Bayesian Filtering for ODEs with Bounded Derivatives

TL;DR

This paper explores probabilistic methods for solving ODEs using Bayesian filtering with different priors, introducing the IOUP prior to better model solutions with bounded derivatives, and compares it to the existing IWP approach.

Contribution

It introduces the IOUP prior for Bayesian ODE filtering, enhancing modeling of solutions with bounded derivatives, and compares its performance to the IWP prior.

Findings

IOUP better models solutions with bounded derivatives

IWP approximates divergent ODE solutions more effectively

Experimental results support the suitability of each prior for different types of ODEs

Abstract

Recently there has been increasing interest in probabilistic solvers for ordinary differential equations (ODEs) that return full probability measures, instead of point estimates, over the solution and can incorporate uncertainty over the ODE at hand, e.g. if the vector field or the initial value is only approximately known or evaluable. The ODE filter proposed in recent work models the solution of the ODE by a Gauss-Markov process which serves as a prior in the sense of Bayesian statistics. While previous work employed a Wiener process prior on the (possibly multiple times) differentiated solution of the ODE and established equivalence of the corresponding solver with classical numerical methods, this paper raises the question whether other priors also yield practically useful solvers. To this end, we discuss a range of possible priors which enable fast filtering and propose a new prior--the Integrated Ornstein Uhlenbeck Process (IOUP)--that complements the existing Integrated Wiener process (IWP) filter by encoding the property that a derivative in time of the solution is bounded in the sense that it tends to drift back to zero. We provide experiments comparing IWP and IOUP filters which support the belief that IWP approximates better divergent ODE's solutions whereas IOUP is a better prior for trajectories with bounded derivatives.