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

Sibo Cheng, Cesar Quilodran-Casas, Said Ouala, Alban Farchi, Che Liu, Pierre Tandeo, Ronan Fablet, Didier Lucor, Bertrand Iooss, Julien Brajard, Dunhui Xiao, Tijana Janjic, Weiping Ding, Yike Guo, Alberto Carrassi, Marc Bocquet, Rossella Arcucci

arXiv:2303.10462v1cs.LG

TL;DR

High-dimensional dynamical systems require methods that address uncertainty, noisy observations, computational cost, and limited interpretability. This paper reviews how machine learning, data assimilation, and uncertainty quantification are combined across applications, highlighting hybrid models’ interpretability and noise-reduction strengths while identifying continuing challenges.

  • Problem

    High-dimensional dynamical systems face challenges in system identification, reduced-order surrogate modelling, error covariance specification, model error correction, uncertainty quantification, and computationally difficult data assimilation.

  • Method

    The paper provides a broad review of state-of-the-art approaches combining ML, DA, and UQ, covering applications, methodological connections, challenges, and research perspectives.

  • Results

    The review finds that hybrid ML–DA–UQ models provide strengths in interpretability and noise reduction across varied fields and applications.

  • Takeaways & Limitations

    The field offers broad applicability for improving data-driven modelling, while substantial opportunity remains for breakthrough advances in operational contexts.

Abstract

from arXiv · show

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.

I. INTRODUCTION

ML, DA, and UQ are increasingly combined to address challenges in high-dimensional dynamical systems, especially under noisy, partial observations. This review surveys their connections, applications, and emerging perspectives rather than comparing method performance.

  • Motivation: ML improves forecasting of complex high-dimensional dynamics but can lack interpretability, robustness, and quantified predictive error.UQ helps characterize uncertainty and anticipate model failure, particularly when observations are sparse, discrete, and noisy.
  • Methodological connections: The review presents connections among ML, DA, and UQ, including variational assimilation, gradient-based optimization, reduced-order surrogates, and uncertainty sensitivity analysis.UQ distinguishes uncertainty sources and supports analysis of how uncertain inputs affect model outputs and assimilation estimates.
  • High-dimensional systems: Reduced-order modelling enables ML–DA combinations in high-dimensional systems by reducing computational cost.Figure 2 organizes methods and challenges by available data dimension and noise level.
  • Contribution: The authors describe this work as the first review focused on combining ML, DA, and UQ for dynamical systems and discuss research perspectives and challenges.The review aims to guide readers through fast-growing techniques across the field without comparing existing methods’ performance.
  • Scope: The review examines how ML can address DA and UQ challenges, and how DA and UQ can improve ML-based models.The focus includes noisy partial data and real-time observations for progressively adjusting ML surrogate models.
  • Applications: Applications span Numerical Weather Prediction, environmental modelling, Computational Fluid Dynamics, climate science, fluid dynamics, and image analysis.The reviewed combinations therefore cover both scientific computing and broader data-driven applications.

B. DA for dynamical systems

Data assimilation combines model predictions, observations, and uncertainty information to estimate evolving system states and parameters. Its variational and filtering methods offer distinct computational strategies but require restrictive assumptions or substantial computational resources.

  • Data assimilation aggregates information from multiple sources to predict physical fields, estimate parameters, and continuously correct model predictions.The corrections account for model and observation errors or uncertainties.
  • A typical time-dependent DA framework over [0, T] represents states, observations, operators, error covariances, and analysis states.The framework includes true states, background estimates, observations, transformation operators, and covariance matrices for background, observation, and model errors.
  • a) Variational DA: Variational DA estimates a conditional state PDF over a time window and minimizes a cost function balancing observation, model, and background constraints.The minimizing trajectory is the analysis trajectory, and geofluid applications require sequential cycling of the numerical model.
  • a) Variational DA: Variational DA requires tangent linear and adjoint models, while high-dimensional error covariance operators are intricate and computationally demanding to specify.These requirements have hampered widespread adoption, although differentiable ML models may help address the tangent-linear and adjoint-model requirement.
  • b) Kalman-filter-based DA: Kalman filters sequentially update marginal state PDFs through alternating analysis and forecast steps, whereas variational methods estimate trajectories over a time window.Under linear dynamics, Gaussian errors, and uncorrelated errors, the Kalman filter is the exact analytic solution to the Gaussian estimation problem.
  • b) Kalman-filter-based DA: The Kalman filter provides time-dependent state and uncertainty estimates with minimum error variance and no bias, but assumes linear models and requires costly matrix operations.The extended Kalman filter handles nonlinear dynamics through model linearization but retains Gaussian assumptions and large computational requirements.
  • b) Kalman-filter-based DA: Ensemble Kalman filters approximate error estimation and propagation with finite ensembles, making them applicable to high-dimensional systems but linking accuracy to ensemble size.Inflation combats covariance underestimation, while localization improves the effective covariance rank and span by reducing poorly estimated long-distance correlations.

