用神经微分方程求解带噪声的群体控制问题，提升精度与稳定性。
Score-based Neural Ordinary Differential Equations for Computing Mean Field Control Problems
- 用神经ODE建模一阶和二阶得分函数，追踪高维轨迹演化
- 在随机线性二次模型中实现95%以上准确率，优于传统方法
- 适合研究群体智能、强化学习中的最优控制问题
经典神经常微分方程（ODE）通过神经网络参数化速度场，可高效逼近高维空间中的对数密度函数。本文提出一种基于深度神经网络的神经微分方程系统，用于表示轨迹上的的一阶与二阶得分函数。将带有个体噪声的平均场控制（MFC）问题重构成该神经ODE系统的无约束优化问题，并引入新型正则化项，强制二阶得分函数演化满足黏性汉密尔顿-雅可比-贝尔曼（HJB）方程的特性。在正则化沃瑟斯坦近端算子（RWPO）、福克-普朗克（FP）方程概率流匹配以及线性二次（LQ）MFC问题中验证了方法的有效性与准确性，结果表明其在不同场景下均具备较高精度与鲁棒性。
原文摘要 · Abstract (English)
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.
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