Prediction of discretization of online GMsFEM using deep learning for Richards equation

TL;DR

This paper introduces a novel approach combining online GMsFEM with deep learning to efficiently predict multiscale basis functions for Richards equation, enabling rapid and adaptive nonlinear flow simulations in heterogeneous media.

Contribution

The paper presents a new deep learning-based method to quickly compute online multiscale basis functions, improving the efficiency of nonlinear Richards equation simulations.

Findings

Deep learning accurately predicts basis functions.

Method effectively captures nonlinearity and time-dependence.

Numerical experiments demonstrate good performance.

Abstract

We develop a new coarse-scale approximation strategy for the nonlinear single-continuum Richards equation as an unsaturated flow over heterogeneous non-periodic media, using the online generalized multiscale finite element method (online GMsFEM) together with deep learning. A novelty of this approach is that local online multiscale basis functions are computed rapidly and frequently by utilizing deep neural networks (DNNs). More precisely, we employ the training set of stochastic permeability realizations and the computed relating online multiscale basis functions to train neural networks. The nonlinear map between such permeability fields and online multiscale basis functions is developed by our proposed deep learning algorithm. That is, in a new way, the predicted online multiscale basis functions incorporate the nonlinearity treatment of the Richards equation and refect any time-dependent changes in the problem's properties. Multiple numerical experiments in two-dimensional model problems show the good performance of this technique, in terms of predictions of the online multiscale basis functions and thus finding solutions.