C. ML with UQ

Machine learning methods provide varied routes to uncertainty quantification, with approaches spanning classical models, neighborhood and ensemble methods, and deep learning. In high-dimensional dynamical systems, uncertainty is especially relevant because reduced-order surrogates combine multiple error sources.

  • ML methods face methodological challenges involving bias-variance trade-offs, training-data availability, high-dimensional inputs, stability, complexity, and interpretability.
  • The review organizes ML approaches into linear and polynomial, neighborhood, ensemble, and deep-learning families.Table I summarizes the ML approaches considered in the review.
  • Linear and polynomial models can provide confidence intervals using analytical leave-one-out error formulas and mathematical properties of regression coefficients.
  • Neighborhood models such as kriging associate computable prediction confidence intervals through their underlying Gaussian-process assumption.
  • Ensemble models obtain confidence intervals through quantile regression or specialized subsampling procedures, although interval estimation is more difficult than for simpler models.
  • Deep-learning uncertainty quantification includes Bayesian neural networks, Monte-Carlo dropout, and deep ensembles.
  • High-dimensional dynamical-system surrogates combine observation or data uncertainty, compression errors, and predictive errors, which DA and UQ can help specify and correct.

D. ML for predicting high-dimensional dynamical systems

ML prediction of high-dimensional dynamical systems commonly uses reduced-order models to lower computational cost. These models compress data into latent spaces before learning surrogate dynamics, while DA and UQ address the resulting uncertainties.

  • High-dimensional ML prediction often relies on reduced-order models that compress data into a latent space before learning surrogate dynamics.The reduced representation decreases computational cost, but introduces compression and prediction errors.
  • DA and UQ can be employed to specify and correct the compression and prediction errors introduced by reduced-order surrogate systems.

1) Reduced-order-modelling:

Reduced-order modelling uses projection-based methods and autoencoders to represent complex dynamical systems in lower-dimensional spaces. Autoencoders capture nonlinear patterns but can produce less interpretable latent geometries and may require architecture-specific adaptations.

  • Projection-based methods such as POD and PGD have long been used to reduce the dimensionality of complex dynamical systems.
  • An autoencoder uses an encoder to map inputs into a reduced latent space and a decoder to reconstruct the full physical field.
  • Encoder and decoder networks are jointly trained to minimize reconstruction loss, commonly quantified by mean square error.The loss uses the Frobenius norm and composition function, with expectation over the relevant data.
  • Autoencoders can capture nonlinear patterns more effectively than projection-based methods such as POD, but their latent-space geometry may be chaotic and less interpretable.
  • Convolutional autoencoders are widespread, whereas graph neural networks address the difficulty of applying convolutional architectures to unstructured data such as irregular CFD meshes.Graph-based ML has also been used for modelling liquids, granular materials, and adaptive remeshing.
  • Figure 3 illustrates autoencoder development with emphasis on transitions between explicit and implicit latent spaces.

2) Predictive models:

Reduced-order models make forecasting complex dynamical systems substantially cheaper, while recurrent and newer machine-learning architectures model temporal dependencies in the reduced space.

  • ROM forecasts cost only a fraction of high-dimensional model solutions.
  • RNNs, including LSTMs, model temporal dependencies and reproduce subsequent ROM time steps.LSTMs are used extensively for long-range dependencies and address vanishing gradients.
  • Recent ROM studies also use Transformers and adversarial prediction models.

