Neural ODE Control for Trajectory Approximation of Continuity Equation

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