# Machine learning with data assimilation and uncertainty quantification for dynamical systems: a review

arXiv: https://arxiv.org/abs/2303.10462 · HTML: https://mlsanity.com/paper/2303.10462/

## Key facts

- Cited by 75 papers in the map (data through 14 Oct 2026).
- Builds on 105 of its 331 references that are in the map.
- First posted on arXiv 18 Mar 2023.

## Builds on (most cited first, 30 of 105)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Masked Autoencoders Are Scalable Vision Learners](https://mlsanity.com/paper/2111.06377/) (2021)
- [Neural Discrete Representation Learning](https://mlsanity.com/paper/1711.00937/) (2017)
- [Dropout as a Bayesian Approximation: Representing Model Uncertainty in Deep Learning](https://mlsanity.com/paper/1506.02142/) (2015)
- [Auto-Encoding Variational Bayes](https://mlsanity.com/paper/1312.6114/) (2013)
- [Simple and Scalable Predictive Uncertainty Estimation using Deep Ensembles](https://mlsanity.com/paper/1612.01474/) (2016)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [3D U-Net: Learning Dense Volumetric Segmentation from Sparse Annotation](https://mlsanity.com/paper/1606.06650/) (2016)
- [Convolutional LSTM Network: A Machine Learning Approach for Precipitation Nowcasting](https://mlsanity.com/paper/1506.04214/) (2015)
- [A Comprehensive Survey on Transfer Learning](https://mlsanity.com/paper/1911.02685/) (2019)
- [Meta-Learning in Neural Networks: A Survey](https://mlsanity.com/paper/2004.05439/) (2020)
- [Aleatoric and Epistemic Uncertainty in Machine Learning: An Introduction to Concepts and Methods](https://mlsanity.com/paper/1910.09457/) (2019)
- [Learning to Simulate Complex Physics with Graph Networks](https://mlsanity.com/paper/2002.09405/) (2020)
- [Learning Mesh-Based Simulation with Graph Networks](https://mlsanity.com/paper/2010.03409/) (2020)
- [A Review of Uncertainty Quantification in Deep Learning: Techniques, Applications and Challenges](https://mlsanity.com/paper/2011.06225/) (2020)
- [A Survey of Uncertainty in Deep Neural Networks](https://mlsanity.com/paper/2107.03342/) (2021)
- [Learning to learn by gradient descent by gradient descent](https://mlsanity.com/paper/1606.04474/) (2016)
- [A Simple Baseline for Bayesian Uncertainty in Deep Learning](https://mlsanity.com/paper/1902.02476/) (2019)
- [A Survey of Transformers](https://mlsanity.com/paper/2106.04554/) (2021)
- [Hands-on Bayesian Neural Networks -- a Tutorial for Deep Learning Users](https://mlsanity.com/paper/2007.06823/) (2020)
- [LSTM: A Search Space Odyssey](https://mlsanity.com/paper/1503.04069/) (2015)
- [Skillful Precipitation Nowcasting using Deep Generative Models of Radar](https://mlsanity.com/paper/2104.00954/) (2021)
- [WeatherBench: A benchmark dataset for data-driven weather forecasting](https://mlsanity.com/paper/2002.00469/) (2020)
- [Recent Trends in Deep Learning Based Natural Language Processing](https://mlsanity.com/paper/1708.02709/) (2017)
- [hp-VPINNs: Variational Physics-Informed Neural Networks With Domain Decomposition](https://mlsanity.com/paper/2003.05385/) (2020)
- [Deep Hidden Physics Models: Deep Learning of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1801.06637/) (2018)
- [PDE-Net 2.0: Learning PDEs from Data with A Numeric-Symbolic Hybrid Deep Network](https://mlsanity.com/paper/1812.04426/) (2018)
- [Towards Physics-informed Deep Learning for Turbulent Flow Prediction](https://mlsanity.com/paper/1911.08655/) (2019)
- [FourCastNet: Accelerating Global High-Resolution Weather Forecasting using Adaptive Fourier Neural Operators](https://mlsanity.com/paper/2208.05419/) (2022)

