  Inspired by normalizing flows, we analyze the bilinear control of neural
transport equations by means of time-dependent velocity fields restricted to
fulfill, at any time instance, a simple neural network ansatz. The L^1
approximate controllability property is proved, showing that any probability
density can be driven arbitrarily close to any other one in any time horizon.
The control vector fields are built explicitly and inductively and this
provides quantitative estimates on their complexity and amplitude. This also
leads to statistical error bounds when only random samples of the target
probability density are available.
