# Latent Space Data Assimilation by using Deep Learning

arXiv: https://arxiv.org/abs/2104.00430 · HTML: https://mlsanity.com/paper/2104.00430/

## Key facts

- Cited by 30 papers in the map (data through 14 Oct 2026).
- Builds on 25 of its 71 references that are in the map.
- First posted on arXiv 1 Apr 2021.

## Builds on (most cited first, 25 of 25)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Very Deep Convolutional Networks for Large-Scale Image Recognition](https://mlsanity.com/paper/1409.1556/) (2014)
- [PointNet: Deep Learning on Point Sets for 3D Classification and Segmentation](https://mlsanity.com/paper/1612.00593/) (2016)
- [Intriguing properties of neural networks](https://mlsanity.com/paper/1312.6199/) (2013)
- [Towards Evaluating the Robustness of Neural Networks](https://mlsanity.com/paper/1608.04644/) (2016)
- [Sequence to Sequence Learning with Neural Networks](https://mlsanity.com/paper/1409.3215/) (2014)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Google's Neural Machine Translation System: Bridging the Gap between Human and Machine Translation](https://mlsanity.com/paper/1609.08144/) (2016)
- [Solving high-dimensional partial differential equations using deep learning](https://mlsanity.com/paper/1707.02568/) (2017)
- [Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1711.10561/) (2017)
- [AllenNLP: A Deep Semantic Natural Language Processing Platform](https://mlsanity.com/paper/1803.07640/) (2018)
- [Stable Architectures for Deep Neural Networks](https://mlsanity.com/paper/1705.03341/) (2017)
- [ReZero is All You Need: Fast Convergence at Large Depth](https://mlsanity.com/paper/2003.04887/) (2020)
- [Informed Machine Learning -- A Taxonomy and Survey of Integrating Knowledge into Learning Systems](https://mlsanity.com/paper/1903.12394/) (2019)
- [Physics Informed Deep Learning (Part II): Data-driven Discovery of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1711.10566/) (2017)
- [Data-Driven Forecasting of High-Dimensional Chaotic Systems with Long Short-Term Memory Networks](https://mlsanity.com/paper/1802.07486/) (2018)
- [Latent-space Physics: Towards Learning the Temporal Evolution of Fluid Flow](https://mlsanity.com/paper/1802.10123/) (2018)
- [Deep Learning of Vortex Induced Vibrations](https://mlsanity.com/paper/1808.08952/) (2018)
- [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)
- [Differentiable Physics-informed Graph Networks](https://mlsanity.com/paper/1902.02950/) (2019)
- [IMEXnet: A Forward Stable Deep Neural Network](https://mlsanity.com/paper/1903.02639/) (2019)
- [Towards a Robust Parameterization for Conditioning Facies Models Using Deep Variational Autoencoders and Ensemble Smoother](https://mlsanity.com/paper/1812.06900/) (2018)
- [Attention-based Convolutional Autoencoders for 3D-Variational Data Assimilation](https://mlsanity.com/paper/2101.02121/) (2021)
- [Latent Space Subdivision: Stable and Controllable Time Predictions for Fluid Flow](https://mlsanity.com/paper/2003.08723/) (2020)
- [Detection of Iterative Adversarial Attacks via Counter Attack](https://mlsanity.com/paper/2009.11397/) (2020)

