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We consider the controllability problem for the continuity equation,corresponding to neural ordinary differential equations (ODEs), which describeshow a probability measure is pushedforward by the flow. We show that thecontrolled continuity equation has very strong controllability properties.Particularly, a given solution of the continuity equation corresponding to abounded Lipschitz vector field defines a trajectory on the set of probabilitymeasures. For this trajectory, we show that there exist piecewise constanttraining weights for a neural ODE such that the solution of the continuityequation corresponding to the neural ODE is arbitrarily close to it. As acorollary to this result, we establish that the continuity equation of theneural ODE is approximately controllable on the set of compactly supportedprobability measures that are absolutely continuous with respect to theLebesgue measure.

Neural ODE Control for Trajectory Approximation of Continuity Equation