  Generally, reduced order models of fluid flows are obtained by projecting the
Navier-Stokes equations onto a reduced subspace spanned by vector functions
that carry the meaningful information of the dynamics. A common method to
generate such subspace is the Proper Orthogonal Decomposition. This projection
strategy is intrusive since it assumes that the approximation model used to
solve the high fidelity flow problem is known and accessible. In the present
article, we propose a non-intrusive paradigm based on Machine Learning to build
Closed Parametric Reduced Order Models (ML-PROM) relevant to fluid dynamics.
With no prior knowledge requirement of the approximation model, this method is
purely data-driven as it operates directly on data regardless their origin,
DNS, RANS simulations or experiment. The key idea to build the ML-CPROM is to
use the generally known form of Galerkin ROMs and assimilate the time
variations of the temporal POD modes to a nonlinear quadratic form taking a
closure right hand side member. This last term is introduced to account for
uncaptured dynamics due to data noise, POD truncation errors and time
integration schemes. For parameter variation, the ML-CPROM is updated by
interpolation on the quotient manifold of the set of fixed rank matrices by the
orthogonal group. The predicted closure term is afterwards calculated by a
Long-Short-Term-Memory neural network. We assess the potential of the proposed
approach on the parametric examples of the lid-driven cavity flow and the flow
past a cylinder by varying the Reynolds number through the viscosity, and the
Ahmed-Body flow by varying the geometry through the rear slant angle. We show
on these examples that our method enables to recover the dynamics with a good
accuracy, not only for the training parameter points, but also for unseen
parameter values for which the model was not priory trained.
