  The lack of anisotropic kernels in graph neural networks (GNNs) strongly
limits their expressiveness, contributing to well-known issues such as
over-smoothing. To overcome this limitation, we propose the first globally
consistent anisotropic kernels for GNNs, allowing for graph convolutions that
are defined according to topologicaly-derived directional flows. First, by
defining a vector field in the graph, we develop a method of applying
directional derivatives and smoothing by projecting node-specific messages into
the field. Then, we propose the use of the Laplacian eigenvectors as such
vector field. We show that the method generalizes CNNs on an $n$-dimensional
grid and is provably more discriminative than standard GNNs regarding the
Weisfeiler-Lehman 1-WL test. We evaluate our method on different standard
benchmarks and see a relative error reduction of 8% on the CIFAR10 graph
dataset and 11% to 32% on the molecular ZINC dataset, and a relative increase
in precision of 1.6% on the MolPCBA dataset. An important outcome of this work
is that it enables graph networks to embed directions in an unsupervised way,
thus allowing a better representation of the anisotropic features in different
physical or biological problems.
