{
  "abstract": "Fractional partial differential equations couple a memory-dependent time derivative with a nonlocal fractional Laplacian, and their repeated solution under varying fractional orders, initial data, or diffusivity fields is computationally expensive. Neural operators offer a fast surrogate, but standard architectures must relearn the dominant linear fractional evolution from data. We propose the \\emph{Fractional Fourier Neural Operator} (FrFNO), a resolution-robust conditional operator. It injects the analytic Mittag--Leffler response of the frozen linear part as a once-precomputed, resolution-independent propagator table, and trains a spectral convolutional network only on the residual induced by variable coefficients and nonlinear advection, conditioned continuously on both fractional orders. The central theoretical result is that under zero-shot super-resolution the injected propagator fills the out-of-band modes that a standard Fourier neural operator sets to zero, replacing the out-of-band full-field tail by the smaller residual tail (a $K$-independent constant-factor reduction under weak perturbation). The residual and the full solution carry the same Sobolev order, so the advantage is amplitude reduction rather than a steeper tail. Mesh refinement drives the error to a residual-controlled floor. On a nonlinear two-dimensional space--time fractional Burgers problem FrFNO achieves the lowest relative $L^2$ error among six baselines (FNO, PINO, PDNO, CNO, DeepONet, U-Net) at the training resolution and under zero-shot super-resolution, yields the smallest spectral phase error, and remains the best at the integer-order limit and for long integration windows. The same construction extends to coupled fractional systems such as fractional Allen-Cahn and Navier--Stokes equations. Code, training scripts, and reference outputs are available at https://github.com/Derek2021Pang/FrFNO.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.21512",
  "author_total": 1,
  "authors": [
    {
      "name": "Guofei Pang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6hd11j62m1zew.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.21512",
  "fulltext_url": null,
  "id": "6hd11j62m1zew",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.21512",
  "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.NA"
    ],
    [
      "arxiv",
      "math-ph"
    ],
    [
      "arxiv",
      "math.MP"
    ],
    [
      "arxiv",
      "math.NA"
    ],
    [
      "openalex",
      "3109|Model Reduction and Neural Networks"
    ]
  ],
  "openalex": "W7213909784",
  "pdf_url": "https://arxiv.org/pdf/2609.21512",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-18",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6hd11j62m1zew.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:03Z",
      "native_id": "2609.21512",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.21512"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    }
  ],
  "title": "FrFNO:Injecting the analytic Mittag-Leffler propagator into a resolution-robust neural operator for space-time fractional PDEs",
  "title_text": "FrFNO:Injecting the analytic Mittag-Leffler propagator into a resolution-robust neural operator for space-time fractional PDEs",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:07Z",
  "url": "https://research.draconitic.com/paper/6hd11j62m1zew",
  "venue": "arXiv",
  "versions": []
}