  Self-Organized Criticality (SOC) is a ubiquitous dynamical phenomenon
believed to be responsible for the emergence of universal scale-invariant
behavior in many, seemingly unrelated systems, such as forest fires, virus
spreading or atomic excitation dynamics. SOC describes the buildup of
large-scale and long-range spatio-temporal correlations as a result of only
local interactions and dissipation. The simulation of SOC dynamics is typically
based on Monte-Carlo (MC) methods, which are however numerically expensive and
do not scale beyond certain system sizes. We investigate the use of Graph
Neural Networks (GNNs) as an effective surrogate model to learn the dynamics
operator for a paradigmatic SOC system, inspired by an experimentally
accessible physics example: driven Rydberg atoms. To this end, we generalize
existing GNN simulation approaches to predict dynamics for the internal state
of the node. We show that we can accurately reproduce the MC dynamics as well
as generalize along the two important axes of particle number and particle
density. This paves the way to model much larger systems beyond the limits of
traditional MC methods. While the exact system is inspired by the dynamics of
Rydberg atoms, the approach is quite general and can readily be applied to
other systems.
