  We present a reduced order modeling (ROM) technique for subsurface
multi-phase flow problems building on the recently introduced deep residual
recurrent neural network (DR-RNN) [1]. DR-RNN is a physics aware recurrent
neural network for modeling the evolution of dynamical systems. The DR-RNN
architecture is inspired by iterative update techniques of line search methods
where a fixed number of layers are stacked together to minimize the residual
(or reduced residual) of the physical model under consideration. In this
manuscript, we combine DR-RNN with proper orthogonal decomposition (POD) and
discrete empirical interpolation method (DEIM) to reduce the computational
complexity associated with high-fidelity numerical simulations. In the
presented formulation, POD is used to construct an optimal set of reduced basis
functions and DEIM is employed to evaluate the nonlinear terms independent of
the full-order model size.
  We demonstrate the proposed reduced model on two uncertainty quantification
test cases using Monte-Carlo simulation of subsurface flow with random
permeability field. The obtained results demonstrate that DR-RNN combined with
POD-DEIM provides an accurate and stable reduced model with a fixed
computational budget that is much less than the computational cost of standard
POD-Galerkin reduced model combined with DEIM for nonlinear dynamical systems.
