  While standard generative adversarial networks (GANs) rely solely on training
data to learn unknown probability distributions, physics-informed GANs
(PI-GANs) encode physical laws in the form of stochastic partial differential
equations (PDEs) using auto differentiation. By relating observed data to
unobserved quantities of interest through PDEs, PI-GANs allow for the
estimation of underlying probability distributions without their direct
measurement (i.e. inverse problems). The scalable nature of GANs allows
high-dimensional, spatially-dependent probability distributions (i.e., random
fields) to be inferred, while incorporating prior information through PDEs
allows the training datasets to be relatively small.
  In this work, PI-GANs are demonstrated for the application of elastic modulus
estimation in mechanical testing. Given measured deformation data, the
underlying probability distribution of spatially-varying elastic modulus
(stiffness) is learned. Two feed-forward deep neural network generators are
used to model the deformation and material stiffness across a two dimensional
domain. Wasserstein GANs with gradient penalty are employed for enhanced
stability. In the absence of explicit training data, it is demonstrated that
the PI-GAN learns to generate realistic, physically-admissible realizations of
material stiffness by incorporating the PDE that relates it to the measured
deformation. It is shown that the statistics (mean, standard deviation,
point-wise distributions, correlation length) of these generated stiffness
samples have good agreement with the true distribution.
