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 the true density $\rho^\dagger$; 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对动力系统（可能为随机系统）进行滤波的问题。因此，计算任务是在获得真实密度$\rho^\dagger$的含噪观测的前提下，估计随时间演化的密度$\rho(v, t)$；这与基于状态$v$观测的标准滤波问题形成对比。该问题自然可表述为密度空间$\rho$上的无穷维滤波问题。然而，为追求可计算性，我们寻求状态空间中的算法：具体而言，我们引入均值场状态空间模型，并利用该模型的相互作用粒子系统逼近，提出一种集合方法。我们将所得方法称为集合Fokker-Planck滤波器（EnFPF）。在特定限制性假设下，我们证明EnFPF可逼近Fokker-Planck方程的Kalman-Bucy滤波器，而后者正是无穷维滤波问题的精确解；数值实验表明，该方法在更广泛的非限制性设定中仍具实用性。具体实验结果显示，EnFPF能够校正集合统计量、加速自治系统向不变密度的收敛，并加速非自治系统向时变不变密度的收敛。我们讨论了EnFPF在气候集合及湍流建模中的潜在应用。