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Full information about a many-body quantum system is usually out-of-reach dueto the exponential growth -- with the size of the system -- of the number ofparameters needed to encode its state. Nonetheless, in order to understand thecomplex phenomenology that can be observed in these systems, it is oftensufficient to consider dynamical or stationary properties of local observablesor, at most, of few-body correlation functions. These quantities are typicallystudied by singling out a specific subsystem of interest and regarding theremainder of the many-body system as an effective bath. In the simplestscenario, the subsystem dynamics, which is in fact an open quantum dynamics,can be approximated through Markovian quantum master equations. Here, weformulate the problem of finding the generator of the subsystem dynamics as avariational problem, which we solve using the standard toolbox of machinelearning for optimization. This dynamical or ``Lindblad" generator provides therelevant dynamical parameters for the subsystem of interest. Importantly, thealgorithm we develop is constructed such that the learned generator implementsa physically consistent open quantum time-evolution. We exploit this to learnthe generator of the dynamics of a subsystem of a many-body system subject to aunitary quantum dynamics. We explore the capability of our method to recoverthe time-evolution of a two-body subsystem and exploit the physical consistencyof the generator to make predictions on the stationary state of the subsystemdynamics.

Inferring Markovian quantum master equations of few-body observables in interacting spin chains