  We explore a supervised machine learning approach to estimate the
entanglement entropy of multi-qubit systems from few experimental samples. We
put a particular focus on estimating both aleatoric and epistemic uncertainty
of the network's estimate and benchmark against the best known conventional
estimation algorithms. For states that are contained in the training
distribution, we observe convergence in a regime of sample sizes in which the
baseline method fails to give correct estimates, while extrapolation only seems
possible for regions close to the training regime. As a further application of
our method, highly relevant for quantum simulation experiments, we estimate the
quantum mutual information for non-unitary evolution by training our model on
different noise strengths.
