  The integration of Graph Neural Networks (GNNs) and Neural Ordinary and
Partial Differential Equations has been extensively studied in recent years.
GNN architectures powered by neural differential equations allow us to reason
about their behavior, and develop GNNs with desired properties such as
controlled smoothing or energy conservation. In this paper we take inspiration
from Turing instabilities in a Reaction Diffusion (RD) system of partial
differential equations, and propose a novel family of GNNs based on neural RD
systems. We \textcolor{black}{demonstrate} that our RDGNN is powerful for the
modeling of various data types, from homophilic, to heterophilic, and
spatio-temporal datasets. We discuss the theoretical properties of our RDGNN,
its implementation, and show that it improves or offers competitive performance
to state-of-the-art methods.
