  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.
