论文标题
通过梯度-LOG密度估计，Fokker-Planck方程的相互作用粒子溶液
Interacting particle solutions of Fokker-Planck equations through gradient-log-density estimation
论文作者
论文摘要
Fokker-Planck方程在各种科学领域都广泛使用，因为它们表征了概率密度函数级别的随机系统的行为。尽管广泛使用，但它们仅在有限的环境中才能进行分析治疗，并且通常不可避免地求助于数值解决方案。在这里，我们开发了一种计算方法，用于模拟Fokker-Planck解决方案的时间演变，以相互作用的粒子系统的平均场极限。颗粒之间的相互作用取决于粒子密度对数的梯度，在此近似于新型的统计估计量。我们方法的性能显示出令人鼓舞的结果，与可比较粒子数的直接随机模拟相比，统计数据更加准确，波动较小。综上所述，我们的框架允许在低和中等尺寸中轻松且可靠的粒子模拟Fokker-Planck方程。所提出的梯度-LOG密度估计器也具有独立的兴趣，例如，在最佳控制的背景下。
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 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.
