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地变换为后验分布的样本，从而解决贝叶斯推断问题。粒子在McKean-Vlasov-Ito过程的驱动下于状态空间中运动。本文提出了用于数据同化的变分Fokker-Planck（VFP）框架——一种将此前已知的粒子流滤波器作为特例的通用方法。变换粒子的McKean-Vlasov-Ito过程通过依赖于所选扩散项的最优漂移来定义。研究表明，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。对于有限数量的粒子，最优漂移包含一个正则化项，该项促使粒子成为独立随机变量。基于这一分析，我们推导出计算可行的近似正则化方法，该方法通过惩罚粒子对之间的互信息避免粒子坍缩。此外，扩散项起到类似粒子重生技术的作用，旨在缓解粒子坍缩问题。VFP框架具有高度灵活性。可利用先验和中间概率分布的不同假设来实现最优漂移，并可应用定位和协方差收缩来缓解维数灾难。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。通过在三个难度递增的测试问题（即Lorenz '63、Lorenz '96和准地转方程）上的数值实验，验证了VFP框架的有效性。