Neural ODE Control for Trajectory Approximation of Continuity Equation

TL;DR

This paper demonstrates that neural ODEs can approximate the trajectories of probability measures governed by the continuity equation, showing strong controllability properties and the ability to closely match desired measure evolutions.

Contribution

It establishes the approximate controllability of the continuity equation for neural ODEs and shows how neural network weights can be trained to replicate measure trajectories.

Findings

Neural ODEs can approximate solutions of the continuity equation arbitrarily closely.

The controlled continuity equation exhibits strong controllability properties.

Neural ODEs are approximately controllable on certain sets of probability measures.

Abstract

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.