# Amortized Filtering and Smoothing with Conditional Normalizing Flows

arXiv: https://arxiv.org/abs/2604.07169 · HTML: https://mlsanity.com/paper/2604.07169/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 21 of its 37 references that are in the map.
- First posted on arXiv 8 Apr 2026.

## Builds on (most cited first, 21 of 21)

- [Normalizing Flows for Probabilistic Modeling and Inference](https://mlsanity.com/paper/1912.02762/) (2019)
- [The frontier of simulation-based inference](https://mlsanity.com/paper/1911.01429/) (2019)
- [KalmanNet: Neural Network Aided Kalman Filtering for Partially Known Dynamics](https://mlsanity.com/paper/2107.10043/) (2021)
- [Score-based Data Assimilation](https://mlsanity.com/paper/2306.10574/) (2023)
- [Machine learning with data assimilation and uncertainty quantification for dynamical systems: a review](https://mlsanity.com/paper/2303.10462/) (2023)
- [DiffDA: a Diffusion Model for Weather-scale Data Assimilation](https://mlsanity.com/paper/2401.05932/) (2024)
- [Coupling techniques for nonlinear ensemble filtering](https://mlsanity.com/paper/1907.00389/) (2019)
- [Neural Methods for Amortized Inference](https://mlsanity.com/paper/2404.12484/) (2024)
- [An Ensemble Score Filter for Tracking High-Dimensional Nonlinear Dynamical Systems](https://mlsanity.com/paper/2309.00983/) (2023)
- [Simulation-Based Inference: A Practical Guide](https://mlsanity.com/paper/2508.12939/) (2025)
- [Appa: Bending Weather Dynamics with Latent Diffusion Models for Global Data Assimilation](https://mlsanity.com/paper/2504.18720/) (2025)
- [Solving time dependent Fokker-Planck equations via temporal normalizing flow](https://mlsanity.com/paper/2112.14012/) (2021)
- [Learning Enhanced Ensemble Filters](https://mlsanity.com/paper/2504.17836/) (2025)
- [Machine Learning for Inverse Problems and Data Assimilation](https://mlsanity.com/paper/2410.10523/) (2024)
- [Deep Bayesian Filter for Bayes-faithful Data Assimilation](https://mlsanity.com/paper/2405.18674/) (2024)
- [BI-EqNO: Generalized Approximate Bayesian Inference with an Equivariant Neural Operator Framework](https://mlsanity.com/paper/2410.16420/) (2024)
- [Adaptive deep density approximation for stochastic dynamical systems](https://mlsanity.com/paper/2405.02810/) (2024)
- [An Optimal Transport Formulation of the Ensemble Kalman Filter](https://mlsanity.com/paper/1910.02338/) (2019)
- [Data-Driven Approximation of Stationary Nonlinear Filters with Optimal Transport Maps](https://mlsanity.com/paper/2403.15630/) (2024)
- [Flow-based Bayesian filtering for high-dimensional nonlinear stochastic dynamical systems](https://mlsanity.com/paper/2502.16232/) (2025)
- [Deep Normalizing Flows for State Estimation](https://mlsanity.com/paper/2306.15605/) (2023)

## Built on by (most cited first, 2 of 2)

- [One Operator for Many Densities: Amortized Approximation of Conditioning by Neural Operators](https://mlsanity.com/paper/2605.06873/) (2026)
- [PR-Smoother: Simulator-Preserving Non-Gaussian Smoothing for Data Assimilation](https://mlsanity.com/paper/2609.26890/) (2026)

## Abstract (arXiv)

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.