Bayesian numerical methods for nonlinear partial differential equations

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.

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

“…The papers [10,20] cited earlier also belong to this context. For PDEs, methods based on Bayesian inference and Gaussian processes [6,8,10,28,31,39], multiscale techniques [29], and random meshes [2] have been studied. The research area of 'information field dynamics' [11,14] also considers probabilistic simulation schemes for PDEs by using Gaussian processes and information theoretic ideas.…”

Sparse Gaussian processes for solving nonlinear PDEs

“…The authors of [1] extend these ideas and propose a Gaussian process (GP) framework for solving general nonlinear PDEs. In particular, for time-dependent PDEs, GP methods based on time-discretization are considered in [11,23,32]. The computational costs of the above GP methods grow cubically with respect to the number of samples due to the inversion of covariance matrices.…”