  Safe operation of multi-robot systems is critical, especially in
communication-degraded environments such as underwater for seabed mapping,
underground caves for navigation, and in extraterrestrial missions for assembly
and construction. We address safety of networked autonomous systems where the
information exchanged between robots incurs communication delays. We formalize
a notion of distributed control barrier function (CBF) for multi-robot systems,
a safety certificate amenable to a distributed implementation, which provides
formal ground to using graph neural networks to learn safe distributed
controllers. Further, we observe that learning a distributed controller
ignoring delays can severely degrade safety. Our main contribution is a
predictor-based framework to train a safe distributed controller under
communication delays, where the current state of nearby robots is predicted
from received data and age-of-information. Numerical experiments on multi-robot
collision avoidance show that our predictor-based approach can significantly
improve the safety of a learned distributed controller under communication
delays
