  Bayesian inference and uncertainty quantification in a general class of
non-linear inverse regression models is considered. Analytic conditions on the
regression model $\{\mathscr G(\theta): \theta \in \Theta\}$ and on Gaussian
process priors for $\theta$ are provided such that semi-parametrically
efficient inference is possible for a large class of linear functionals of
$\theta$. A general semi-parametric Bernstein-von Mises theorem is proved that
shows that the (non-Gaussian) posterior distributions are approximated by
certain Gaussian measures centred at the posterior mean. As a consequence
posterior-based credible sets are valid and optimal from a frequentist point of
view. The theory is illustrated with two applications with PDEs that arise in
non-linear tomography problems: an elliptic inverse problem for a Schr\"odinger
equation, and inversion of non-Abelian X-ray transforms. New analytical
techniques are deployed to show that the relevant Fisher information operators
are invertible between suitable function spaces
