  We derive a Bernstein von-Mises theorem in the context of misspecified,
non-i.i.d., hierarchical models parametrized by a finite-dimensional parameter
of interest. We apply our results to hierarchical models containing non-linear
operators, including the squared integral operator, and PDE-constrained inverse
problems. More specifically, we consider the elliptic, time-independent
Schr\"odinger equation with parametric boundary condition and general parabolic
PDEs with parametric potential and boundary constraints. Our theoretical
results are complemented with numerical analysis on synthetic data sets,
considering both the square integral operator and the Schr\"odinger equation.
