基于基于得分的归一化流的平均场控制模拟福克-普朗克方程
Simulating Fokker-Planck equations via mean field control of score-based normalizing flows
AI总结：
本研究将FP方程转化为连续性方程，设计MFC问题匹配速度场，用基于得分的归一化流高效计算得分函数，经多类系统数值实验及收敛分析验证了算法的有效性与可扩展性。
摘要：
福克-普朗克（FP）方程描述了物理系统随机动力学的密度演化，如朗之万动力学和洛伦兹系统。本研究通过平均场控制（MFC）问题模拟FP方程：首先将FP方程表述为连续性方程，其速度场由漂移函数和得分函数（即密度函数对数的梯度）组成；接着设计MFC问题，使连续性方程中的速度场与FP方程中的速度场匹配。沿确定性轨迹的得分函数可通过仅依赖参数化速度场导数的基于得分的归一化流高效计算。数值结果涵盖朗之万动力学、欠阻尼朗之万动力学、混沌系统及高维相互作用粒子系统，验证了所提算法的有效性与可扩展性，还对该算法在奥恩斯坦-乌伦贝克过程的FP方程上进行了收敛性分析。
英文摘要：
The Fokker--Planck (FP) equation governs the evolution of densities for stochastic dynamics of physical systems, such as the Langevin dynamics and the Lorenz system. This work simulates FP equations through a mean field control (MFC) problem. We first formulate the FP equation as a continuity equation, where the velocity field consists of the drift function and the score function, i.e., the gradient of the logarithm of the density function. Next, we design a MFC problem that matches the velocity fields in a continuity equation with the ones in the FP equation. The score functions along deterministic trajectories are computed efficiently through the score-based normalizing flow, which only relies on the derivatives of the parameterized velocity fields. Numerical results, including Langevin dynamics, underdamped Langevin dynamics, chaotic systems, and high-dimensional interacting particle systems validate the effectiveness and scalability of our proposed algorithm. A convergence analysis is conducted for our algorithm on the FP equation of Ornstein--Uhlenbeck processes.
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