POD-Galerkin reduced order model coupled with neural networks to solve flow in porous media

Abstract

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. Indeed, 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 term, 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 Reynolds numbers that are not included in the sampling and for times longer than sampling times. In the final part of the paper, the ROM MLP method was compared with purely data driven methods. It was shown that the MLP method is superior to the purely data driven methods.