# High-dimensional Optimal Density Control with Wasserstein Metric Matching

arXiv: https://arxiv.org/abs/2307.13135 · HTML: https://mlsanity.com/paper/2307.13135/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 24 Jul 2023.

## Built on by (most cited first, 2 of 2)

- [Maximum Principle of Optimal Probability Density Control](https://mlsanity.com/paper/2505.18362/) (2025)
- [GradNetOT: Learning Optimal Transport Maps with GradNets](https://mlsanity.com/paper/2507.13191/) (2025)

## Abstract (arXiv)

We present a novel computational framework for density control in high-dimensional state spaces. The considered dynamical system consists of a large number of indistinguishable agents whose behaviors can be collectively modeled as a time-evolving probability distribution. The goal is to steer the agents from an initial distribution to reach (or approximate) a given target distribution within a fixed time horizon at minimum cost. To tackle this problem, we propose to model the drift as a nonlinear reduced-order model, such as a deep network, and enforce the matching to the target distribution at terminal time either strictly or approximately using the Wasserstein metric. The resulting saddle-point problem can be solved by an effective numerical algorithm that leverages the excellent representation power of deep networks and fast automatic differentiation for this challenging high-dimensional control problem. A variety of numerical experiments were conducted to demonstrate the performance of our method.