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 FLUID, a flow-based unified amortized inference framework for filtering and smoothing dynamics. 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, FLUID also supports extrapolation beyond the training horizon. Moreover, by coupling the two flows through shared summary statistics, FLUID 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 FLUID provides accurate approximations of both filtering distributions and smoothing paths.

翻译：在高维非线性动力系统中，贝叶斯滤波与平滑是众多科学与工程领域中的基础性难题。本文提出FLUID——一个基于流的统一摊销推断框架，用于处理滤波与平滑动力学过程。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滤波分布的前向流与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，该统计量的维度不依赖于时间序列长度。在此共享汇总统计量的条件下，前向流近似滤波分布，后向流近似后向转移核。通过将终端滤波分布与所学的后向流结合，经由标准后向递推即可恢复整个轨迹的平滑分布。通过学习潜在的时间演化结构，FLUID还支持超出训练时域的外推。此外，通过共享汇总统计量耦合两股流，FLUID在潜在状态轨迹间引入隐式正则化，从而提升轨迹级平滑效果。本文进一步开发了基于流的粒子滤波变体，该变体提供替代性滤波过程，并在明确模型因子可用时实现基于有效样本量的诊断。数值实验表明，FLUID能够精确逼近滤波分布与平滑路径。