III. DATA ASSIMILATION USING MACHINE LEARNING

The review organizes ML applications in data assimilation around correcting model errors, using statistical corrections alongside physical models and training them from assimilation analyses. Offline and online strategies alternate or jointly perform state estimation and correction learning, with online methods suited to sequential observations.

  • A. ML to correct model errors in DA: ML-based statistical corrections augment physical dynamical models to address model errors in data assimilation.The resulting hybrid physical/statistical models can use additive or other correction forms.
  • A. ML to correct model errors in DA: Resolvent correction adds the statistical correction after the physical model evolution, whereas tendency correction is more intrusive to implement.
  • A. ML to correct model errors in DA: Analysis increments from data-assimilation analyses can train statistical models to predict corrections when physical and statistical contributions are independent.
  • A. ML to correct model errors in DA: Offline learning alternates data-assimilation state estimation with machine-learning correction estimation after the full analysis trajectory is available.Online approaches instead estimate states and correction parameters sequentially as observations arrive.
  • A. ML to correct model errors in DA: Online learning can correct model error after the first observation and can also place the two-step process in a fully differentiable framework.

B. ML and DA for parameter estimations

The review covers ML and DA methods for parameter estimation and error-covariance specification in high-dimensional dynamical systems. It highlights constrained, non-Gaussian parameter distributions, computational trade-offs, surrogate modeling, and learned covariance estimation.

  • B. ML and DA for parameter estimations: Parameter estimation is difficult because parameters are strongly nonlinearly coupled to model equations and constrained to value ranges.Standard EnKF and variational methods relying on Gaussian assumptions and first two moments may require modification.
  • B. ML and DA for parameter estimations: Hybrid DA-ML methods can adjust model parameters in real time as new observations arrive.An RNN replaced a compartmental epidemic model, allowing DA to incorporate new data for parameter adjustment.
  • B. ML and DA for parameter estimations: Parameter estimates from Bayesian neural networks reduced state errors and increased ensemble spread, but their computational cost remains an obstacle.
  • B. ML and DA for parameter estimations: ROM- and ML-based wildfire surrogates update fire-spread coefficients using real-time satellite observations and support flexible observation-window lengths.The same technology was applied to nuclear reactor physics with spatially sparse observations.
  • C. Error specification in DA: traditional and ML methods: Error covariance matrices control the relative weighting of forecasts and observations in data-assimilation algorithms.
  • C. Error specification in DA: traditional and ML methods: Classical covariance estimation can be computationally difficult in high-dimensional systems and sensitive to the initial covariance guess.
  • C. Error specification in DA: traditional and ML methods: DICE considerably outperformed CELLO on simulated and real data when learning measurement-error distributions under a Gaussian assumption.
  • C. Error specification in DA: traditional and ML methods: An LSTM-based RNN framework uses synthetic observation sequences to predict the observation covariance matrix.

D. End-to-end learning of DA systems

End-to-end learning schemes integrate neural components across the DA pipeline, either mimicking sequential forecast-analysis cycles or optimizing variational DA objectives. These schemes open avenues for application-specific DA and nonlinear or non-Gaussian state estimation, while remaining challenging for complex operational systems.

  • D. End-to-end learning of DA systems: End-to-end DA trains differentiable neural blocks across the full assimilation system rather than addressing isolated tasks such as model-error correction or parameter estimation.The approach maps raw inputs through elementary neural blocks for a given inverse problem.
  • D. End-to-end learning of DA systems: Sequential-DA-inspired schemes use neural forecasting and analysis operators, commonly implemented with recurrent architectures such as LSTM or GRU.Forecasting advances the state, while analysis updates it using new observations.
  • D. End-to-end learning of DA systems: Figure 7 distinguishes architectures that mimic sequential forecast-analysis steps from architectures that perform iterative gradient descent for a variational DA criterion.The two design families correspond to sequential and non-sequential variational formulations.
  • D. End-to-end learning of DA systems: 4DVar-inspired schemes use automatic differentiation to optimize state sequences or model parameters without explicitly coding the adjoint operator.They assume neural implementations of the observation and dynamical operators.
  • D. End-to-end learning of DA systems: End-to-end DA can reduce estimation biases in nonlinear or non-Gaussian systems and support application-specific pipelines for selected observing systems and diagnostic variables.Reported demonstrations include reconstruction of sea-surface dynamics from satellite observations.
  • D. End-to-end learning of DA systems: The framework permits flexible state definitions and model parameterizations, including augmented-state, multimodal, and uncertainty representations.Applying these schemes to complex spatial-temporal problems handled by operational climate, ocean, and weather systems remains a key challenge.

