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 to the infinite-dimensional filtering problem. Furthermore, 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 modeling.

翻译：我们研究了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。具体而言，计算任务是根据真实密度$\rho^\dagger$的含噪观测值，估计随时间演化的密度$\rho(v, t)$；这与基于状态$v$观测值的标准滤波问题不同。该任务自然地被表述为密度空间$\rho$中的无穷维滤波问题。然而，为了计算的可行性，我们寻求状态空间中的算法：具体地，引入平均场状态空间模型，并基于该模型的相互作用粒子系统近似，提出一种集成方法。我们将所得方法称为集成福克-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，后者正是无穷维滤波问题的精确解。此外，数值实验表明，该方法在更广泛的场景中依然有效：实验显示EnFPF能够修正集成统计量、加速自治系统向不变密度的收敛，以及加速非自治系统向时变不变密度的收敛。我们讨论了EnFPF在气候集成和湍流建模中的潜在应用。