{
  "id": "2206.00934",
  "version": "v1",
  "published": "2022-06-02T08:51:46.000Z",
  "updated": "2022-06-02T08:51:46.000Z",
  "title": "Deep neural networks can stably solve high-dimensional, noisy, non-linear inverse problems",
  "authors": [
    "Andrés Felipe Lerma Pineda",
    "Philipp Christian Petersen"
  ],
  "categories": [
    "math.NA",
    "cs.NA",
    "math.AP",
    "stat.ML"
  ],
  "abstract": "We study the problem of reconstructing solutions of inverse problems with neural networks when only noisy data is available. We assume the problem can be modeled with an infinite-dimensional forward operator that is not continuously invertible. Then, we restrict this forward operator to finite-dimensional spaces so that the inverse is Lipschitz continuous. For the inverse operator, we demonstrate that there exists a neural network which is a robust-to-noise approximation of the function. In addition, we show that these neural networks can be learned from appropriately perturbed training data. We demonstrate the admissibility of this approach to a wide range of inverse problems of practical interest. Numerical examples are given that support the theoretical findings.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-06-02T08:51:46.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35R30",
      "41A25",
      "68T05"
    ],
    "keywords": [
      "deep neural networks",
      "non-linear inverse problems",
      "high-dimensional",
      "wide range",
      "finite-dimensional spaces"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}