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

• URL: http://arxiv.org/abs/2505.18362v1
• Date: Fri, 23 May 2025 20:41:37 GMT
• Title: Hamiltonian Theory and Computation of Optimal Probability Density Control in High Dimensions
• Authors: Nathan Gaby, Xiaojing Ye,
• Abstract summary: We establish the Pontryagin Maximum Principle (PMP) for optimal density control and construct the Hamilton-Jacobi-Bellman equation of the value functional.<n>We propose to use reduced-order models, such as deep neural networks (DNNs), to parameterize the control vector-field and the adjoint function.<n> Numerical results demonstrate promising performances of our algorithm on a variety of density control problems with obstacles and nonlinear interaction challenges in high dimensions.
• Score: 1.9534129819019077
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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