  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.
