  The fractional advection-dispersion equation (FADE) has attracted increased
attention from researchers as it provides an accurate description for
challenging phenomenas with long-range time memory and spatial interactions,
such as the anomalous diffusion behavior in the solute transport in porous
media. Practically, a full characterization of the model parameters, such as
the fluid velocity, dispersion coefficient and the order of the fractional
derivative, often implies a huge amount of experiments and measurements and
thus are hard to be determined. In this paper, we employ the framework of
feedforward deep neural networks (DNNs) to develop an efficient data-driven
deep learning algorithm for inferring the aforementioned parameters of the
FADE, such as the time-dependent space-fractional advection-dispersion equation
(sFADE) and the variable-order fractional mobile/immobile equation (VoFMIE), in
which the feedforward DNNs are trained to minimize the mean square error loss
function formulated by means of the finite difference approximations of sFADE
and VoFMIE, respectively. Several numerical experiments, in which we discover
the model parameters by the feedforward DNNs for both the synthetic and field
data, are presented to demonstrate the effectiveness and robustness of the
proposed data-driven deep learning algorithm.
