{
  "id": "2312.14276",
  "version": "v1",
  "published": "2023-12-21T19:57:29.000Z",
  "updated": "2023-12-21T19:57:29.000Z",
  "title": "Deep Neural Networks and Finite Elements of Any Order on Arbitrary Dimensions",
  "authors": [
    "Juncai He",
    "Jinchao Xu"
  ],
  "comment": "23 pages, 2 figures",
  "categories": [
    "math.NA",
    "cs.LG",
    "cs.NA"
  ],
  "abstract": "In this study, we establish that deep neural networks employing ReLU and ReLU$^2$ activation functions are capable of representing Lagrange finite element functions of any order on simplicial meshes across arbitrary dimensions. We introduce a novel global formulation of the basis functions for Lagrange elements, grounded in a geometric decomposition of these elements and leveraging two essential properties of high-dimensional simplicial meshes and barycentric coordinate functions. This representation theory facilitates a natural approximation result for such deep neural networks. Our findings present the first demonstration of how deep neural networks can systematically generate general continuous piecewise polynomial functions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-12-21T19:57:29.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "68T07",
      "65D40"
    ],
    "keywords": [
      "deep neural networks",
      "arbitrary dimensions",
      "lagrange finite element functions",
      "neural networks employing relu",
      "general continuous piecewise polynomial"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 23,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}