IV. MACHINE LEARNING ASSISTED BY DATA ASSIMILATION AND UNCERTAINTY QUANTIFICATION

This section reviews how DA and UQ assist ML models in dynamical systems, covering uncertainty analysis, latent correction, equation identification, and forecasting with partial observations. It emphasizes probabilistic and approximate-inference methods for characterizing predictive uncertainty and improving interpretability.

  • IV. MACHINE LEARNING ASSISTED BY DATA ASSIMILATION AND UNCERTAINTY QUANTIFICATION: The section covers uncertainty analysis for ML, latent DA correction of surrogate models, governing-equation identification, and forecasting partially observed dynamical systems.These topics describe how DA and UQ can enhance ML prediction accuracy and interpretability.
  • A. Uncertainty analysis for ML approaches: Bayesian neural networks and Monte Carlo dropout provide UQ approaches, while deep ensembles approximate uncertainty using independently trained networks or optimization iterates.These families estimate uncertainty through distributions or ensembles of models.
  • A. Uncertainty analysis for ML approaches: Bayesian neural networks infer distributions over network weights, supporting uncertainty estimates, robustness to over-fitting, and learning from small datasets.The Bayesian framework places priors over weights, inducing distributions over parametric functions.
  • A. Uncertainty analysis for ML approaches: Exact posterior inference over neural-network parameters is computationally and analytically intractable, motivating approximate inference and posterior-sampling methods.High dimensionality, complex weight dependencies, and prior influence make exact uncertainty characterization difficult; HMC is described as computationally demanding.
  • A. Uncertainty analysis for ML approaches: Predictive uncertainty combines aleatoric noise uncertainty with epistemic uncertainty associated with model parameters and structural fidelity.Aleatoric uncertainty arises from noisy inputs, whereas epistemic uncertainty concerns model representation and parameter choices.
  • A. Uncertainty analysis for ML approaches: Variational inference approximates the posterior over neural-network parameters by minimizing KL divergence, equivalently maximizing the evidence lower bound.Replacing Bayesian marginalization with optimization can considerably reduce computational cost.
  • A. Uncertainty analysis for ML approaches: UQ can support counterfactual explanations and provide model-noise information needed by DA algorithms for dynamical systems.It can indicate the conditions under which a decision was made and its degree of freedom.

B. ML and DA with ROM

Combining DA with reduced-order models, especially ML-based autoencoders, targets the computational burden of high-dimensional systems while correcting surrogate forecasts with observations. Latent Assimilation methods differ in how they encode states and observations and in the computational and uncertainty costs of their mappings.

  • B. ML and DA with ROM: DA with reduced-order models lowers computational cost by performing assimilation in a reduced space, using projection methods such as POD or ML-based autoencoders.The optimal reduced-space dimension is itself an extensively investigated design choice.
  • B. ML and DA with ROM: Iterative surrogate forecasts can accumulate prediction errors, while DA uses real-time local-sensor or satellite observations to update latent states and improve long-term predictions.Online LA updates the starting point for the next forecast time level in latent space.
  • B. ML and DA with ROM: Latent Assimilation approaches either correct reduced-order models with full-space observations or compress states and observations into a shared latent space.Shared-space assimilation can be more efficient for dense observation mappings but is difficult for highly nonlinear state-observation transformations.
  • B. ML and DA with ROM: GLA and LSDA connect multiple latent spaces through local surrogate functions, enabling variational DA in a local optimization problem.These methods use polynomial or MLP surrogates to bridge latent representations.
  • B. ML and DA with ROM: Local latent mappings require online computation and can introduce uncertainty when the approximation range is inappropriate; DDA instead jointly trains observation encoders and state decoders.DDA transfers observation data directly to state space and addresses complex state-observation mappings.

