Source-linked AI summary
Combining data assimilation and machine learning to infer unresolved scale parametrisation
Julien Brajard, Alberto Carrassi, Marc Bocquet, Laurent Bertino
TL;DR
The paper addresses how to learn unresolved-scale parametrisations without relying on dense, noiseless high-resolution targets. It combines data assimilation to reconstruct system states from noisy, sparse observations with neural-network emulation of unresolved processes, producing a hybrid model that improves forecasts and attractor representation over a truncated model.
Problem
Existing ML parametrisation approaches commonly rely on expensive parametrisation schemes, high- and low-resolution simulations, or complete state sequences, motivating learning from incomplete and noisy observations.
Method
Data assimilation estimates a complete state sequence from noisy, sparse observations and a truncated model, after which a neural network learns unresolved-scale model error for a hybrid model.
Results
The hybrid model has better forecast skill and significantly better attractor representation than the truncated model in both the two-scale Lorenz model and MAOOAM.
Takeaways & Limitations
The method supports learning unresolved parametrisations directly from noisy and sparse observations while retaining a physics-based model core.
Takeaways & Limitations
The approach assumes filtering can separate data uncertainty from unresolved model error, but filtering may remove fast unresolved scales and hamper forecast skill.
Abstract
from arXiv · showhide
In recent years, machine learning (ML) has been proposed to devise data-driven parametrisations of unresolved processes in dynamical numerical models. In most cases, the ML training leverages high-resolution simulations to provide a dense, noiseless target state. Our goal is to go beyond the use of high-resolution simulations and train ML-based parametrisation using direct data, in the realistic scenario of noisy and sparse observations. The algorithm proposed in this work is a two-step process. First, data assimilation (DA) techniques are applied to estimate the full state of the system from a truncated model. The unresolved part of the truncated model is viewed as a model error in the DA system. In a second step, ML is used to emulate the unresolved part, a predictor of model error given the state of the system. Finally, the ML-based parametrisation model is added to the physical core truncated model to produce a hybrid model. The DA component of the proposed method relies on an ensemble Kalman filter while the ML parametrisation is represented by a neural network. The approach is applied to the two-scale Lorenz model and to MAOOAM, a reduced-order coupled ocean-atmosphere model. We show that in both cases the hybrid model yields forecasts with better skill than the truncated model. Moreover, the attractor of the system is significantly better represented by the hybrid model than by the truncated model.
1. Introduction
This work proposes combining data assimilation with machine learning to learn unresolved-scale parametrisations from noisy and sparse observations. It extends prior approaches that rely on expensive parametrisations, high- and low-resolution simulations, or observations to represent model processes.
- Physical ODEs and PDEs underpin numerical models that represent the time evolution of observed natural systems such as Earth’s climate.
- Prior ML approaches learned subgrid parametrisations from expensive physics-based schemes or differences between high- and low-resolution simulations.
- The method combines a physics-based resolved model with a data-driven parametrisation of unresolved scales, using data assimilation to handle noisy and sparse observations.The resulting approach is designed to retain the physical core while learning unresolved processes from direct observations.
2. Objectives and definitions
The framework projects a full dynamical system into a reduced space, identifies unresolved-scale effects as model error, and learns a neural-network correction for the truncated model. Because the ideal loss depends on unavailable full-state trajectories, the method uses data assimilation with incomplete, noisy observations and tractable approximations.
- Objectives and definitions: The unresolved-scale model error is the difference between one-step projected truth and the reduced model prediction.This error is defined when the reduced model starts from the truth’s projection, without initial-condition error.
- Objectives and definitions: The study complements a reduced, scale-truncated model with a neural-network representation of unresolved scales, producing a hybrid model.The neural network g(x, θ) is trained to represent unresolved processes while retaining the resolved physical model.
- Objectives and definitions: The target hybrid model is Mh(x, δt) = Mr(x, δt) + g(x, θ), with θ chosen to make it accurately represent the underlying dynamics.The parameter vector θ contains the trainable neural-network parameters.
- Objectives and definitions: The ideal loss is generally unavailable because the true next state and invariant state distribution cannot be computed directly.An ergodic Monte Carlo approximation uses samples from a time series, but these samples can be dependent, biased, and require a long attractor-covering trajectory.
- Objectives and definitions: The learning setup uses an adjoint-free approach because the truncated model’s gradient is assumed unavailable analytically.The paper notes that gradient-based alternatives exist when this gradient can be computed.
- Objectives and definitions: The observations are incomplete and noisy, with a linear observation operator and regularly spaced sampling under the stated simplifying assumptions.The perfect-observation case instead observes the full reduced state without error.
3. Method
The method estimates unresolved-scale model error from observations using data assimilation, then trains a neural network to parameterize it within the truncated model. It evaluates hybrid forecasts against truncated and true-model references using held-out twin experiments and relative RMSE.
- Loss function approximation: The linear superposition assumption treats model error as an additive forecast term over a sufficiently short observation interval with limited within-interval variability.This approximation makes the loss function tractable when model error is unavailable at every integration step.
- Method overview: The algorithm combines data assimilation state estimates with neural-network training to construct a hybrid model for unresolved-scale effects.Noisy and incomplete observations are processed through DA before the neural network is trained to predict model error.
