Particle flow filters solve Bayesian inference problems by smoothly transforming a set of particles into samples from the posterior distribution. Particles move in state space under the flow of an McKean-Vlasov-Ito process. This work introduces the Variational Fokker-Planck (VFP) framework for data assimilation, a general approach that includes previously known particle flow filters as special cases. The McKean-Vlasov-Ito process that transforms particles is defined via an optimal drift that depends on the selected diffusion term. It is established that the underlying probability density - sampled by the ensemble of particles - converges to the Bayesian posterior probability density. For a finite number of particles the optimal drift contains a regularization term that nudges particles toward becoming independent random variables. Based on this analysis, we derive computationally-feasible approximate regularization approaches that penalize the mutual information between pairs of particles, and avoid particle collapse. Moreover, the diffusion plays a role akin to a particle rejuvenation approach that aims to alleviate particle collapse. The VFP framework is very flexible. Different assumptions on prior and intermediate probability distributions can be used to implement the optimal drift, and localization and covariance shrinkage can be applied to alleviate the curse of dimensionality. A robust implicit-explicit method is discussed for the efficient integration of stiff McKean-Vlasov-Ito processes. The effectiveness of the VFP framework is demonstrated on three progressively more challenging test problems, namely the Lorenz '63, Lorenz '96 and the quasi-geostrophic equations.

翻译：粒子流滤波器通过将一组粒子平滑变换为后验分布样本，实现了贝叶斯推断问题的求解。粒子在状态空间中受McKean-Vlasov-Ito过程驱动而运动。本文提出了用于数据同化的变分福克-普朗克（VFP）框架，该通用方法将已有的粒子流滤波器作为特例包含在内。定义粒子变换的McKean-Vlasov-Ito过程通过依赖于所选扩散项的最优漂移项确定。研究证明，由粒子集合采样的底层概率密度收敛于贝叶斯后验概率密度。对于有限粒子数目，最优漂移包含一个正则化项，促使粒子向独立随机变量方向发展。基于此分析，我们推导出计算上可行的近似正则化方法，通过惩罚粒子对间的互信息来避免粒子坍缩。此外，扩散项起到类似于粒子再生方法的作用，旨在缓解粒子坍缩问题。VFP框架具有高度灵活性。通过采用先验分布和中间概率分布的不同假设可实现最优漂移，同时可采用定位与协方差收缩方法缓解维度灾难。针对刚性McKean-Vlasov-Ito过程的高效积分，本文讨论了一种稳健的隐式-显式方法。通过三个难度递增的测试问题——Lorenz '63、Lorenz '96及准地转方程——验证了VFP框架的有效性。