## Built on by (most cited first, 30 of 75)

- [Med-UniC: Unifying Cross-Lingual Medical Vision-Language Pre-Training by Diminishing Bias](https://mlsanity.com/paper/2305.19894/) (2023)
- [Generative Text-Guided 3D Vision-Language Pretraining for Unified Medical Image Segmentation](https://mlsanity.com/paper/2306.04811/) (2023)
- [Continuum Attention for Neural Operators](https://mlsanity.com/paper/2406.06486/) (2024)
- [Towards an end-to-end artificial intelligence driven global weather forecasting system](https://mlsanity.com/paper/2312.12462/) (2023)
- [FuXi Weather: A data-to-forecast machine learning system for global weather](https://mlsanity.com/paper/2408.05472/) (2024)
- [Appa: Bending Weather Dynamics with Latent Diffusion Models for Global Data Assimilation](https://mlsanity.com/paper/2504.18720/) (2025)
- [Fuxi-DA: A Generalized Deep Learning Data Assimilation Framework for Assimilating Satellite Observations](https://mlsanity.com/paper/2404.08522/) (2024)
- [Efficient deep data assimilation with sparse observations and time-varying sensors](https://mlsanity.com/paper/2310.16187/) (2023)
- [Uncertainty Quantification in Large Language Models Through Convex Hull Analysis](https://mlsanity.com/paper/2406.19712/) (2024)
- [Spatially-Aware Diffusion Models with Cross-Attention for Global Field Reconstruction with Sparse Observations](https://mlsanity.com/paper/2409.00230/) (2024)
- [Machine learning for modelling unstructured grid data in computational physics: a review](https://mlsanity.com/paper/2502.09346/) (2025)
- [Learning Enhanced Ensemble Filters](https://mlsanity.com/paper/2504.17836/) (2025)
- [Neural Koopman prior for data assimilation](https://mlsanity.com/paper/2309.05317/) (2023)
- [Data Assimilation with Machine Learning Surrogate Models: A Case Study with FourCastNet](https://mlsanity.com/paper/2405.13180/) (2024)
- [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)
- [Conformal Approach To Gaussian Process Surrogate Evaluation With Coverage Guarantees](https://mlsanity.com/paper/2401.07733/) (2024)
- [Observation-Guided Meteorological Field Downscaling at Station Scale: A Benchmark and a New Method](https://mlsanity.com/paper/2401.11960/) (2024)
- [Accurate deep learning-based filtering for chaotic dynamics by identifying instabilities without an ensemble](https://mlsanity.com/paper/2408.04739/) (2024)
- [CGKN: A Deep Learning Framework for Modeling Complex Dynamical Systems and Efficient Data Assimilation](https://mlsanity.com/paper/2410.20072/) (2024)
- [CGNSDE: Conditional Gaussian Neural Stochastic Differential Equation for Modeling Complex Systems and Data Assimilation](https://mlsanity.com/paper/2404.06749/) (2024)
- [4D-Var using Hessian approximation and backpropagation applied to automatically-differentiable numerical and machine learning models](https://mlsanity.com/paper/2408.02767/) (2024)
- [Modeling Partially Observed Nonlinear Dynamical Systems and Efficient Data Assimilation via Discrete-Time Conditional Gaussian Koopman Network](https://mlsanity.com/paper/2507.08749/) (2025)
- [Joint Parameter and Parameterization Inference with Uncertainty Quantification through Differentiable Programming](https://mlsanity.com/paper/2403.02215/) (2024)
- [TorchDA: A Python package for performing data assimilation with deep learning forward and transformation functions](https://mlsanity.com/paper/2409.00244/) (2024)
- [Dynamical system prediction from sparse observations using deep neural networks with Voronoi tessellation and physics constraint](https://mlsanity.com/paper/2409.00458/) (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)
- [Physics-informed neural networks modeling for systems with moving immersed boundaries: application to an unsteady flow past a plunging foil](https://mlsanity.com/paper/2306.13395/) (2023)
- [OceanBench: The Sea Surface Height Edition](https://mlsanity.com/paper/2309.15599/) (2023)

## Abstract (arXiv)

Data Assimilation (DA) and Uncertainty quantification (UQ) are extensively used in analysing and reducing error propagation in high-dimensional spatial-temporal dynamics. Typical applications span from computational fluid dynamics (CFD) to geoscience and climate systems. Recently, much effort has been given in combining DA, UQ and machine learning (ML) techniques. These research efforts seek to address some critical challenges in high-dimensional dynamical systems, including but not limited to dynamical system identification, reduced order surrogate modelling, error covariance specification and model error correction. A large number of developed techniques and methodologies exhibit a broad applicability across numerous domains, resulting in the necessity for a comprehensive guide. This paper provides the first overview of the state-of-the-art researches in this interdisciplinary field, covering a wide range of applications. This review aims at ML scientists who attempt to apply DA and UQ techniques to improve the accuracy and the interpretability of their models, but also at DA and UQ experts who intend to integrate cutting-edge ML approaches to their systems. Therefore, this article has a special focus on how ML methods can overcome the existing limits of DA and UQ, and vice versa. Some exciting perspectives of this rapidly developing research field are also discussed.