  We propose a novel iterative numerical method to solve the three-dimensional
inverse obstacle scattering problem of recovering the shape of the obstacle
from far-field measurements. To address the inherent ill-posed nature of the
inverse problem, we advocate the use of a trained latent representation of
surfaces as the generative prior. This prior enjoys excellent expressivity
within the given class of shapes, and meanwhile, the latent dimensionality is
low, which greatly facilitates the computation. Thus, the admissible manifold
of surfaces is realistic and the resulting optimization problem is less
ill-posed. We employ the shape derivative to evolve the latent surface
representation, by minimizing the loss, and we provide a local convergence
analysis of a gradient descent type algorithm to a stationary point of the
loss. We present several numerical examples, including also backscattered and
phaseless data, to showcase the effectiveness of the proposed algorithm.
