Bayesian filtering and smoothing for high-dimensional nonlinear dynamical systems are fundamental yet challenging problems in many areas of science and engineering. In this work, 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 demonstrate that AFSF provides accurate approximations of both filtering distributions and smoothing paths.

翻译：贝叶斯滤波与平滑是科学和工程众多领域中针对高维非线性动态系统的基础且具挑战性的问题。本文提出AFSF，一个基于条件归一化流的滤波与平滑统一摊销框架。其核心思想是将每个观测历史编码为固定维度的汇总统计量，并利用该共享表示同时学习用于滤波分布的前向流和用于后向转移核的后向流。具体而言，循环编码器将每个观测历史映射为固定维度的汇总统计量，其维度不依赖于时间序列长度。基于这一共享汇总统计量，前向流逼近滤波分布，而后向流逼近后向转移核。通过标准后向递归，将终端的滤波分布与学习到的后向流结合，即可恢复整个轨迹上的平滑分布。通过学习潜在的时域演化结构，AFSF还支持超越训练时间范围的外推。此外，通过共享汇总统计量耦合两个流，AFSF对潜在状态轨迹施加了隐式正则化，从而改善了轨迹级平滑。同时，我们开发了一种基于流的粒子滤波变体，该变体提供了另一种滤波过程，并在存在显式模型因子时支持基于有效样本量（ESS）的诊断。数值实验表明，AFSF能够为滤波分布和平滑路径提供精确的近似。