  We present a non-intrusive model reduction framework for linear
poroelasticity problems in heterogeneous porous media using proper orthogonal
decomposition (POD) and neural networks, based on the usual offline-online
paradigm. As the conductivity of porous media can be highly heterogeneous and
span several orders of magnitude, we utilize the interior penalty discontinuous
Galerkin (DG) method as a full order solver to handle discontinuity and ensure
local mass conservation during the offline stage. We then use POD as a data
compression tool and compare the nested POD technique, in which time and
uncertain parameter domains are compressed consecutively, to the classical POD
method in which all domains are compressed simultaneously. The neural networks
are finally trained to map the set of uncertain parameters, which could
correspond to material properties, boundary conditions, or geometric
characteristics, to the collection of coefficients calculated from an $L^2$
projection over the reduced basis. We then perform a non-intrusive evaluation
of the neural networks to obtain coefficients corresponding to new values of
the uncertain parameters during the online stage. We show that our framework
provides reasonable approximations of the DG solution, but it is significantly
faster. Moreover, the reduced order framework can capture sharp discontinuities
of both displacement and pressure fields resulting from the heterogeneity in
the media conductivity, which is generally challenging for intrusive reduced
order methods. The sources of error are presented, showing that the nested POD
technique is computationally advantageous and still provides comparable
accuracy to the classical POD method. We also explore the effect of different
choices of the hyperparameters of the neural network on the framework
performance.
