# Optimal Control of Agent-Based Dynamics under Deep Galerkin Feedback Laws

arXiv: https://arxiv.org/abs/2406.09141 · HTML: https://mlsanity.com/paper/2406.09141/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 10 of its 22 references that are in the map.
- First posted on arXiv 13 Jun 2024.

## Builds on (most cited first, 10 of 10)

- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
- [The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems](https://mlsanity.com/paper/1710.00211/) (2017)
- [Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations](https://mlsanity.com/paper/1706.04702/) (2017)
- [Forward-Backward Stochastic Neural Networks: Deep Learning of High-dimensional Partial Differential Equations](https://mlsanity.com/paper/1804.07010/) (2018)
- [Mitigating the Curse of Dimensionality: Sparse Grid Characteristics Method for Optimal Feedback Control and HJB Equations](https://mlsanity.com/paper/1507.04769/) (2015)
- [Learning Optimal Control via Forward and Backward Stochastic Differential Equations](https://mlsanity.com/paper/1509.02195/) (2015)
- [Boltzmann type control of opinion consensus through leaders](https://mlsanity.com/paper/1405.0736/) (2014)
- [Error analysis for POD Approximations of infinite horizon problems via the Dynamic Programming approach](https://mlsanity.com/paper/1511.00265/) (2015)
- [Mean field control hierarchy](https://mlsanity.com/paper/1608.01728/) (2016)
- [Meanfield games and model predictive control](https://mlsanity.com/paper/1412.7517/) (2014)

## Abstract (arXiv)

Ever since the concepts of dynamic programming were introduced, one of the most difficult challenges has been to adequately address high-dimensional control problems. With growing dimensionality, the utilisation of Deep Neural Networks promises to circumvent the issue of an otherwise exponentially increasing complexity. The paper specifically investigates the sampling issues the Deep Galerkin Method is subjected to. It proposes a drift relaxation-based sampling approach to alleviate the symptoms of high-variance policy approximations. This is validated on mean-field control problems; namely, the variations of the opinion dynamics presented by the Sznajd and the Hegselmann-Krause model. The resulting policies induce a significant cost reduction over manually optimised control functions and show improvements on the Linear-Quadratic Regulator problem over the Deep FBSDE approach.