Neural ODE Control for Trajectory Approximation of Continuity Equation

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. Par...

Full description

Saved in:
Bibliographic Details
Published in:IEEE control systems letters 2022, Vol.6, p.3152-3157
Main Authors: Elamvazhuthi, Karthik, Gharesifard, Bahman, Bertozzi, Andrea L., Osher, Stanley
Format: Article
Language:English
Subjects:
Controllability
Distributed parameter systems
Estimation
machine learning
Neural networks
Particle measurements
Topology
Training
Trajectory
Citations: Items that this one cites
Items that cite this one
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary: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.
ISSN:2475-1456
2475-1456
DOI:10.1109/LCSYS.2022.3182284