{
  "id": "2211.06299",
  "version": "v1",
  "published": "2022-11-06T21:59:40.000Z",
  "updated": "2022-11-06T21:59:40.000Z",
  "title": "Principled interpolation of Green's functions learned from data",
  "authors": [
    "Harshwardhan Praveen",
    "Nicolas Boulle",
    "Christopher Earls"
  ],
  "categories": [
    "math.NA",
    "cs.NA"
  ],
  "abstract": "We present a data-driven approach to mathematically model physical systems whose governing partial differential equations are unknown, by learning their associated Green's function. The subject systems are observed by collecting input-output pairs of system responses under excitations drawn from a Gaussian process. Two methods are proposed to learn the Green's function. In the first method, we use the proper orthogonal decomposition (POD) modes of the system as a surrogate for the eigenvectors of the Green's function, and subsequently fit the eigenvalues, using data. In the second, we employ a generalization of the randomized singular value decomposition (SVD) to operators, in order to construct a low-rank approximation to the Green's function. Then, we propose a manifold interpolation scheme, for use in an offline-online setting, where offline excitation-response data, taken at specific model parameter instances, are compressed into empirical eigenmodes. These eigenmodes are subsequently used within a manifold interpolation scheme, to uncover other suitable eigenmodes at unseen model parameters. The approximation and interpolation numerical techniques are demonstrated on several examples in one and two dimensions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-11-06T21:59:40.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "greens function",
      "principled interpolation",
      "manifold interpolation scheme",
      "specific model parameter instances",
      "governing partial differential equations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}