  Achieving ultimate bounds in estimation processes is the main objective of
quantum metrology. In this context, several problems require measurement of
multiple parameters by employing only a limited amount of resources. To this
end, adaptive protocols, exploiting additional control parameters, provide a
tool to optimize the performance of a quantum sensor to work in such limited
data regime. Finding the optimal strategies to tune the control parameters
during the estimation process is a non-trivial problem, and machine learning
techniques are a natural solution to address such task. Here, we investigate
and implement experimentally for the first time an adaptive Bayesian
multiparameter estimation technique tailored to reach optimal performances with
very limited data. We employ a compact and flexible integrated photonic
circuit, fabricated by femtosecond laser writing, which allows to implement
different strategies with high degree of control. The obtained results show
that adaptive strategies can become a viable approach for realistic sensors
working with a limited amount of resources.
