  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.
