A data-driven approach for a macroscopic conductivity model utilizing finite element approximation

Macroscopic modeling is useful in many application areas, such as flow simulation in porous media and reduced order approximation for fast solvers and multi-physics simulations. The focus of this work is to propose an algorithm for macroscopic modeling for elliptic problems with coefficients of random and high variations, which can often be used to describe microscopic structures in porous media applications, composite materials, and medical imaging applications. The proposed method approximates a general tensor type macroscopic conductivity with a deep neural network and the parameters in the deep neural network are then optimized using the measured data on the boundary of a microscopic model. Numerical results for various test examples are presented and they show that the proposed scheme is promising for macroscopic modeling.