Solving flows in porous media with a POD-Galerkin reduced order model coupled with multilayer perceptron

TL;DR

This paper introduces a novel reduced order model for porous media flow that integrates neural networks to efficiently handle nonlinear Forchheimer terms and pressure effects, improving accuracy and stability over standard methods.

Contribution

The paper proposes a combined POD-Galerkin reduced order model with neural networks to accurately model nonlinear and pressure effects in porous media flows, enhancing computational efficiency and predictive capability.

Findings

The ROM coupled with MLP accurately predicts flow dynamics.

The approach outperforms standard ROM in stability and long-term predictions.

Neural network modeling reduces computational cost for nonlinear terms.

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. 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.