  In this work we present an application of modern deep learning methodologies
to the numerical solution of partial differential equations in transport
models. More specifically, we employ a supervised deep neural network that
takes into account the equation and initial conditions of the model. We apply
it to the Riemann problems over the inviscid nonlinear Burger's equation, whose
solutions might develop discontinuity (shock wave) and rarefaction, as well as
to the classical one-dimensional Buckley-Leverett two-phase problem. The
Buckley-Leverett case is slightly more complex and interesting because it has a
non-convex flux function with one inflection point. Our results suggest that a
relatively simple deep learning model was capable of achieving promising
results in such challenging tasks, providing numerical approximation of entropy
solutions with very good precision and consistent to classical as well as to
recently novel numerical methods in these particular scenarios.
