# Conditional Normalizing Flows for Forward and Backward Joint State and Parameter Estimation

arXiv: https://arxiv.org/abs/2601.07013 · HTML: https://mlsanity.com/paper/2601.07013/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 24 of its 59 references that are in the map.
- First posted on arXiv 11 Jan 2026.

## Builds on (most cited first, 24 of 24)

- [Generative Adversarial Networks](https://mlsanity.com/paper/2203.00667/) (2022)
- [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016)
- [Masked Autoregressive Flow for Density Estimation](https://mlsanity.com/paper/1705.07057/) (2017)
- [Pyro: Deep Universal Probabilistic Programming](https://mlsanity.com/paper/1810.09538/) (2018)
- [Scalable Gradients for Stochastic Differential Equations](https://mlsanity.com/paper/2001.01328/) (2020)
- [Flow++: Improving Flow-Based Generative Models with Variational Dequantization and Architecture Design](https://mlsanity.com/paper/1902.00275/) (2019)
- [OT-Flow: Fast and Accurate Continuous Normalizing Flows via Optimal Transport](https://mlsanity.com/paper/2006.00104/) (2020)
- [Deep Transformer Models for Time Series Forecasting: The Influenza Prevalence Case](https://mlsanity.com/paper/2001.08317/) (2020)
- [Noise Flow: Noise Modeling with Conditional Normalizing Flows](https://mlsanity.com/paper/1908.08453/) (2019)
- [Equivariant flow-based sampling for lattice gauge theory](https://mlsanity.com/paper/2003.06413/) (2020)
- [Targeted free energy estimation via learned mappings](https://mlsanity.com/paper/2002.04913/) (2020)
- [Equivariant Flows: sampling configurations for multi-body systems with symmetric energies](https://mlsanity.com/paper/1910.00753/) (2019)
- [Coupling techniques for nonlinear ensemble filtering](https://mlsanity.com/paper/1907.00389/) (2019)
- [Dynamic Conditional Optimal Transport through Simulation-Free Flows](https://mlsanity.com/paper/2404.04240/) (2024)
- [Bridging Mean-Field Games and Normalizing Flows with Trajectory Regularization](https://mlsanity.com/paper/2206.14990/) (2022)
- [Structural Consistency and Controllability for Diverse Colorization](https://mlsanity.com/paper/1809.02129/) (2018)
- [Learning Enhanced Ensemble Filters](https://mlsanity.com/paper/2504.17836/) (2025)
- [Normalizing Flow-based Differentiable Particle Filters](https://mlsanity.com/paper/2403.01499/) (2024)
- [Multiple Object Tracking with Mixture Density Networks for Trajectory Estimation](https://mlsanity.com/paper/2106.10950/) (2021)
- [Ensemble transport smoothing. Part II: Nonlinear updates](https://mlsanity.com/paper/2210.17435/) (2022)
- [Optimal Transport Particle Filters](https://mlsanity.com/paper/2304.00392/) (2023)
- [Optimal Transportation Methods in Nonlinear Filtering: The feedback particle filter](https://mlsanity.com/paper/2102.10712/) (2021)
- [Fast filtering of non-Gaussian models using Amortized Optimal Transport Maps](https://mlsanity.com/paper/2503.12633/) (2025)
- [XRMDN: An Extended Recurrent Mixture Density Network for Short-Term Probabilistic Rider Demand Forecasting with High Volatility](https://mlsanity.com/paper/2310.09847/) (2023)

## Abstract (arXiv)

Traditional filtering algorithms for state estimation -- such as classical Kalman filtering, unscented Kalman filtering, and particle filters -- show performance degradation when applied to nonlinear systems whose uncertainty follows arbitrary non-Gaussian, and potentially multi-modal distributions. This study reviews recent approaches to state estimation via nonlinear filtering based on conditional normalizing flows, where the conditional embedding is generated by standard MLP architectures, transformers or selective state-space models (like Mamba-SSM). In addition, we test the effectiveness of an optimal-transport-inspired kinetic loss term in mitigating overparameterization in flows consisting of a large collection of transformations. We investigate the performance of these approaches on applications relevant to autonomous driving and patient population dynamics, paying special attention to how they handle time inversion and chained predictions. Finally, we assess the performance of various conditioning strategies for an application to real-world COVID-19 joint SIR system forecasting and parameter estimation.