A Role for Symmetry in the Bayesian Solution of Differential Equations

Abstract

The interpretation of numerical methods, such as finite difference methods for differential equations, as point estimators suggests that formal uncertainty quantification can also be performed in this context. Competing statistical paradigms can be considered and Bayesian probabilistic numerical methods (PNMs) are obtained when Bayesian statistical principles are deployed. Bayesian PNM have the appealing property of being closed under composition, such that uncertainty due to different sources of discretisation in a numerical method can be jointly modelled and rigorously propagated. Despite recent attention, no exact Bayesian PNM for the numerical solution of ordinary differential equations (ODEs) has been proposed. This raises the fundamental question of whether exact Bayesian methods for (in general nonlinear) ODEs even exist. The purpose of this paper is to provide a positive answer for a limited class of ODE. To this end, we work at a foundational level, where a novel Bayesian PNM is proposed as a proof-of-concept. Our proposal is a synthesis of classical Lie group methods, to exploit underlying symmetries in the gradient field, and non-parametric regression in a transformed solution space for the ODE. The procedure is presented in detail for first and second order ODEs and relies on a certain strong technical condition -existence of a solvable Lie algebra -being satisfied. Numerical illustrations are provided.

Convergence rates of Gaussian ODE filters

“…Here, the word 'Bayesian' describes the algorithm in the sense that it employs a prior over the quantity of interest and updates it by Bayes rule according to a prespecified measurement model (as also used inSkilling (1991);Chkrebtii et al (2016);Kersting and Hennig (2016)). The ODE filter is not Bayesian in the stronger sense ofCockayne et al (2019), and it remains an open problem to construct a Bayesian solver in this strong sense without restrictive assumptions, as discussed inWang et al (2018).…”

Randomised one-step time integration methods for deterministic operator differential equations

“…For some reviews of research in this area, see [9,17,27]. In probabilistic numerics, ODEs have been considered from many perspectives, including structure-or symmetry-preserving methods [1,40], Bayesian modelling of the unknown solution with Gaussian processes [5,10,33,36,38,40], data-based statistical estimation of discretisation error [24,35], and filtering [19,38]. The papers [10,20] cited earlier also belong to this context.…”

A Probabilistic Taylor Expansion with Applications in Filtering and Differential Equations

“…We now use a first-order Taylor kernel to develop a probabilistic Euler method which is capable of uncertainty quantification for the unknown solution. A number of Gaussian process based (Schober et al, 2014(Schober et al, , 2019Teymur et al, 2016;Tronarp et al, 2019;Wang et al, 2020) and other probabilistic (Conrad et al, 2016) differential equation solvers have been proposed previously.…”