{
  "id": "2106.01078",
  "version": "v1",
  "published": "2021-06-02T11:24:49.000Z",
  "updated": "2021-06-02T11:24:49.000Z",
  "title": "KO-PDE: Kernel Optimized Discovery of Partial Differential Equations with Varying Coefficients",
  "authors": [
    "Yingtao Luo",
    "Qiang Liu",
    "Yuntian Chen",
    "Wenbo Hu",
    "Jun Zhu"
  ],
  "comment": "Preprint. Under review",
  "categories": [
    "cs.LG",
    "cs.NA",
    "math.NA"
  ],
  "abstract": "Partial differential equations (PDEs) fitting scientific data can represent physical laws with explainable mechanisms for various mathematically-oriented subjects. Most natural dynamics are expressed by PDEs with varying coefficients (PDEs-VC), which highlights the importance of PDE discovery. Previous algorithms can discover some simple instances of PDEs-VC but fail in the discovery of PDEs with coefficients of higher complexity, as a result of coefficient estimation inaccuracy. In this paper, we propose KO-PDE, a kernel optimized regression method that incorporates the kernel density estimation of adjacent coefficients to reduce the coefficient estimation error. KO-PDE can discover PDEs-VC on which previous baselines fail and is more robust against inevitable noise in data. In experiments, the PDEs-VC of seven challenging spatiotemporal scientific datasets in fluid dynamics are all discovered by KO-PDE, while the three baselines render false results in most cases. With state-of-the-art performance, KO-PDE sheds light on the automatic description of natural phenomenons using discovered PDEs in the real world.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-06-02T11:24:49.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "partial differential equations",
      "kernel optimized discovery",
      "varying coefficients",
      "seven challenging spatiotemporal scientific datasets",
      "baselines render false results"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}