On the Bayesian Solution of Differential Equations

The interpretation of numerical methods, such as finite difference methods for differential equations, as point estimators allows for formal statistical quantification of the error due to discretisation in the numerical context. Competing statistical paradigms can be considered and Bayesian probabilistic numerical methods (PNMs) are obtained when Bayesian statistical principles are deployed. Bayesian PNM are closed under composition, such that uncertainty due to different sources of discretisation can be jointly modelled and rigorously propagated. However, we argue that no strictly Bayesian PNM for the numerical solution of ordinary differential equations (ODEs) have yet been developed. To address this gap, 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 the underlying structure of the gradient field, and non-parametric regression in a transformed solution space for the ODE. The procedure is presented in detail for first order ODEs and relies on a certain technical condition -- existence of a solvable Lie algebra -- being satisfied. Numerical illustrations are provided.