  We present a deep learning framework for quantifying and propagating
uncertainty in systems governed by non-linear differential equations using
physics-informed neural networks. Specifically, we employ latent variable
models to construct probabilistic representations for the system states, and
put forth an adversarial inference procedure for training them on data, while
constraining their predictions to satisfy given physical laws expressed by
partial differential equations. Such physics-informed constraints provide a
regularization mechanism for effectively training deep generative models as
surrogates of physical systems in which the cost of data acquisition is high,
and training data-sets are typically small. This provides a flexible framework
for characterizing uncertainty in the outputs of physical systems due to
randomness in their inputs or noise in their observations that entirely
bypasses the need for repeatedly sampling expensive experiments or numerical
simulators. We demonstrate the effectiveness of our approach through a series
of examples involving uncertainty propagation in non-linear conservation laws,
and the discovery of constitutive laws for flow through porous media directly
from noisy data.
