  We present a Parametrization of the Physics Informed Neural Network (P-PINN)
approach to tackle the problem of uncertainty quantification in reservoir
engineering problems. We demonstrate the approach with the immiscible two phase
flow displacement (Buckley-Leverett problem) in heterogeneous porous medium.
The reservoir properties (porosity, permeability) are treated as random
variables. The distribution of these properties can affect dynamic properties
such as the fluids saturation, front propagation speed or breakthrough time. We
explore and use to our advantage the ability of networks to interpolate complex
high dimensional functions. We observe that the additional dimensions resulting
from a stochastic treatment of the partial differential equations tend to
produce smoother solutions on quantities of interest (distributions parameters)
which is shown to improve the performance of PINNS. We show that provided a
proper parameterization of the uncertainty space, PINN can produce solutions
that match closely both the ensemble realizations and the stochastic moments.
We demonstrate applications for both homogeneous and heterogeneous fields of
properties. We are able to solve problems that can be challenging for classical
methods. This approach gives rise to trained models that are both more robust
to variations in the input space and can compete in performance with
traditional stochastic sampling methods.
