  This paper deals with the numerical modeling of flow around and through a
porous obstacle by a reduced order model (ROM) obtained by Galerkin projection
of the Navier-Stokes equations onto a Proper Orthogonal Decomposition (POD)
reduced basis. In the few existing works dealing with model reduction
techniques applied to flows in porous media, flows were described by Darcy's
law and the non linear Forchheimer term was neglected. This last term cannot be
expressed in reduced form during the Galerkin projection phase. Indeeed, at
each new time step, the norm of the velocity needs to be recalculated and
projected, which significantly increases the computational cost, rendering the
reduced model inefficient. To overcome this difficulty, we propose to model the
projected Forchheimer term with artificial neural networks. Moreover in order
to build a stable ROM, the influence of unresolved modes and pressure
variations are also modeled using a neural network. Instead of separately
modeling each term, these terms were combined into a single residual, which was
modeled using the multilayer perceptron method (MLP). The validation of this
approach was carried out for laminar flow past a porous obstacle in an
unconfined channel. The proposed ROM coupled with MLP approach is able to
accurately predict the dynamics of the flow while the standard ROM yields wrong
results. Moreover, the ROM MLP method improves the prediction of flow for
Reynols number that are not included in the sampling and for times longer than
sampling times.
