{
  "id": "2201.11967",
  "version": "v1",
  "published": "2022-01-28T07:22:32.000Z",
  "updated": "2022-01-28T07:22:32.000Z",
  "title": "Pseudo-Differential Integral Operator for Learning Solution Operators of Partial Differential Equations",
  "authors": [
    "Jin Young Shin",
    "Jae Yong Lee",
    "Hyung Ju Hwang"
  ],
  "comment": "16 pages, 12 figures. This paper is under review on the Thirty-ninth International Conference on Machine Learning",
  "categories": [
    "cs.LG",
    "cs.NA",
    "math.NA"
  ],
  "abstract": "Learning mapping between two function spaces has attracted considerable research attention. However, learning the solution operator of partial differential equations (PDEs) remains a challenge in scientific computing. Therefore, in this study, we propose a novel pseudo-differential integral operator (PDIO) inspired by a pseudo-differential operator, which is a generalization of a differential operator and characterized by a certain symbol. We parameterize the symbol by using a neural network and show that the neural-network-based symbol is contained in a smooth symbol class. Subsequently, we prove that the PDIO is a bounded linear operator, and thus is continuous in the Sobolev space. We combine the PDIO with the neural operator to develop a pseudo-differential neural operator (PDNO) to learn the nonlinear solution operator of PDEs. We experimentally validate the effectiveness of the proposed model by using Burgers' equation, Darcy flow, and the Navier-Stokes equation. The results reveal that the proposed PDNO outperforms the existing neural operator approaches in most experiments.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-01-28T07:22:32.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35S05",
      "47G30",
      "68U07"
    ],
    "keywords": [
      "partial differential equations",
      "learning solution operators",
      "novel pseudo-differential integral operator",
      "smooth symbol class",
      "pseudo-differential neural operator"
    ],
    "tags": [
      "conference paper"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}