numerical derivatives, or per-equation symbolic search, making them fragile under sparse observations, noisy data, broad equation families, and unseen symbolic compositions. Here, we introduce FoundPDE, a pretrained data-to-symbol foundation model for PDE discovery. FoundPDE maps discretized spatio-temporal trajectories to symbolic equations with a transformer, uses Fourier and first-order spatial-derivative augmentation to expose complementary trajectory information, refines numerical coefficients, and supports downstream extension with improved generalization through few-shot fine-tuning and hierarchical discovery. On a clean in-domain benchmark spanning 20 PDE families, our model reaches 97.50% type recognition. Under coefficient shifts beyond the training support, type recognition remains 90.53%; under multiplicative Gaussian noise on the model input, it is 94.43% at a 2% noise level but falls to 52.09% at 10%. For unseen composite equations, few-shot fine-tuning reaches over 90% exact recovery on four held-out families, and we further develop a hierarchical discovery algorithm that recovers two-term composites without fine-tuning FoundPDE. By amortizing symbolic discovery across equation families, FoundPDE shifts PDE discovery from per-dataset search toward a reusable pretrained model that extends to new physics with limited additional data, a step toward foundation models that uncover interpretable governing laws directly from scientific data.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.26963",
  "author_total": 5,
  "authors": [
    {
      "name": "Langchen Liu"
    },
    {
      "name": "Junlin Luo"
    },
    {
      "name": "Min Zhu"
    },
    {
      "name": "Zhuoran Yang"
    },
    {
      "name": "Lu Lu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8hh3r84zcgpga.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.26963",
  "fulltext_url": null,
  "id": "8hh3r84zcgpga",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.26963",
  "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.AI"
    ],
    [
      "arxiv",
      "cs.LG"
    ],
    [
      "arxiv",
      "physics.comp-ph"
    ],
    [
      "openalex",
      "3109|Model Reduction and Neural Networks"
    ]
  ],
  "openalex": "W7223129229",
  "pdf_url": "https://arxiv.org/pdf/2610.26963",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-18",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8hh3r84zcgpga.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:04Z",
      "native_id": "2610.26963",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.26963"
    }
  ],
  "subjects": [
    {
      "field": "physics",
      "origin": "arxiv:physics.comp-ph",
      "subfield": "computational"
    }
  ],
  "title": "FoundPDE: A generative foundation model for data-driven symbolic discovery of partial differential equations",
  "title_text": "FoundPDE: A generative foundation model for data-driven symbolic discovery of partial differential equations",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:04Z",
  "url": "https://research.draconitic.com/paper/8hh3r84zcgpga",
  "venue": "arXiv",
  "versions": []
}