{
  "id": "1905.09076",
  "version": "v1",
  "published": "2019-05-22T11:16:58.000Z",
  "updated": "2019-05-22T11:16:58.000Z",
  "title": "Selection dynamics for deep neural networks",
  "authors": [
    "Hailiang Liu",
    "Peter Markowich"
  ],
  "comment": "27",
  "categories": [
    "math.AP",
    "math.OC"
  ],
  "abstract": "This paper introduces the mathematical formulation of deep residual neural networks as a PDE optimal control problem. We study the wellposedness, the large time solution behavior, and the characterization of the steady states for the forward problem. Several useful time-uniform estimates and stability/instability conditions are presented. We state and prove optimality conditions for the inverse deep learning problem, using the Hamilton-Jacobi-Bellmann equation and the Pontryagin maximum principle. This serves to establish a mathematical foundation for investigating the algorithmic and theoretical connections between optimal control and deep learning.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-05-22T11:16:58.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "49K20",
      "49L20"
    ],
    "keywords": [
      "deep neural networks",
      "selection dynamics",
      "large time solution behavior",
      "deep residual neural networks",
      "pde optimal control problem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}