{
  "abstract": "A physics-informed loss adds a discretised PDE residual to the data term, but is only strictly valid if the equations being enforced are closed on the fields being fitted. This fails on coarse-grained, stochastically forced data: nonlinear terms in the PDE generally do not commute with the filter, so the coarse-grained fields do not satisfy the original equations, and driving their residual to zero encodes an implicit closure into the learned operator. Under normalised, translation-invariant filtering on a periodic domain, constant-coefficient linear constraints remain valid on the filtered grid. For the 2D incompressible Navier-Stokes equations, two such constraints are continuity and global momentum balance. We enforce these exactly by applying a closed-form projection on the output of a Fourier neural operator (FNO) at $O(N \\log N)$ cost. On two-dimensional isotropic turbulence, truncated so that its forcing lies entirely beyond the cutoff wavenumber, the projection achieves a continuity error of $5.0\\times10^{-7}$ against 0.51 for a plain FNO and 0.17 for a physics-informed FNO, and a global momentum balance error of $8.6\\times10^{-11}$ against $7.5\\times10^{-4}$ and $8.3\\times10^{-4}$. The projection only costs an extra 2% of training time, and takes the fraction of rollouts whose energy remains bounded at 448 steps from 0.18 to 0.87. On resolved Kolmogorov flow, test-time optimisation costs approximately 3,300$\\times$ as much per trajectory as the projection, while leaving maximum divergence three orders of magnitude higher. Nonlinear constraints can also be enforced once their unclosed terms are modelled: closing the global energy balance with a constant subgrid flux, fitted from the coarse training data alone, removes the 20.9% energy deficit that enforcing the unclosed balance produces and keeps every free-running rollout bounded over 2,000 steps.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.07811",
  "author_total": 2,
  "authors": [
    {
      "name": "Michael Groom"
    },
    {
      "name": "Rafael Oliveira"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/f0gjr925f5pd9.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.07811",
  "fulltext_url": null,
  "id": "f0gjr925f5pd9",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.07811",
  "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",
      "physics.comp-ph"
    ],
    [
      "arxiv",
      "physics.flu-dyn"
    ],
    [
      "openalex",
      "3109|Model Reduction and Neural Networks"
    ]
  ],
  "openalex": "W7220918689",
  "pdf_url": "https://arxiv.org/pdf/2610.07811",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-06",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/f0gjr925f5pd9.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:22Z",
      "native_id": "2610.07811",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.07811"
    }
  ],
  "subjects": [
    {
      "field": "physics",
      "origin": "arxiv:physics.comp-ph",
      "subfield": "computational"
    },
    {
      "field": "physics",
      "origin": "arxiv:physics.flu-dyn",
      "subfield": "fluid-dynamics"
    }
  ],
  "title": "Only Linear Constraints Survive Coarse-Graining: Evaluating Physics-Constrained Neural Operators on Stochastically Forced Turbulence",
  "title_text": "Only Linear Constraints Survive Coarse-Graining: Evaluating Physics-Constrained Neural Operators on Stochastically Forced Turbulence",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:22Z",
  "url": "https://research.draconitic.com/paper/f0gjr925f5pd9",
  "venue": "arXiv",
  "versions": []
}