Amortized Filtering and Smoothing with Conditional Normalizing Flows
stat.ML, cs.LG, cs.NA, math.NA
Submitted: 2026-04-08
Updated: 2026-09-16
Comments: 26 pages
License: http://creativecommons.org/licenses/by/4.0/
The gist: Bayesian filtering and smoothing are central to data assimilation in nonlinear dynamical systems.
Terminology
Abstract
Bayesian filtering and smoothing are central to data assimilation in nonlinear dynamical systems. Recent advances in deep generative models provide flexible approximations of the associated non-Gaussian posterior distributions. However, several existing approaches require the score to be re-estimated or a transport map to be constructed at each assimilation step. We propose an amortized framework for filtering and smoothing that reuses trained conditional models across observation sequences and assimilation times. The framework jointly learns a shared recurrent summary network and two conditional normalizing flows from simulated state and observation trajectories. The recurrent network represents each observation history by a fixed-dimensional summary that conditions the filtering approximation and, together with the next state, the backward transition approximation. An information-theoretic analysis shows that, under the Markov assumption, a summary sufficient for filtering is also sufficient for the backward kernel. Combining the terminal filtering approximation with the backward kernels yields an approximate joint smoothing distribution. Numerical experiments on advection-diffusion, Burgers, and Lorenz systems demonstrate the accuracy of the proposed filtering and smoothing approximations and characterize the evolution of their errors beyond the training horizon.
Sources
- Machine Learning for Inverse Problems and Data Assimilation
- BI-EqNO: Generalized Approximate Bayesian Inference with an Equivariant Neural Operator Framework
- Learning Optimal Filters Using Variational Inference
- Learning Enhanced Ensemble Filters
- Latent Autoencoder Ensemble Kalman Filter for Nonlinear Data assimilation
- A friendly introduction to triangular transport
- Simulation-Based Inference: A Practical Guide
- Flow-based Bayesian filtering for high-dimensional nonlinear stochastic dynamical systems
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