  The solution of probabilistic inverse problems for which the corresponding
forward problem is constrained by physical principles is challenging. This is
especially true if the dimension of the inferred vector is large and the prior
information about it is in the form of a collection of samples. In this work, a
novel deep learning based approach is developed and applied to solving these
types of problems. The approach utilizes samples of the inferred vector drawn
from the prior distribution and a physics-based forward model to generate
training data for a conditional Wasserstein generative adversarial network
(cWGAN). The cWGAN learns the probability distribution for the inferred vector
conditioned on the measurement and produces samples from this distribution. The
cWGAN developed in this work differs from earlier versions in that its critic
is required to be 1-Lipschitz with respect to both the inferred and the
measurement vectors and not just the former. This leads to a loss term with the
full (and not partial) gradient penalty. It is shown that this rather simple
change leads to a stronger notion of convergence for the conditional density
learned by the cWGAN and a more robust and accurate sampling strategy. Through
numerical examples it is shown that this change also translates to better
accuracy when solving inverse problems. The numerical examples considered
include illustrative problems where the true distribution and/or statistics are
known, and a more complex inverse problem motivated by applications in
biomechanics.
