Maoutsa Dimitra, Reich Sebastian, Opper Manfred
Artificial Intelligence Group, Technische Universität Berlin, Marchstraße 23, 10587 Berlin, Germany.
Institute of Mathematics, University of Potsdam, Karl-Liebknecht-Str. 24/25, 14476 Potsdam, Germany.
Entropy (Basel). 2020 Jul 22;22(8):802. doi: 10.3390/e22080802.
Fokker-Planck equations are extensively employed in various scientific fields as they characterise the behaviour of stochastic systems at the level of probability density functions. Although broadly used, they allow for analytical treatment only in limited settings, and often it is inevitable to resort to numerical solutions. Here, we develop a computational approach for simulating the time evolution of Fokker-Planck solutions in terms of a mean field limit of an interacting particle system. The interactions between particles are determined by the gradient of the logarithm of the particle density, approximated here by a novel statistical estimator. The performance of our method shows promising results, with more accurate and less fluctuating statistics compared to direct stochastic simulations of comparable particle number. Taken together, our framework allows for effortless and reliable particle-based simulations of Fokker-Planck equations in low and moderate dimensions. The proposed gradient-log-density estimator is also of independent interest, for example, in the context of optimal control.
福克-普朗克方程在各个科学领域中被广泛应用，因为它们在概率密度函数层面描述了随机系统的行为。尽管被广泛使用，但它们仅在有限的情况下允许进行解析处理，而且通常不可避免地要采用数值解。在此，我们开发了一种计算方法，用于根据相互作用粒子系统的平均场极限来模拟福克-普朗克解的时间演化。粒子之间的相互作用由粒子密度对数的梯度决定，这里通过一种新颖的统计估计器进行近似。我们方法的性能显示出有前景的结果，与具有可比粒子数的直接随机模拟相比，统计数据更准确且波动更小。总体而言，我们的框架允许在低维和中等维度中轻松且可靠地基于粒子模拟福克-普朗克方程。所提出的梯度对数密度估计器本身也具有独立的研究价值，例如在最优控制的背景下。
