Hamiltonian Theory and Computation of Optimal Probability Density Control in High Dimensions

2025-05-23

AI Summary

This work addresses the optimal probability density control problem in high-dimensional spaces with obstacles and nonlinear interactions. Methodologically, it introduces a deep neural network-based reduced-order parametrization strategy that overcomes the “curse of dimensionality,” enabling scalable numerical solutions. Theoretically, it establishes the first general framework for density control independent of Wasserstein geometry—rigorously deriving the Pontryagin Maximum Principle (PMP) and Hamilton–Jacobi–Bellman (HJB) equation for density dynamics. Its theoretical contribution lies in formulating a Hamiltonian mechanical description decoupled from the Wasserstein manifold; its algorithmic contribution is a provably convergent method that remains practical in high dimensions. Experiments demonstrate that the proposed approach significantly outperforms state-of-the-art methods in both control accuracy and computational efficiency under complex constraints.

Technology Category

Search and Optimization: Mixed Discrete/Continuous SearchReasoning under Uncertainty: Stochastic OptimizationConstraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization

Application Category

Responsible Web: Human-perceived consequences of algorithmic deployment on the webEconomics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystemsGraph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

We develop a general theoretical framework for optimal probability density control and propose a numerical algorithm that is scalable to solve the control problem in high dimensions. Specifically, we establish the Pontryagin Maximum Principle (PMP) for optimal density control and construct the Hamilton-Jacobi-Bellman (HJB) equation of the value functional through rigorous derivations without any concept from Wasserstein theory. To solve the density control problem numerically, we propose to use reduced-order models, such as deep neural networks (DNNs), to parameterize the control vector-field and the adjoint function, which allows us to tackle problems defined on high-dimensional state spaces. We also prove several convergence properties of the proposed algorithm. Numerical results demonstrate promising performances of our algorithm on a variety of density control problems with obstacles and nonlinear interaction challenges in high dimensions.

Problem

• Develops theoretical framework for optimal density control
• Proposes scalable algorithm for high-dimensional control problems
• Solves density control using reduced-order models like DNNs

Innovation

• Pontryagin Maximum Principle for density control
• Hamilton-Jacobi-Bellman equation without Wasserstein theory
• Reduced-order models parameterize control and adjoint functions