  Large-scale outflows driven by active galactic nuclei (AGN) can have a
profound influence on their host galaxies. The outflow properties themselves
depend sensitively on the history of AGN energy injection during the lifetime
of the outflow. Most observed outflows have dynamical timescales longer than
the typical AGN episode duration, i.e. they have been inflated by multiple AGN
episodes. Here, we present a neural-network based approach to inferring the
most likely duty cycle and other properties of AGN based on the observable
properties of their massive outflows. Our model recovers the AGN parameters of
simulated outflows with typical errors $< 25\%$. We apply the method to a
sample of 59 real molecular outflows and show that a large fraction of them
have been inflated by AGN shining with a rather high duty cycle $\delta_{\rm
AGN} > 0.2$. This result suggests that nuclear activity in galaxies is
clustered hierarchically in time, with long phases of more frequent activity
composed of many short activity episodes. We predict that $\sim \! 19\%$ of
galaxies should have AGN-driven outflows, but half of them are fossils - this
is consistent with currently available data. We discuss the possibilities to
investigate AGN luminosity histories during outflow lifetimes and suggest ways
to use our software to test other physical models of AGN outflows. The source
code of all of the software used here is made public.