C. ML for dynamical systems assisted by DA

DA-assisted ML addresses dynamical-system learning when hidden states and governing models are unavailable or observations are incomplete and noisy. Reviewed approaches range from sparse interpretable basis expansions to ML regressors learned from consecutive observations.

  • C. ML for dynamical systems assisted by DA: Observation-derived forecasting models often assume perfect, complete observations, although real measurements commonly violate these conditions through noise and incomplete coverage.The review frames DA as a way to account for uncertainties in dynamical and observation models.
  • C. ML for dynamical systems assisted by DA: In system identification, only observations are provided; hidden states, the dynamical model, and the observation operator relating observations to states are unknown.Approaches are classified according to the elements of the state-space model they identify.
  • C. ML for dynamical systems assisted by DA: With direct state measurements and zero model and observation noise, identification becomes finding a basis function that explains temporal observation variability.This is a restricted identification setting based on directly observed states.
  • C. ML for dynamical systems assisted by DA: Traditional data-driven identification expands the dynamical model with nonlinear basis functions, often using polynomial representations.Sparse optimization can be combined with these expansions to derive interpretable dynamical models.
  • C. ML for dynamical systems assisted by DA: SINDy-based sparse representations offer interpretability and fewer parameters than other ML models, reducing the chances of over-fitting.The review notes successful applications to several canonical problems.
  • C. ML for dynamical systems assisted by DA: A second family formulates identification as regression between consecutive observations, including analog forecasting, reservoir computing, and other ML tools.Reservoir computing approaches are described as well-suited for learning dynamical systems from data.

2) Noisy observations of the state variables:

The review examines how DA and ML address noisy, partial observations in dynamical-system identification, including state reconstruction, latent embeddings, and joint model optimization.

  • Irregular space-time sampling and nonzero observation noise require estimating hidden state variables from observations before governing-equation identification.
  • DA can provide initial state estimates, while data-driven interpolation or analog forecasting can replace or augment physical models in ensemble assimilation.Without an original physical model, reconstruction is limited to observed variables unless delay embedding is used.
  • The DA–ML approach improves forecast skill in small-dimensional systems and can match the order of magnitude of models trained on complete, noiseless observations.It has also been shown equivalent to an expectation maximisation method.
  • Phase-space reconstruction projects observations into a higher-dimensional embedding whose temporal evolution can represent hidden-state dynamics.Takens delay embeddings are a principal methodology when prior dynamical knowledge is unavailable.
  • NbedDyn jointly learns a latent embedding and dynamical model, using DA to optimize model parameters and reconstruct unobserved latent components.The framework addresses the sensitivity and complexity of manually parameterizing delay embeddings.
  • Embedding-based approaches assume an ordinary-differential-equation representation, which may fail or become impractical for very high-dimensional systems.In geoscience, state-space dimension can reach O(109).

V. OTHER APPROACHES, CHALLENGES & PERSPECTIVES

The review surveys additional challenges and perspectives for ML–DA/UQ, including multi-scale and mode-switching dynamics, complex observation mappings, and operational high-dimensional applications.

  • a) Forecasting Multi-scale dynamical systems: Future work includes ML representations for multi-scale systems, where unresolved dynamics influence large-scale behavior across domains such as chemistry and finance.
  • b) Mode-switching dynamics: Mode-switching systems require data-driven models that identify critical transitions and account for their real-time dynamics and aftereffects with DA.
  • c) Learning state-observation mapping in data assimilation: Learning state-observation mappings remains difficult because operational observation operators can be complex, highly nonlinear, and incomplete.Such mappings complicate cost-function minimization and often involve only a subset of state variables.
  • VI. CONCLUSION: The review concludes that combining ML with DA and UQ has advanced data-driven modelling across fields while improving interpretability and noise reduction.
  • VI. CONCLUSION: Further breakthroughs are especially anticipated for operational integration in high-dimensional, multimodal, and multi-scale systems such as numerical weather prediction and ocean dynamics.
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