A probabilistic model for the numerical solution of initial value problems

Michael Schober*, Simo Särkkä, Philipp Hennig

*Corresponding author for this work

Research output: Contribution to journalArticleScientificpeer-review

39 Citations (Scopus)
259 Downloads (Pure)


We study connections between ordinary differential equation (ODE) solvers and probabilistic regression methods in statistics. We provide a new view of probabilistic ODE solvers as active inference agents operating on stochastic differential equation models that estimate the unknown initial value problem (IVP) solution from approximate observations of the solution derivative, as provided by the ODE dynamics. Adding to this picture, we show that several multistep methods of Nordsieck form can be recasted as Kalman filtering on q-times integrated Wiener processes. Doing so provides a family of IVP solvers that return a Gaussian posterior measure, rather than a point estimate. We show that some such methods have low computational overhead, nontrivial convergence order, and that the posterior has a calibrated concentration rate. Additionally, we suggest a step size adaptation algorithm which completes the proposed method to a practically useful implementation, which we experimentally evaluate using a representative set of standard codes in the DETEST benchmark set.

Original languageEnglish
Pages (from-to)99-122
Number of pages24
Issue number1
Publication statusPublished - 15 Jan 2019
MoE publication typeA1 Journal article-refereed


  • Filtering
  • Gaussian processes
  • Initial value problems
  • Markov processes
  • Nordsieck methods
  • Probabilistic numerics
  • Runge–Kutta methods


Dive into the research topics of 'A probabilistic model for the numerical solution of initial value problems'. Together they form a unique fingerprint.

Cite this