  We consider the controllability problem for the continuity equation,
corresponding to neural ordinary differential equations (ODEs), which describes
how a probability measure is pushedforward by the flow. We show that the
controlled continuity equation has very strong controllability properties.
Particularly, a given solution of the continuity equation corresponding to a
bounded Lipschitz vector field defines a trajectory on the set of probability
measures. For this trajectory, we show that there exist piecewise constant
training weights for a neural ODE such that the solution of the continuity
equation corresponding to the neural ODE is arbitrarily close to it. As a
corollary to this result, we establish that the continuity equation of the
neural ODE is approximately controllable on the set of compactly supported
probability measures that are absolutely continuous with respect to the
Lebesgue measure.
