  Entangled quantum many-body systems can be used as sensors that enable the
estimation of parameters with a precision larger than that achievable with
ensembles of individual quantum detectors. Typically, the parameter estimation
strategy requires the microscopic modelling of the quantum many-body system, as
well as a an accurate description of its dynamics. This entails a complexity
that can hinder the applicability of Bayesian inference techniques. In this
work we show how to circumvent these issues by using neural networks that
faithfully reproduce the dynamics of quantum many-body sensors, thus allowing
for an efficient Bayesian analysis. We exemplify with an XXZ model driven by
magnetic fields, and show that our method is capable to yield an estimation of
field parameters beyond the standard quantum limit scaling. Our work paves the
way for the practical use of quantum many-body systems as black-box sensors
exploiting quantum resources to improve precision estimation.
