A Role for Symmetry in the Bayesian Solution of Differential Equations

TL;DR

This paper introduces a novel Bayesian probabilistic numerical method for solving certain ordinary differential equations by leveraging symmetry properties and Lie group techniques, enabling exact Bayesian solutions under specific conditions.

Contribution

It proposes the first exact Bayesian PNM for a class of nonlinear ODEs by combining classical symmetry methods with non-parametric regression, filling a key gap in the field.

Findings

The method provides exact Bayesian solutions for first and second order ODEs.

Numerical experiments demonstrate the effectiveness of the approach.

The approach relies on the existence of a solvable Lie algebra for the ODE.

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.