Bayesian Numerical Methods for Nonlinear Partial Differential Equations

TL;DR

This paper develops a Bayesian numerical approach for solving nonlinear PDEs, enabling probabilistic uncertainty quantification while efficiently managing computational costs.

Contribution

It extends Bayesian methods from linear to nonlinear PDEs, introducing an approximate likelihood based on discretization and analyzing prior models with novel theoretical insights.

Findings

Probabilistic solutions with quantified uncertainty are feasible.

The method reduces the number of costly PDE evaluations.

Theoretical analysis of Matérn processes informs prior selection.

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 the PDE is identified using novel theoretical analysis of the sample path properties of Mat\'{e}rn processes, which may be of independent interest.