Bayesian numerical methods for nonlinear partial differential equations

T. J. Sullivan,Oksana Chkrebtii,Junyang Wang,Jon Cockayne,Chris. J. Oates

doi:10.1007/s11222-021-10030-w

Abstract

The numerical solution of differential equations can be formulated as an inference problem to which formal statistical approaches can be applied. However, nonlinear partial differential equations (PDEs) pose substantial challenges from an inferential perspective, most notably the absence of explicit conditioning formula. This paper extends earlier work on linear PDEs to a general class of initial value problems specified by nonlinear PDEs, motivated by problems for which evaluations of the right-hand-side, initial conditions, or boundary conditions of the PDE have a high computational cost. The proposed method can be viewed as exact Bayesian inference under an approximate likelihood, which is based on discretisation of the nonlinear differential operator. Proof-of-concept experimental results demonstrate that meaningful probabilistic uncertainty quantification for the unknown solution of the PDE can be performed, while controlling the number of times the right-hand-side, initial and boundary conditions are evaluated. A suitable prior model for the solution of PDEs is identified using novel theoretical analysis of the sample path properties of Matérn processes, which may be of independent interest.

Highlights

IntroductionClassical numerical methods for differential equations produce an approximation to the solution of the differential equation whose error (called numerical error) is uncertain in general
Classical numerical methods for differential equations produce an approximation to the solution of the differential equation whose error is uncertain in general
This paper focuses on partial differential equations (PDEs); in particular, we focus on PDEs whose governing equations must be evaluated pointwise at high computational cost

Summary

Introduction

Classical numerical methods for differential equations produce an approximation to the solution of the differential equation whose error (called numerical error) is uncertain in general. Probability theory provides a natural language in which uncertainty can be expressed and, since the solution of a differential equation is unknown, it is interesting to ask whether probability theory can be applied to quantify numerical uncertainty. This perspective, in which numerical tasks are cast as problems of statistical inference, is pursued in the field of probabilistic numerics (Larkin 1972; Diaconis 1988; O‘Hagan 1992; Hennig et al 2015; Oates and Sullivan 2019). Wang et al (2020) argued that a rigorous Bayesian treatment of ordinary differential equations (ODEs) may only be possible in a limited context

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