Probabilistic ODE Solvers with Runge-Kutta Means
Abstract
Runge-Kutta methods are the classic family of solvers for ordinary differential equations (ODEs), and the basis for the state of the art. Like most numerical methods, they return point estimates. We construct a family of probabilistic numerical methods that instead return a Gauss-Markov process defining a probability distribution over the ODE solution. In contrast to prior work, we construct this family such that posterior means match the outputs of the Runge-Kutta family exactly, thus inheriting their proven good properties. Remaining degrees of freedom not identified by the match to Runge-Kutta are chosen such that the posterior probability measure fits the observed structure of the ODE. Our results shed light on the structure of Runge-Kutta solvers from a new direction, provide a richer, probabilistic output, have low computational cost, and raise new research questions.
Cite
@article{arxiv.1406.2582,
title = {Probabilistic ODE Solvers with Runge-Kutta Means},
author = {Michael Schober and David Duvenaud and Philipp Hennig},
journal= {arXiv preprint arXiv:1406.2582},
year = {2014}
}
Comments
18 pages (9 page conference paper, plus supplements); appears in Advances in Neural Information Processing Systems (NIPS), 2014