  Natural convection in porous media is a highly nonlinear multiphysical
problem relevant to many engineering applications (e.g., the process of
$\mathrm{CO_2}$ sequestration). Here, we present a non-intrusive reduced order
model of natural convection in porous media employing deep convolutional
autoencoders for the compression and reconstruction and either radial basis
function (RBF) interpolation or artificial neural networks (ANNs) for mapping
parameters of partial differential equations (PDEs) on the corresponding
nonlinear manifolds. To benchmark our approach, we also describe linear
compression and reconstruction processes relying on proper orthogonal
decomposition (POD) and ANNs. We present comprehensive comparisons among
different models through three benchmark problems. The reduced order models,
linear and nonlinear approaches, are much faster than the finite element model,
obtaining a maximum speed-up of $7 \times 10^{6}$ because our framework is not
bound by the Courant-Friedrichs-Lewy condition; hence, it could deliver
quantities of interest at any given time contrary to the finite element model.
Our model's accuracy still lies within a mean squared error of 0.07 (two-order
of magnitude lower than the maximum value of the finite element results) in the
worst-case scenario. We illustrate that, in specific settings, the nonlinear
approach outperforms its linear counterpart and vice versa. We hypothesize that
a visual comparison between principal component analysis (PCA) or t-Distributed
Stochastic Neighbor Embedding (t-SNE) could indicate which method will perform
better prior to employing any specific compression strategy.
