{
  "id": "2110.11847",
  "version": "v2",
  "published": "2021-10-22T15:26:05.000Z",
  "updated": "2022-03-09T15:00:49.000Z",
  "title": "Probabilistic Numerical Method of Lines for Time-Dependent Partial Differential Equations",
  "authors": [
    "Nicholas Krämer",
    "Jonathan Schmidt",
    "Philipp Hennig"
  ],
  "categories": [
    "math.NA",
    "cs.NA",
    "stat.ML"
  ],
  "abstract": "This work develops a class of probabilistic algorithms for the numerical solution of nonlinear, time-dependent partial differential equations (PDEs). Current state-of-the-art PDE solvers treat the space- and time-dimensions separately, serially, and with black-box algorithms, which obscures the interactions between spatial and temporal approximation errors and misguides the quantification of the overall error. To fix this issue, we introduce a probabilistic version of a technique called method of lines. The proposed algorithm begins with a Gaussian process interpretation of finite difference methods, which then interacts naturally with filtering-based probabilistic ordinary differential equation (ODE) solvers because they share a common language: Bayesian inference. Joint quantification of space- and time-uncertainty becomes possible without losing the performance benefits of well-tuned ODE solvers. Thereby, we extend the toolbox of probabilistic programs for differential equation simulation to PDEs.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2022-03-09T15:00:49.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "time-dependent partial differential equations",
      "probabilistic numerical method",
      "probabilistic ordinary differential equation",
      "current state-of-the-art pde solvers treat"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}