  Graph Neural Networks (GNNs) have been recently leveraged to solve several
logical reasoning tasks. Nevertheless, counting problems such as propositional
model counting (#SAT) are still mostly approached with traditional solvers.
Here we tackle this gap by presenting an architecture based on the GNN
framework for belief propagation (BP) of Kuch et al., extended with
self-attentive GNN and trained to approximately solve the #SAT problem. We ran
a thorough experimental investigation, showing that our model, trained on a
small set of random Boolean formulae, is able to scale effectively to much
larger problem sizes, with comparable or better performances of state of the
art approximate solvers. Moreover, we show that it can be efficiently
fine-tuned to provide good generalization results on different formulae
distributions, such as those coming from SAT-encoded combinatorial problems.
