  A fundamental question regarding the Galactic Center Excess (GCE) is whether
the underlying structure is point-like or smooth. This debate, often framed in
terms of a millisecond pulsar or annihilating dark matter (DM) origin for the
emission, awaits a conclusive resolution. In this work we weigh in on the
problem using Bayesian graph convolutional neural networks. In simulated data,
our neural network (NN) is able to reconstruct the flux of inner Galaxy
emission components to on average $\sim$0.5%, comparable to the non-Poissonian
template fit (NPTF). When applied to the actual $\textit{Fermi}$-LAT data, we
find that the NN estimates for the flux fractions from the background templates
are consistent with the NPTF; however, the GCE is almost entirely attributed to
smooth emission. While suggestive, we do not claim a definitive resolution for
the GCE, as the NN tends to underestimate the flux of point-sources peaked near
the 1$\sigma$ detection threshold. Yet the technique displays robustness to a
number of systematics, including reconstructing injected DM, diffuse
mismodeling, and unmodeled north-south asymmetries. So while the NN is hinting
at a smooth origin for the GCE at present, with further refinements we argue
that Bayesian Deep Learning is well placed to resolve this DM mystery.
