汇报标题 (Title)：Amortized filtering and smoothing with conditional normalizing flow
（基于前提归一化流的摊销滤波与滑润步骤）
汇报人 (Speaker)：冯晓东 （北师香港浸会大学）
汇报功夫 (Time)：2026年4月25日（周六）9:30-10:00
汇报地址 (Place)：校本部GJ303
约请人(Inviter)：纪丽洁
主办部门：理学院数学系
提要：Bayesian filtering and smoothing for high-dimensional nonlinear dynamical systems are fundamental yet challenging problems in many areas of science and engineering. Gaussian-based approximations often break down when posterior distributions are highly nonlinear or non-Gaussian, while sequential Monte Carlo methods can be computationally demanding and often suffer from particle degeneracy, especially over long-time horizons or when smoothing distributions are required. Recent deep generative models are capable of representing complex high-dimensional posteriors, but they typically treat filtering and smoothing separately and often rely on costly per-instance optimization, which limits their applicability in online and large-scale settings. To address these challenges, we propose AFSF, a unified amortized framework for filtering and smoothing with conditional normalizing flows. The core idea is to encode each observation history into a fixed-dimensional summary statistic and use this shared representation to learn both a forward flow for the filtering distribution and a backward flow for the backward transition kernel. Specifically, a recurrent encoder maps each observation history to a fixed-dimensional summary statistic whose dimension does not depend on the length of the time series. Conditioned on this shared summary statistic, the forward flow approximates the filtering distribution, while the backward flow approximates the backward transition kernel. The smoothing distribution over an entire trajectory is then recovered by combining the terminal filtering distribution with the learned backward flow through the standard backward recursion. By learning the underlying temporal evolution structure, AFSF also supports extrapolation beyond the training horizon. Moreover, by coupling the two flows through shared summary statistics, AFSF induces an implicit regularization across latent state trajectories and improves trajectory-level smoothing. In addition, we develop a flow-based particle filtering variant that provides an alternative filtering procedure and enables ESS-based diagnostics when explicit model factors are available. Numerical experiments on a high-dimensional advection-diffusion system, a strongly nonlinear stochastic volatility model, a high-dimensional PDE system, and Lorenz systems in both single-scale and two-scale settings demonstrate that AFSF provides accurate approximations of both filtering distributions and smoothing paths.
