2023/07/24 by Shaojun Ma, Mengxue Hou, Ma, Shaojun +5 · 1 citation
Economics, Econometrics and Finance · Mathematics · #49L99 #76N25 #93E20 #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Optimization and Control (math.OC) #Stochastic processes and financial applications
paper · pdf · doi:10.48550/arxiv.2307.13135
openalex publication_date 2023/07/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.