  We introduce a priori Sobolev-space error estimates for the solution of
nonlinear, and possibly parametric, PDEs using Gaussian process and kernel
based methods. The primary assumptions are: (1) a continuous embedding of the
reproducing kernel Hilbert space of the kernel into a Sobolev space of
sufficient regularity; and (2) the stability of the differential operator and
the solution map of the PDE between corresponding Sobolev spaces. The proof is
articulated around Sobolev norm error estimates for kernel interpolants and
relies on the minimizing norm property of the solution. The error estimates
demonstrate dimension-benign convergence rates if the solution space of the PDE
is smooth enough. We illustrate these points with applications to
high-dimensional nonlinear elliptic PDEs and parametric PDEs. Although some
recent machine learning methods have been presented as breaking the curse of
dimensionality in solving high-dimensional PDEs, our analysis suggests a more
nuanced picture: there is a trade-off between the regularity of the solution
and the presence of the curse of dimensionality. Therefore, our results are in
line with the understanding that the curse is absent when the solution is
regular enough.
