Probabilistic Solutions To Ordinary Differential Equations As Non-Linear Bayesian Filtering: A New Perspective

TL;DR

This paper introduces a novel approach to solving ordinary differential equations by framing them as non-linear Bayesian filtering problems using Gaussian processes, leading to new stable and flexible probabilistic solvers.

Contribution

It formulates probabilistic ODE solutions as non-linear Bayesian filtering problems and develops new Gaussian and non-Gaussian solvers with improved stability and flexibility.

Findings

New Gaussian solvers with favorable stability properties.

Non-Gaussian solvers via particle filtering.

Comparative experiments showing advantages over existing methods.

Abstract

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.