{
  "abstract": "We study fractional parabolic initial-value problems with lower-order drift and potential terms in anisotropic spectral Barron spaces, defined by weighted space--time Fourier $L^1$ norms adapted to parabolic scaling. We prove existence, uniqueness, and maximal regularity with a gain of one derivative in time and $γ$ derivatives in space, where $γ>0$ is the order of the fractional Laplacian. The evolution is defined only for $t\\geq0$, whereas the finite-time norm requires a global extension with sufficient temporal Fourier decay. We construct a finite reflected semigroup extension using a Vandermonde system to match derivatives at $t=0$, obtaining temporal Fourier estimates uniform in the semigroup parameter. Combined with Fourier multiplier estimates for the damped principal operator, it yields maximal regularity. Dimension-independent multiplication estimates support a finite regularity bootstrap, while interpolation and sufficient damping absorb the lower-order terms in the base estimate. The a priori estimate and the method of continuity yield maximal regularity without smallness assumptions on the lower-order coefficients. A frequency-localized counterexample shows that a uniform-in-time spatial Barron bound on the forcing does not imply the corresponding two-derivative solution bound, even for the one-dimensional heat equation. Using this regularity, Fourier sampling yields $n^{-1/2}$ approximation rates for the solution in mixed space--time Sobolev norms using shallow networks with suitable activations. Sampling in a product Hilbert space yields a population-level PINN consistency estimate for shallow cosine networks on a bounded cylinder. There exists a single width-$n$ network for which the sum of the squared mixed-Sobolev solution error, the squared $L^2$-norm of the residual for the whole-space fractional equation, and the squared initial-data error is $O(n^{-1})$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.27781",
  "author_total": 5,
  "authors": [
    {
      "name": "Jae-Hwan Choi"
    },
    {
      "name": "Hyojae Lim"
    },
    {
      "name": "Jinsol Seo"
    },
    {
      "name": "Young-Jin Sim"
    },
    {
      "name": "Changhoon Song"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0ngy4pcp1b8bg.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.27781",
  "fulltext_url": null,
  "id": "0ngy4pcp1b8bg",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.27781",
  "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",
      "cs.LG"
    ],
    [
      "arxiv",
      "math.AP"
    ],
    [
      "openalex",
      "2604|Differential Equations and Boundary Problems"
    ]
  ],
  "openalex": "W7171931598",
  "pdf_url": "https://arxiv.org/pdf/2607.27781",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0ngy4pcp1b8bg.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:17Z",
      "native_id": "2607.27781",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.27781"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Fractional Parabolic Partial Differential Equations in Anisotropic Spectral Barron Spaces: Regularity and Neural Approximation",
  "title_text": "Fractional Parabolic Partial Differential Equations in Anisotropic Spectral Barron Spaces: Regularity and Neural Approximation",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:17Z",
  "url": "https://research.draconitic.com/paper/0ngy4pcp1b8bg",
  "venue": "arXiv",
  "versions": []
}