  We present an application of deep generative models in the context of
partial-differential equation (PDE) constrained inverse problems. We combine a
generative adversarial network (GAN) representing an a priori model that
creates subsurface geological structures and their petrophysical properties,
with the numerical solution of the PDE governing the propagation of acoustic
waves within the earth's interior. We perform Bayesian inversion using an
approximate Metropolis-adjusted Langevin algorithm (MALA) to sample from the
posterior given seismic observations. Gradients with respect to the model
parameters governing the forward problem are obtained by solving the adjoint of
the acoustic wave equation. Gradients of the mismatch with respect to the
latent variables are obtained by leveraging the differentiable nature of the
deep neural network used to represent the generative model. We show that
approximate MALA sampling allows efficient Bayesian inversion of model
parameters obtained from a prior represented by a deep generative model,
obtaining a diverse set of realizations that reflect the observed seismic
response.
