High-dimensional Optimal Density Control with Wasserstein Metric Matching

TL;DR

This paper introduces a new computational approach for controlling the distribution of large agent populations in high-dimensional spaces, using deep networks and Wasserstein metric for distribution matching.

Contribution

It proposes a novel framework combining deep network-based reduced-order models with Wasserstein metric for high-dimensional density control.

Findings

Effective numerical algorithm developed for high-dimensional control.

Demonstrated ability to steer distributions with minimal cost.

Applicable to complex, high-dimensional agent systems.

Abstract

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.