{
  "id": "2210.08140",
  "version": "v1",
  "published": "2022-10-14T22:33:28.000Z",
  "updated": "2022-10-14T22:33:28.000Z",
  "title": "A Kernel Approach for PDE Discovery and Operator Learning",
  "authors": [
    "Da Long",
    "Nicole Mrvaljevic",
    "Shandian Zhe",
    "Bamdad Hosseini"
  ],
  "categories": [
    "stat.ML",
    "cs.LG"
  ],
  "abstract": "This article presents a three-step framework for learning and solving partial differential equations (PDEs) using kernel methods. Given a training set consisting of pairs of noisy PDE solutions and source/boundary terms on a mesh, kernel smoothing is utilized to denoise the data and approximate derivatives of the solution. This information is then used in a kernel regression model to learn the algebraic form of the PDE. The learned PDE is then used within a kernel based solver to approximate the solution of the PDE with a new source/boundary term, thereby constituting an operator learning framework. The proposed method is mathematically interpretable and amenable to analysis, and convenient to implement. Numerical experiments compare the method to state-of-the-art algorithms and demonstrate its superior performance on small amounts of training data and for PDEs with spatially variable coefficients.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-10-14T22:33:28.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "62F30",
      "68T99"
    ],
    "keywords": [
      "pde discovery",
      "operator learning",
      "kernel approach",
      "source/boundary term",
      "kernel regression model"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}