  The transport of traffic flow can be modeled by the advection equation.
Finite difference and finite volumes methods have been used to numerically
solve this hyperbolic equation on a mesh. Advection has also been modeled
discretely on directed graphs using the graph advection operator [4, 18]. In
this paper, we first show that we can reformulate this graph advection operator
as a finite difference scheme. We then propose the Directed Graph Advection
Mat\'ern Gaussian Process (DGAMGP) model that incorporates the dynamics of this
graph advection operator into the kernel of a trainable Mat\'ern Gaussian
Process to effectively model traffic flow and its uncertainty as an advective
process on a directed graph.
