  Effective quantification of uncertainty is an essential and still missing
step towards a greater adoption of deep-learning approaches in different
applications, including mission-critical ones. In particular, investigations on
the predictive uncertainty of deep-learning models describing non-linear
dynamical systems are very limited to date. This paper is aimed at filling this
gap and presents preliminary results on uncertainty quantification for system
identification with neural state-space models. We frame the learning problem in
a Bayesian probabilistic setting and obtain posterior distributions for the
neural network's weights and outputs through approximate inference techniques.
Based on the posterior, we construct credible intervals on the outputs and
define a surprise index which can effectively diagnose usage of the model in a
potentially dangerous out-of-distribution regime, where predictions cannot be
trusted.
