# Integrating Recurrent Neural Networks with Data Assimilation for Scalable Data-Driven State Estimation

arXiv: https://arxiv.org/abs/2109.12269 · HTML: https://mlsanity.com/paper/2109.12269/

## Key facts

- Cited by 21 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 69 references that are in the map.
- First posted on arXiv 25 Sep 2021.

## Builds on (most cited first, 10 of 10)

- [Backpropagation Algorithms and Reservoir Computing in Recurrent Neural Networks for the Forecasting of Complex Spatiotemporal Dynamics](https://mlsanity.com/paper/1910.05266/) (2019)
- [Combining data assimilation and machine learning to emulate a dynamical model from sparse and noisy observations: a case study with the Lorenz 96 model](https://mlsanity.com/paper/2001.01520/) (2020)
- [Forecasting Chaotic Systems with Very Low Connectivity Reservoir Computers](https://mlsanity.com/paper/1910.00659/) (2019)
- [Bayesian inference of chaotic dynamics by merging data assimilation, machine learning and expectation-maximization](https://mlsanity.com/paper/2001.06270/) (2020)
- [Latent Space Data Assimilation by using Deep Learning](https://mlsanity.com/paper/2104.00430/) (2021)
- [Online learning of both state and dynamics using ensemble Kalman filters](https://mlsanity.com/paper/2006.03859/) (2020)
- [Machine Learning, Deepest Learning: Statistical Data Assimilation Problems](https://mlsanity.com/paper/1707.01415/) (2017)
- [A comparison of combined data assimilation and machine learning methods for offline and online model error correction](https://mlsanity.com/paper/2107.11114/) (2021)
- [Degenerate Kalman filter error covariances and their convergence onto the unstable subspace](https://mlsanity.com/paper/1604.02578/) (2016)
- [Forecasting Using Reservoir Computing: The Role of Generalized Synchronization](https://mlsanity.com/paper/2102.08930/) (2021)

## Built on by (most cited first, 21 of 21)

- [Towards an end-to-end artificial intelligence driven global weather forecasting system](https://mlsanity.com/paper/2312.12462/) (2023)
- [Deep learning-enhanced ensemble-based data assimilation for high-dimensional nonlinear dynamical systems](https://mlsanity.com/paper/2206.04811/) (2022)
- [LD-EnSF: Synergizing Latent Dynamics with Ensemble Score Filters for Fast Data Assimilation with Sparse Observations](https://mlsanity.com/paper/2411.19305/) (2024)
- [Latent assimilation with implicit neural representations for unknown dynamics](https://mlsanity.com/paper/2309.09574/) (2023)
- [FNP: Fourier Neural Processes for Arbitrary-Resolution Data Assimilation](https://mlsanity.com/paper/2406.01645/) (2024)
- [State-observation augmented diffusion model for nonlinear assimilation with unknown dynamics](https://mlsanity.com/paper/2407.21314/) (2024)
- [Learning dynamical systems with hit-and-run random feature maps](https://mlsanity.com/paper/2501.06661/) (2025)
- [Constraining Chaos: Enforcing dynamical invariants in the training of recurrent neural networks](https://mlsanity.com/paper/2304.12865/) (2023)
- [4D-Var using Hessian approximation and backpropagation applied to automatically-differentiable numerical and machine learning models](https://mlsanity.com/paper/2408.02767/) (2024)
- [TorchDA: A Python package for performing data assimilation with deep learning forward and transformation functions](https://mlsanity.com/paper/2409.00244/) (2024)
- [`Next Generation' Reservoir Computing: an Empirical Data-Driven Expression of Dynamical Equations in Time-Stepping Form](https://mlsanity.com/paper/2201.05193/) (2022)
- [Temporal Subsampling Diminishes Small Spatial Scales in Recurrent Neural Network Emulators of Geophysical Turbulence](https://mlsanity.com/paper/2305.00100/) (2023)
- [Combined Optimization of Dynamics and Assimilation with End-to-End Learning on Sparse Observations](https://mlsanity.com/paper/2409.07137/) (2024)
- [RL-DAUNCE: Reinforcement Learning-Driven Data Assimilation with Uncertainty-Aware Constrained Ensembles](https://mlsanity.com/paper/2505.05452/) (2025)
- [Ensemble data assimilation to diagnose AI-based weather prediction model: A case with ClimaX version 0.3.1](https://mlsanity.com/paper/2407.17781/) (2024)
- [Long-time accuracy of ensemble Kalman filters for chaotic and machine-learned dynamical systems](https://mlsanity.com/paper/2412.14318/) (2024)
- [ADDA: a Modular Framework for Representing, Simulating and Assimilating Dynamics with End-to-end Differentiability](https://mlsanity.com/paper/2608.23297/) (2026)
- [U-Net Kalman Filter (UNetKF): An Example of Machine Learning-assisted Ensemble Data Assimilation](https://mlsanity.com/paper/2403.12366/) (2024)
- [Probabilistic Emulation of the Community Radiative Transfer Model Using Machine Learning](https://mlsanity.com/paper/2504.16192/) (2025)
- [Ensemble Controlled-Flow Filtering for Implicit Data Assimilation](https://mlsanity.com/paper/2607.12975/) (2026)
- [LD-EnFF: Latent-Dynamics Ensemble Flow Filtering for Data Assimilation with Sparse Observations](https://mlsanity.com/paper/2610.04034/) (2026)

## Abstract (arXiv)

Data assimilation (DA) is integrated with machine learning in order to perform entirely data-driven online state estimation. To achieve this, recurrent neural networks (RNNs) are implemented as surrogate models to replace key components of the DA cycle in numerical weather prediction (NWP), including the conventional numerical forecast model, the forecast error covariance matrix, and the tangent linear and adjoint models. It is shown how these RNNs can be initialized using DA methods to directly update the hidden/reservoir state with observations of the target system. The results indicate that these techniques can be applied to estimate the state of a system for the repeated initialization of short-term forecasts, even in the absence of a traditional numerical forecast model. Further, it is demonstrated how these integrated RNN-DA methods can scale to higher dimensions by applying domain localization and parallelization, providing a path for practical applications in NWP.