Neural ODE Control for Trajectory Approximation of Continuity Equation
This addresses theoretical control and approximation capabilities in machine learning, particularly for neural ODEs, but appears incremental as it builds on existing frameworks.
The paper tackles the controllability problem for the continuity equation in neural ODEs, showing that piecewise constant training weights can approximate any trajectory of probability measures arbitrarily closely, establishing approximate controllability on a set of measures.
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.