- State estimation: DA estimates a complete state sequence from noisy or incomplete observations, and low-pass filtering produces smoothed states used to reduce observation uncertainty before training.Filtering may also remove fast unresolved scales, potentially affecting forecast skill.
- Neural-network training: The neural network is trained as a supervised regression from estimated resolved states to unresolved-scale model-error targets, using standard gradient-based optimization.The proposed implementation uses the finite-size EnKF-N, whose inflation factor is automatically estimated.
- Evaluation: Twin experiments generate true states and synthetic observations, while independent attractor initial conditions define a held-out test ensemble evaluated with relative root mean square error.The perfect-observation-derived hybrid model serves as a benchmark for the DA-derived hybrid model.
4. Application to the two-scale Lorenz model
The two-scale Lorenz experiments evaluate a DA-derived neural-network parametrisation under varying observation density, sampling frequency, and noise. The hybrid model improves predictive skill over the truncated model, while performance remains robust to sparse spatial observations but depends on temporal sampling and filtering.
- Forecast performance: The hybrid model remains close to the true model for approximately 2 MTU and retains predictive skill until 3–4 MTU for the illustrated initial condition.
- Forecast performance: The DA-derived hybrid model has significantly better predictive skill than the truncated model until 4 MTU.This comparison is reported for forecasts evaluated with R-RMSE across 20 initial conditions and all 36 resolved components.
- Observation density: Reducing spatial observation density has only a minor effect, with hybrid forecasts showing similar skill across the tested observation densities.For Ny < 36, observation locations are randomly subsampled and vary at each time step.
- Sensitivity studies: Forecast skill is sensitive to observation sampling frequency and degrades for larger observation intervals, consistent with limits of the linear superposition assumption.The assumed additive model-error treatment becomes less reliable when resolved–unresolved coupling and unresolved nonlinear effects are no longer negligible.
- Sensitivity studies: Increasing observation noise twentyfold multiplies the forecast error at t0 + 2 MTU by only 1.3, while filtering can remove predictable fast-scale variability.Without filtering, skill deteriorates as noise increases; filtering does not drastically improve skill except under very strong noise and may harm performance for small noise.
- Overall findings: The Lorenz experiments conclude that the method is robust to spatial data density but sensitive to temporal distribution, and that filtering analyses can remove significant fast processes.
5. Application to a low-order coupled ocean-atmosphere model
MAOOAM experiments show that both perfect-observation-derived and DA-derived hybrid models improve forecasts and long-term attractor statistics over the truncated model, especially for oceanic variables. Improvements are weaker for the fast atmospheric variable, particularly when the hybrid is trained from noisy observations.
- Forecast skill: Forecast skill improves over the truncated model for both hybrid models, with oceanic improvements reaching a factor of 2 to 3.The two hybrid models have similar oceanic skill, while their atmospheric forecast lead time is 1 day and their oceanic lead times are 2 years.
- Forecast skill: The hybrid models better represent the atmosphere-ocean coupling because the truncated model omits fast atmospheric variables that interact with the slow oceanic variables.The pure oceanic processes are represented identically in the true and truncated models, so the improvement is attributed to coupling.
- Limitations: Atmospheric improvement is smaller: the perfect-observation hybrid outperforms the DA-derived hybrid, while fast processes remain incompletely retrieved.The limitation is linked to the assumed slow unresolved-term variation, 27-hour observation sampling, and observation errors.
- Attractor representation: Hybrid attractors reduce the truncated model’s excursions into phase-space regions absent from the true dynamics, although some discrepancies remain.This comparison uses the phase space defined by ψo,2 and ψa,1.
- Attractor representation: For all oceanic key variables, both hybrid models have quartile distributions significantly closer to the true distribution than the truncated model.Table 4 compares quartiles relative to the true model.
6. Conclusion
The paper develops a hybrid model that combines a physics-based truncated model with a data-driven representation of unresolved processes under noisy and incomplete observations. Across the tested low-order systems, the hybrid improves forecast skill and long-term statistics, while practical deployment still presents hardware and software challenges.
- Contribution: The hybrid method combines a physics-based truncated model with a data-driven model of unresolved processes using noisy and incomplete observations.The method is designed around realistic observations available at lower frequency than the model integration time step.
- Results: Hybrid models significantly improve forecast skill and global statistics over the truncated model in the two-scale chaotic system and MAOOAM.The noisy-observation hybrid has oceanic forecast skill comparable to the perfect-observation hybrid in MAOOAM.
- Results: Forecast skill is sensitive to observation frequency but robust to high observational noise and sparse spatial distributions in the two-scale chaotic system.The conclusion attributes the frequency sensitivity to the linear superposition assumption being challenged more strongly than by large observational noise.
- Implications: The algorithm can extend to larger problems because its data-assimilation and neural-network components scale to high dimensions and do not require a truncated-model adjoint.The authors identify code maintenance as an advantage of avoiding the adjoint.
- Limitations: Efficient training and execution may require substantial hardware and software resources because physics-based models and ML algorithms often use different computational architectures and data types.The authors warn that this integration can introduce operational overhead.