  Graph neural networks are increasingly becoming the go-to approach in various
fields such as computer vision, computational biology and chemistry, where data
are naturally explained by graphs. However, unlike traditional convolutional
neural networks, deep graph networks do not necessarily yield better
performance than shallow graph networks. This behavior usually stems from the
over-smoothing phenomenon. In this work, we propose a family of architectures
to control this behavior by design. Our networks are motivated by numerical
methods for solving Partial Differential Equations (PDEs) on manifolds, and as
such, their behavior can be explained by similar analysis. Moreover, as we
demonstrate using an extensive set of experiments, our PDE-motivated networks
can generalize and be effective for various types of problems from different
fields. Our architectures obtain better or on par with the current
state-of-the-art results for problems that are typically approached using
different architectures.
