{
  "abstract": "OptiXDE is a matrix-free spectral operator framework for differential equations on uniform grids and embedded domains. Inspired by angular-spectrum propagation in Fourier optics, it maps transform-diagonal spatial operators to analytical modal multipliers and composes them with physical-space operators for nonlinearities, geometry and boundary enforcement. A common transform--operator--inverse-transform backbone is demonstrated across transient diffusion, periodic and embedded-domain Poisson problems, the cubic nonlinear Schr\"odinger equation, viscous Burgers dynamics, the two-dimensional Allen--Cahn equation and incompressible flows from the Taylor--Green vortex to embedded-cylinder vortex shedding. Transform-compatible linear problems are recovered near the floating-point limit, whereas errors on the singular L-shaped domain remain localized near the re-entrant corner and regularized interface. Nonlinear benchmarks recover second-order temporal convergence and the expected conservative or dissipative behavior, while incompressibility remains near round-off level during long-time vortex shedding. The matrix-free updates require \\(\\mathcal{O}(N\\log N)\\) work and \\(\\mathcal{O}(N)\\) memory. Device-resident transform workloads reach \\(94.9\\times\\) GPU acceleration, and the complete embedded-cylinder solver achieves a \\(42.1\\times\\) CPU--GPU speedup under matched numerical settings. These results establish OptiXDE as a deterministic and extensible operator-centric framework for structured and embedded-domain differential equations.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.01009",
  "author_total": 3,
  "authors": [
    {
      "name": "Yang Yang"
    },
    {
      "name": "Mingjiao Yan"
    },
    {
      "name": "Zongliang Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/e295f3hdt2c4s.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.01009",
  "fulltext_url": null,
  "id": "e295f3hdt2c4s",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.01009",
  "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.NA"
    ],
    [
      "openalex",
      "2206|Advanced Numerical Methods in Computational Mathematics"
    ]
  ],
  "openalex": "W7206216139",
  "pdf_url": "https://arxiv.org/pdf/2609.01009",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/e295f3hdt2c4s.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:04Z",
      "native_id": "2609.01009",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.01009"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    }
  ],
  "title": "OptiXDE: A fast optical-inspired solver for differential equations",
  "title_text": "OptiXDE: A fast optical-inspired solver for differential equations",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:04Z",
  "url": "https://research.draconitic.com/paper/e295f3hdt2c4s",
  "venue": "arXiv",
  "versions": []
}