We consider the problem of filtering dynamical systems, possibly stochastic, using observations of statistics. Thus the computational task is to estimate a time-evolving density $\rho(v, t)$ given noisy observations of $\rho$; this contrasts with the standard filtering problem based on observations of the state $v$. The task is naturally formulated as an infinite-dimensional filtering problem in the space of densities $\rho$. However, for the purposes of tractability, we seek algorithms in state space; specifically we introduce a mean field state space model and, using interacting particle system approximations to this model, we propose an ensemble method. We refer to the resulting methodology as the ensemble Fokker-Planck filter (EnFPF). Under certain restrictive assumptions we show that the EnFPF approximates the Kalman-Bucy filter for the Fokker-Planck equation, which is the exact solution of the infinite-dimensional filtering problem; our numerical experiments show that the methodology is useful beyond this restrictive setting. Specifically the experiments show that the EnFPF is able to correct ensemble statistics, to accelerate convergence to the invariant density for autonomous systems, and to accelerate convergence to time-dependent invariant densities for non-autonomous systems. We discuss possible applications of the EnFPF to climate ensembles and to turbulence modelling.

翻译：我们考虑利用统计观测数据过滤可能含有随机性的动力学系统问题。具体而言，计算任务是在观测到含噪的密度$\rho$后，估计随时间演化的密度$\rho(v, t)$；这与基于状态$v$观测的标准过滤问题形成对比。该问题自然被表述为密度空间$\rho$上的无穷维过滤问题。然而，为了可处理性，我们寻求状态空间中的算法；具体而言，我们引入一个平均场状态空间模型，并利用该模型的相互作用粒子系统近似，提出一种集合方法。我们将该方法称为集合福克-普朗克滤波器（EnFPF）。在特定限制性假设下，我们证明EnFPF可近似福克-普朗克方程的卡尔曼-布西滤波器，而该滤波器正是无穷维过滤问题的精确解；我们的数值实验表明，该方法在更广泛的场景下仍具实用性。具体而言，实验显示EnFPF能够修正集合统计量、加速自治系统收敛至不变密度、以及加速非自治系统收敛至时变不变密度。我们讨论了EnFPF在气候集合与湍流建模中的潜在应用。