{
  "id": "2308.16272",
  "version": "v1",
  "published": "2023-08-30T19:02:24.000Z",
  "updated": "2023-08-30T19:02:24.000Z",
  "title": "A numerical approach for the fractional Laplacian via deep neural networks",
  "authors": [
    "Nicolás Valenzuela"
  ],
  "comment": "32 pages, 21 figures, 3 tables",
  "categories": [
    "math.AP",
    "cs.LG",
    "cs.NA",
    "math.NA"
  ],
  "abstract": "We consider the fractional elliptic problem with Dirichlet boundary conditions on a bounded and convex domain $D$ of $\\mathbb{R}^d$, with $d \\geq 2$. In this paper, we perform a stochastic gradient descent algorithm that approximates the solution of the fractional problem via Deep Neural Networks. Additionally, we provide four numerical examples to test the efficiency of the algorithm, and each example will be studied for many values of $\\alpha \\in (1,2)$ and $d \\geq 2$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-08-30T19:02:24.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35R11",
      "62M45",
      "68T07"
    ],
    "keywords": [
      "deep neural networks",
      "fractional laplacian",
      "numerical approach",
      "stochastic gradient descent algorithm",
      "dirichlet boundary conditions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 32,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}