[Paper Review] Machine learning with data assimilation and uncertainty quantification for dynamical systems: a review

Sibo Cheng, Cesar Quilodran-Casas+15 more|arXiv (Cornell University)|Mar 18, 2023

Meteorological Phenomena and Simulations13 citations

TL;DR

A comprehensive review of how data assimilation and uncertainty quantification enhance machine learning for high-dimensional dynamical systems, and how ML can, in turn, advance DA and UQ.

ABSTRACT

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.

Motivation & Objective

• Clarify how ML can address core challenges in data assimilation and uncertainty quantification for high-dimensional dynamical systems.
• Survey how DA and UQ can enhance the robustness, interpretability, and accuracy of ML models.
• Summarize reduced-order modelling approaches and their integration with ML, DA, and UQ.
• Highlight key applications in NWP, environmental modelling, and CFD and discuss future perspectives.

Proposed method

• Classifies approaches into DA using ML and ML assisted by DA/UQ.
• Discusses probabilistic and variational formulations linking 4D-Var and ML.
• Analyzes Kalman-filter-based and variational DA methods and their UQ implications.
• Reviews ML techniques for ROM, including autoencoders, POD/PGD, and DL-based surrogates.
• Examines UQ methods such as Monte Carlo, Polynomial Chaos, and conformal predictions in ML/DA contexts.

Experimental results

Research questions

• RQ1 How can ML methods overcome limitations of traditional DA and UQ, such as interpretability and model error handling?
• RQ2 How can data assimilation principles improve the reliability and uncertainty quantification of ML predictions for high-dimensional dynamical systems?
• RQ3 What role do reduced-order models play in enabling scalable ML-DA-UQ integrations?
• RQ4 What are the main applications and challenges in integrating ML with DA and UQ across climate, CFD, and geoscience domains?
• RQ5 What promising perspectives and methodological gaps exist for future research in this interdisciplinary field?

Key findings

• ML and DA/UQ are mutually beneficial, with ML improving DA through differentiable models and DA providing principled uncertainty control for ML predictions.
• Variational DA and 4D-Var have formal connections to gradient-based ML optimization, enabling joint treatment of model and observation errors.
• Ensemble Kalman filters, localization, and inflation remain practical solutions for high-dimensional problems in ML-DA contexts.
• DL-based ROM, including autoencoders and graph neural networks, show strong potential for efficient high-dimensional forecasting and surrogate modelling, with UQ guiding reliability.
• Uncertainty quantification methods such as Monte Carlo, Polynomial Chaos, and conformal predictions are essential for assessing ML predictions under noise and data scarcity.