Amortized Filtering and Smoothing with Conditional Normalizing Flows

2026-04-08

AI Summary

This work addresses the challenge of Bayesian filtering and smoothing in high-dimensional nonlinear dynamical systems by introducing the AFSF framework, which unifies forward filtering and backward smoothing through conditional normalizing flows for the first time. The approach employs a recurrent encoder to compress the observation history into a fixed-dimensional summary statistic, which simultaneously conditions both the forward flow and the backward transition kernel. Coupled with a standard backward recursion, this enables trajectory-level smoothing. By sharing the same summary statistic across forward and backward flows, AFSF implicitly introduces regularization that significantly enhances trajectory consistency and supports effective extrapolation beyond the training time horizon. Experimental results demonstrate that the method accurately approximates both filtering distributions and smoothed trajectories.

Abstract

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.

Problem

• Bayesian filtering
• smoothing
• high-dimensional nonlinear dynamical systems
• conditional normalizing flows

Innovation

• amortized inference
• conditional normalizing flows
• Bayesian filtering and smoothing
• recurrent encoder
• trajectory-level smoothing