## Built on by (most cited first, 30 of 30)

- [Machine learning with data assimilation and uncertainty quantification for dynamical systems: a review](https://mlsanity.com/paper/2303.10462/) (2023)
- [Deep Generative Data Assimilation in Multimodal Setting](https://mlsanity.com/paper/2404.06665/) (2024)
- [Generalised Latent Assimilation in Heterogeneous Reduced Spaces with Machine Learning Surrogate Models](https://mlsanity.com/paper/2204.03497/) (2022)
- [FengWu-4DVar: Coupling the Data-driven Weather Forecasting Model with 4D Variational Assimilation](https://mlsanity.com/paper/2312.12455/) (2023)
- [Integrating Recurrent Neural Networks with Data Assimilation for Scalable Data-Driven State Estimation](https://mlsanity.com/paper/2109.12269/) (2021)
- [Towards an end-to-end artificial intelligence driven global weather forecasting system](https://mlsanity.com/paper/2312.12462/) (2023)
- [Efficient deep data assimilation with sparse observations and time-varying sensors](https://mlsanity.com/paper/2310.16187/) (2023)
- [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)
- [Accurate deep learning-based filtering for chaotic dynamics by identifying instabilities without an ensemble](https://mlsanity.com/paper/2408.04739/) (2024)
- [TorchDA: A Python package for performing data assimilation with deep learning forward and transformation functions](https://mlsanity.com/paper/2409.00244/) (2024)
- [Ensemble Kalman filter in latent space using a variational autoencoder pair](https://mlsanity.com/paper/2502.12987/) (2025)
- [Align-DA: Align Score-based Atmospheric Data Assimilation with Multiple Preferences](https://mlsanity.com/paper/2505.22008/) (2025)
- [Tensor-Var: Efficient Four-Dimensional Variational Data Assimilation](https://mlsanity.com/paper/2501.13312/) (2025)
- [The Deep Latent Space Particle Filter for Real-Time Data Assimilation with Uncertainty Quantification](https://mlsanity.com/paper/2406.02204/) (2024)
- [Generating Unseen Nonlinear Evolution in Sea Surface Temperature Using a Deep Learning-Based Latent Space Data Assimilation Framework](https://mlsanity.com/paper/2412.13477/) (2024)
- [Bridging Idealized and Operational Models: An Explainable AI Framework for Earth System Emulators](https://mlsanity.com/paper/2510.13030/) (2025)
- [Latent Autoencoder Ensemble Kalman Filter for Nonlinear Data assimilation](https://mlsanity.com/paper/2603.06752/) (2026)
- [Review on Fault Diagnosis and Fault-Tolerant Control Scheme for Robotic Manipulators: Recent Advances in AI, Machine Learning, and Digital Twin](https://mlsanity.com/paper/2402.02980/) (2024)
- [Neural ensemble Kalman filter: Data assimilation for compressible flows with shocks](https://mlsanity.com/paper/2602.23461/) (2026)
- [Accurate and Efficient Hybrid-Ensemble Atmospheric Data Assimilation in Latent Space with Uncertainty Quantification](https://mlsanity.com/paper/2603.04395/) (2026)
- [ADDA: a Modular Framework for Representing, Simulating and Assimilating Dynamics with End-to-end Differentiability](https://mlsanity.com/paper/2608.23297/) (2026)
- [Generating observation guided ensembles for data assimilation with denoising diffusion probabilistic model](https://mlsanity.com/paper/2308.06708/) (2023)
- [Ensemble Controlled-Flow Filtering for Implicit Data Assimilation](https://mlsanity.com/paper/2607.12975/) (2026)
- [Incomplete Observations Boost Evolutionary Performance in Ocean Modeling](https://mlsanity.com/paper/2607.19147/) (2026)
- [PR-Smoother: Simulator-Preserving Non-Gaussian Smoothing for Data Assimilation](https://mlsanity.com/paper/2609.26890/) (2026)
- [Variational Augmented Invertible Koopman Autoencoder for probabilistic time series forecasting](https://mlsanity.com/paper/2609.37435/) (2026)
- [LD-EnFF: Latent-Dynamics Ensemble Flow Filtering for Data Assimilation with Sparse Observations](https://mlsanity.com/paper/2610.04034/) (2026)

## Abstract (arXiv)

Performing Data Assimilation (DA) at a low cost is of prime concern in Earth system modeling, particularly at the time of big data where huge quantities of observations are available. Capitalizing on the ability of Neural Networks techniques for approximating the solution of PDE's, we incorporate Deep Learning (DL) methods into a DA framework. More precisely, we exploit the latent structure provided by autoencoders (AEs) to design an Ensemble Transform Kalman Filter with model error (ETKF-Q) in the latent space. Model dynamics are also propagated within the latent space via a surrogate neural network. This novel ETKF-Q-Latent (thereafter referred to as ETKF-Q-L) algorithm is tested on a tailored instructional version of Lorenz 96 equations, named the augmented Lorenz 96 system: it possesses a latent structure that accurately represents the observed dynamics. Numerical experiments based on this particular system evidence that the ETKF-Q-L approach both reduces the computational cost and provides better accuracy than state of the art algorithms, such as the ETKF-Q.