  We demonstrate that deep neural networks with the ReLU activation function
can efficiently approximate the solutions of various types of parametric linear
transport equations. For non-smooth initial conditions, the solutions of these
PDEs are high-dimensional and non-smooth. Therefore, approximation of these
functions suffers from a curse of dimension. We demonstrate that through their
inherent compositionality deep neural networks can resolve the characteristic
flow underlying the transport equations and thereby allow approximation rates
independent of the parameter dimension.
