Probabilistic solutions to ordinary differential equations as nonlinear Bayesian filtering: A new perspective

Filip Tronarp, Hans Kersting, Simo Särkkä, Philipp Hennig

Research output: Contribution to journalArticleScientificpeer-review

26 Citations (Scopus)
104 Downloads (Pure)


We formulate probabilistic numerical approximations to solutions of ordinary differential equations (ODEs) as problems in Gaussian process (GP) regression with non-linear measurement functions. This is achieved by defining the measurement sequence to consist of the observations of the difference between the derivative of the GP and the vector field evaluated at the GP---which are all identically zero at the solution of the ODE. When the GP has a state-space representation, the problem can be reduced to a non-linear Bayesian filtering problem and all widely-used approximations to the Bayesian filtering and smoothing problems become applicable. Furthermore, all previous GP-based ODE solvers that are formulated in terms of generating synthetic measurements of the gradient field come out as specific approximations. Based on the non-linear Bayesian filtering problem posed in this paper, we develop novel Gaussian solvers for which we establish favourable stability properties. Additionally, non-Gaussian approximations to the filtering problem are derived by the particle filter approach. The resulting solvers are compared with other probabilistic solvers in illustrative experiments.
Original languageEnglish
Pages (from-to)1297-1315
Number of pages19
Publication statusPublished - 18 Sept 2019
MoE publication typeA1 Journal article-refereed


  • Probabilistic numerics
  • Initial value problems
  • Non-linear Bayesian filtering


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