{
  "id": "2106.02735",
  "version": "v1",
  "published": "2021-06-04T22:00:53.000Z",
  "updated": "2021-06-04T22:00:53.000Z",
  "title": "Data-driven discovery of interacting particle systems using Gaussian processes",
  "authors": [
    "Jinchao Feng",
    "Yunxiang Ren",
    "Sui Tang"
  ],
  "comment": "10 pages; Appendix 19 pages;",
  "categories": [
    "stat.ML",
    "cs.LG",
    "cs.NA",
    "math.NA",
    "math.ST",
    "stat.TH"
  ],
  "abstract": "Interacting particle or agent systems that display a rich variety of collection motions are ubiquitous in science and engineering. A fundamental and challenging goal is to understand the link between individual interaction rules and collective behaviors. In this paper, we study the data-driven discovery of distance-based interaction laws in second-order interacting particle systems. We propose a learning approach that models the latent interaction kernel functions as Gaussian processes, which can simultaneously fulfill two inference goals: one is the nonparametric inference of interaction kernel function with the pointwise uncertainty quantification, and the other one is the inference of unknown parameters in the non-collective forces of the system. We formulate learning interaction kernel functions as a statistical inverse problem and provide a detailed analysis of recoverability conditions, establishing that a coercivity condition is sufficient for recoverability. We provide a finite-sample analysis, showing that our posterior mean estimator converges at an optimal rate equal to the one in the classical 1-dimensional Kernel Ridge regression. Numerical results on systems that exhibit different collective behaviors demonstrate efficient learning of our approach from scarce noisy trajectory data.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-06-04T22:00:53.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "interacting particle systems",
      "gaussian processes",
      "data-driven discovery",
      "learning interaction kernel functions",
      "behaviors demonstrate efficient learning"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}