# Feedback maximum principle for ensemble control of local continuity equations. An application to supervised machine learning

arXiv: https://arxiv.org/abs/2105.04248 · HTML: https://mlsanity.com/paper/2105.04248/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 10 May 2021.

## Built on by (most cited first, 1 of 1)

- [Ensemble based Closed-Loop Optimal Control using Physics-Informed Neural Networks](https://mlsanity.com/paper/2510.18195/) (2025)

## Abstract (arXiv)

We consider an optimal control problem for a system of local continuity equations on a space of probability measures. Such systems can be viewed as macroscopic models of ensembles of non-interacting particles or homotypic individuals, representing several different ``populations''. For the stated problem, we propose a necessary optimality condition, which involves feedback controls inherent to the extremal structure, designed via the standard Pontryagin's Maximum Principle conditions. This optimality condition admits a realization as an iterative algorithm for optimal control. As a motivating case, we discuss an application of the derived optimality condition and the consequent numeric method to a problem of supervised machine learning via dynamic systems.