{
  "id": "2105.12080",
  "version": "v1",
  "published": "2021-05-25T16:59:34.000Z",
  "updated": "2021-05-25T16:59:34.000Z",
  "title": "Operator Compression with Deep Neural Networks",
  "authors": [
    "Fabian Kröpfl",
    "Roland Maier",
    "Daniel Peterseim"
  ],
  "categories": [
    "math.NA",
    "cs.NA"
  ],
  "abstract": "This paper studies the compression of partial differential operators using neural networks. We consider a family of operators, parameterized by a potentially high-dimensional space of coefficients that may vary on a large range of scales. Based on existing methods that compress such a multiscale operator to a finite-dimensional sparse surrogate model on a given target scale, we propose to directly approximate the coefficient-to-surrogate map with a neural network. We emulate local assembly structures of the surrogates and thus only require a moderately sized network that can be trained efficiently in an offline phase. This enables large compression ratios and the online computation of a surrogate based on simple forward passes through the network is substantially accelerated compared to classical numerical upscaling approaches. We apply the abstract framework to a family of prototypical second-order elliptic heterogeneous diffusion operators as a demonstrating example.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-05-25T16:59:34.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "68T07",
      "65N30",
      "35J15"
    ],
    "keywords": [
      "deep neural networks",
      "operator compression",
      "second-order elliptic heterogeneous diffusion operators",
      "enables large compression ratios",
      "finite-dimensional sparse surrogate model"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}