  Inspired by our previous work on mitigating the Kolmogorov barrier using a
quadratic approximation manifold, we propose in this paper a computationally
tractable approach for combining a projection-based reduced-order model (PROM)
and an artificial neural network (ANN) for mitigating the Kolmogorov barrier to
reducibility of convection-dominated flow problems. The main objective the
PROM-ANN concept that we propose is to reduce the dimensionality of the online
approximation of the solution beyond what is possible using affine and
quadratic approximation manifolds. In contrast to previous approaches for
constructing arbitrarily nonlinear manifold approximations for nonlinear model
reduction that exploited one form or another of ANN, the training of the
PROM-ANN we propose in this paper does not involve data whose dimension scales
with that of the high-dimensional model; and this PROM-ANN is hyperreducible
using any well-established hyperreduction method. Hence, unlike many other
ANN-based approaches, the PROM-ANN concept we propose in this paper is
practical for large-scale and industry-relevant CFD problems. Its potential is
demonstrated here for a parametric, shock-dominated, benchmark problem.
