{
  "abstract": "High-dimensional partial differential equations (PDEs) remain challenging for spectral methods due to the exponential growth of multidimensional coefficient spaces. We propose TensorSpectra, a low-rank spectral framework that directly parameterizes spectral coefficient tensors in canonical polyadic (CP) form, reducing representation complexity from $\\mathcal{O}(N^d)$ to $\\mathcal{O}(dNr)$. TensorSpectra combines structure-preserving spectral calculus, de-aliased nonlinear interactions, operator-aware residual weighting, and progressive low-rank corrections to enable efficient PDEs solving in compressed coefficient space. We further establish a theoretical framework connecting spectral truncation, low-rank approximation, and optimization errors. Extensive experiments across five PDE families and 24 benchmark configurations demonstrate State-Of-The-Art accuracy among five representative baselines, with TensorSpectra consistently achieving the lowest relative $L^2$ errors. More broadly, our findings offer a new perspective on high-dimensional PDEs solving, suggesting that exploiting intrinsic low-rank structure in spectral coefficient space provides a viable alternative to conventional physical-space approaches.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.38087",
  "author_total": 4,
  "authors": [
    {
      "name": "Bingcheng Hu"
    },
    {
      "name": "Tianchi Yu"
    },
    {
      "name": "Zhaoxiang Li"
    },
    {
      "name": "Ivan Oseledets"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3fqn3sjhm0wra.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.38087",
  "fulltext_url": null,
  "id": "3fqn3sjhm0wra",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.38087",
  "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.NA"
    ],
    [
      "arxiv",
      "math.NA"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.38087",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3fqn3sjhm0wra.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:32Z",
      "native_id": "2610.38087",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.38087"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    }
  ],
  "title": "TensorSpectra: Bridging the Spectral and Tensor Worlds for High-Dimensional PDEs",
  "title_text": "TensorSpectra: Bridging the Spectral and Tensor Worlds for High-Dimensional PDEs",
  "type": "preprint",
  "updated_at": "2026-11-01T12:20:50Z",
  "url": "https://research.draconitic.com/paper/3fqn3sjhm0wra",
  "venue": "arXiv",
  "versions": []
}