Prediction of numerical homogenization using deep learning for the Richards equation

TL;DR

This paper introduces a deep learning-based method to efficiently predict macroscopic parameters for the nonlinear Richards equation, improving the accuracy and speed of numerical homogenization in heterogeneous media.

Contribution

The paper presents a novel deep neural network approach that models nonlinear maps between permeability fields and macroscopic properties, including the treatment of Richards equation's nonlinearity.

Findings

Deep learning accurately predicts effective permeability tensors.

The method significantly reduces computational time for homogenization.

Numerical tests confirm the approach's effectiveness in 2D problems.

Abstract

For the nonlinear Richards equation as an unsaturated flow through heterogeneous media, we build a new coarse-scale approximation algorithm utilizing numerical homogenization. This approach follows deep neural networks (DNNs) to quickly and frequently calculate macroscopic parameters. More specifically, we train neural networks with a training set consisting of stochastic permeability realizations and corresponding computed macroscopic targets (effective permeability tensor, homogenized stiffness matrix, and right-hand side vector). Our proposed deep learning scheme develops nonlinear maps between such permeability fields and macroscopic characteristics, and the treatment for Richards equation's nonlinearity is included in the predicted coarse-scale homogenized stiffness matrix, which is a novelty. This strategy's good performance is demonstrated by several numerical tests in two-dimensional model problems, for predictions of the macroscopic properties and consequently solutions.