2024/09/24 by Mo Zhou, Stanley Osher, Zhou, Mo +3 · 2 citations
Computer Science · Physics and Astronomy · #34H05 #FOS: Computer and information sciences #FOS: Mathematics #G.1.7 #Machine Learning (cs.LG) #Model Reduction and Neural Networks #Neural Networks and Applications #Optimization and Control (math.OC)
paper · pdf · doi:10.48550/arxiv.2409.16471
openalex publication_date 2024/09/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Classical neural ordinary differential equations (ODEs) are powerful tools for approximating the log-density functions in high-dimensional spaces along trajectories, where neural networks parameterize the velocity fields. This paper proposes a system of neural differential equations representing first- and second-order score functions along trajectories based on deep neural networks. We reformulate the mean field control (MFC) problem with individual noises into an unconstrained optimization problem framed by the proposed neural ODE system. Additionally, we introduce a novel regularization term to enforce characteristics of viscous Hamilton--Jacobi--Bellman (HJB) equations to be satisfied based on the evolution of the second-order score function. Examples include regularized Wasserstein proximal operators (RWPOs), probability flow matching of Fokker--Planck (FP) equations, and linear quadratic (LQ) MFC problems, which demonstrate the effectiveness and accuracy of the proposed method.