{
  "id": "2103.13330",
  "version": "v1",
  "published": "2021-03-24T16:30:23.000Z",
  "updated": "2021-03-24T16:30:23.000Z",
  "title": "Convergence Rate Analysis for Deep Ritz Method",
  "authors": [
    "Chenguang Duan",
    "Yuling Jiao",
    "Yanming Lai",
    "Xiliang Lu",
    "Zhijian Yang"
  ],
  "categories": [
    "math.NA",
    "cs.NA"
  ],
  "abstract": "Using deep neural networks to solve PDEs has attracted a lot of attentions recently. However, why the deep learning method works is falling far behind its empirical success. In this paper, we provide a rigorous numerical analysis on deep Ritz method (DRM) \\cite{wan11} for second order elliptic equations with Neumann boundary conditions. We establish the first nonasymptotic convergence rate in $H^1$ norm for DRM using deep networks with $\\mathrm{ReLU}^2$ activation functions. In addition to providing a theoretical justification of DRM, our study also shed light on how to set the hyper-parameter of depth and width to achieve the desired convergence rate in terms of number of training samples. Technically, we derive bounds on the approximation error of deep $\\mathrm{ReLU}^2$ network in $H^1$ norm and on the Rademacher complexity of the non-Lipschitz composition of gradient norm and $\\mathrm{ReLU}^2$ network, both of which are of independent interest.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-03-24T16:30:23.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "deep ritz method",
      "convergence rate analysis",
      "first nonasymptotic convergence rate",
      "second order elliptic equations",
      "neumann boundary conditions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}