# Data Assimilation: A Mathematical Introduction

arXiv: https://arxiv.org/abs/1506.07825 · HTML: https://mlsanity.com/paper/1506.07825/

## Key facts

- Cited by 27 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 25 Jun 2015.

## Built on by (most cited first, 27 of 27)

- [Using Data Assimilation to Train a Hybrid Forecast System that Combines Machine-Learning and Knowledge-Based Components](https://mlsanity.com/paper/2102.07819/) (2021)
- [Discrepancy Modeling Framework: Learning missing physics, modeling systematic residuals, and disambiguating between deterministic and random effects](https://mlsanity.com/paper/2203.05164/) (2022)
- [Leveraging arbitrary mobile sensor trajectories with shallow recurrent decoder networks for full-state reconstruction](https://mlsanity.com/paper/2307.11793/) (2023)
- [Learning Enhanced Ensemble Filters](https://mlsanity.com/paper/2504.17836/) (2025)
- [Scaling Up Bayesian Uncertainty Quantification for Inverse Problems using Deep Neural Networks](https://mlsanity.com/paper/2101.03906/) (2021)
- [Learning Optimal Filters Using Variational Inference](https://mlsanity.com/paper/2406.18066/) (2024)
- [Nonlinear Assimilation via Score-based Sequential Langevin Sampling](https://mlsanity.com/paper/2411.13443/) (2024)
- [CEBoosting: Online Sparse Identification of Dynamical Systems with Regime Switching by Causation Entropy Boosting](https://mlsanity.com/paper/2304.07863/) (2023)
- [Discovery of interpretable structural model errors by combining Bayesian sparse regression and data assimilation: A chaotic Kuramoto-Sivashinsky test case](https://mlsanity.com/paper/2110.00546/) (2021)
- [Phi-DVAE: Physics-Informed Dynamical Variational Autoencoders for Unstructured Data Assimilation](https://mlsanity.com/paper/2209.15609/) (2022)
- [Low-rank statistical finite elements for scalable model-data synthesis](https://mlsanity.com/paper/2109.04757/) (2021)
- [Flow-based Bayesian filtering for high-dimensional nonlinear stochastic dynamical systems](https://mlsanity.com/paper/2502.16232/) (2025)
- [Machine learning-based conditional mean filter: a generalization of the ensemble Kalman filter for nonlinear data assimilation](https://mlsanity.com/paper/2106.07908/) (2021)
- [Ensemble Transport Filter via Optimized Maximum Mean Discrepancy](https://mlsanity.com/paper/2407.11518/) (2024)
- [ProFlow: Zero-Shot Physics-Consistent Sampling via Proximal Flow Guidance](https://mlsanity.com/paper/2601.20227/) (2026)
- [LatentFlow: A General Framework for Conditioning Stochastic Processes](https://mlsanity.com/paper/2607.12922/) (2026)
- [Bridging Prediction and Attribution: Identifying Forward and Backward Causal Influence Ranges Using Assimilative Causal Inference](https://mlsanity.com/paper/2510.21889/) (2025)
- [Reinforcement learning based data assimilation for unknown state model](https://mlsanity.com/paper/2511.02286/) (2025)
- [ISALT: Inference-based schemes adaptive to large time-stepping for locally Lipschitz ergodic systems](https://mlsanity.com/paper/2102.12669/) (2021)
- [Unsupervised learning of observation functions in state-space models by nonparametric moment methods](https://mlsanity.com/paper/2207.05242/) (2022)
- [Preserving linear invariants in ensemble filtering methods](https://mlsanity.com/paper/2404.14328/) (2024)
- [Machine Learning for Complex Systems with Abnormal Pattern by Exception Maximization Outlier Detection Method](https://mlsanity.com/paper/2407.04248/) (2024)
- [Hierarchical Inference and Closure Learning via Adaptive Surrogates for ODEs and PDEs](https://mlsanity.com/paper/2603.03922/) (2026)
- [A Continuous-Time Ensemble Kalman-Bucy Smoother for Causal Inference and Model Discovery](https://mlsanity.com/paper/2604.25157/) (2026)
- [Dynestyx: A Probabilistic Programming Library for Dynamical Systems](https://mlsanity.com/paper/2606.16985/) (2026)
- [FREESIA: Covariance-Aware Posterior Transport for Expressive and Scalable Data Assimilation](https://mlsanity.com/paper/2609.25085/) (2026)
- [LD-EnFF: Latent-Dynamics Ensemble Flow Filtering for Data Assimilation with Sparse Observations](https://mlsanity.com/paper/2610.04034/) (2026)

## Abstract (arXiv)

These notes provide a systematic mathematical treatment of the subject of data assimilation.