Many approaches have recently been proposed to combine data assimilation with machine learning for improved forecasts with geophysical models. Here, we focus on combining advanced sequential data assimilation schemes with a generative forecasting model which accounts for complex model errors. This generative model provides a stochastic approximation of the conditional distribution of the successor state given the current state. The proposed variational formulation is a novel one-step weak-constraint 4D-Var scheme in which a learned conditional stochastic forecast replaces the prescribed additive Gaussian model error representation. We derive and implement this formulation as an ensemble variational algorithm, the generative iterative ensemble Kalman filter (IEnKF-G), which generalises the iterative ensemble Kalman filter with additive Gaussian model error (IEnKF-Q). We extend the method to a smoother with an assimilation window spanning multiple time steps and propose a simplified variant. The IEnKF-G is evaluated on a stochastic variant of the Lorenz 1963 model to assess its robustness. It is then tested on a multi-scale variant of the one-dimensional Lorenz 1996 model, in which the resolved variables are affected by model error originating from the hidden small-scale variables. The algorithm is then compared to an ensemble Kalman filter that adaptively mitigates model and sampling errors, and to the IEnKF-Q. The IEnKF-G is generally more accurate in the tested configurations, with substantial gains under partial observations and significant model error. This demonstrates the potential of generative data-driven models when paired with classical data assimilation methods, particularly when model error is substantial.
Figures & tables
Figure 1: Schematic of the IEnKF-G/IEnKS-G cycling, characterised by a data assimilation window of L time steps Δt , and shifting the window by S time steps Δt between analyses. The left panel corresponds to the IEnKF-G with L=1 , S=1 ; the right panel corresponds to the IEnKS-G with L=2 , S=1 .
Model
Nx (full dim.)
Parameters
Lyap. time
Δt
H
σobs
L63
3 (3)
s=10 , r=28 , b=8/3 , σ=0.25
1.17
0.10 , 0.25 , 0.50
xyz , xz
2
GL15
40 (160)
J=4 , h=0.3 , F=8
1.14
0.05
all var.
1
Table 1: Models’ specifications. The Lyapunov time of L63 is for our stochastic L63. The Lyapunov time of the large-scale GL15 is estimated using the approximate tangent dynamics in Appendix .6 .
Method
DAW
Addressing model error
Parameter range
Coping mechanism
Further tuning
L63
GL15
EnKF-AdInf
adaptive mult. infl.
IEnKF-Q
[t0,t1]
opt. Gaussian add. err.
λ
λ∈[1,2.60]
λ∈[1,1.05]
–
σq
σq∈[0,3.90]
σq∈[0,0.05]
SDA IEnKS
[t0,tL]
λ
λ∈[1,1.10]
IEnKF-G
[t0,t1]
opt. generative seed
λ
λ∈[1,1.10]
λ∈[1,1.10]
Table 2: Comparison of hyperparameters for the DA methods. σq is the amplitude of Gaussian additive noise, λ is the multiplicative inflation parameter. opt. means optimised, mult. infl. means multiplicative inflation, add. err. means additive error.
Figure 4: Averaged RMSEs (a) and averaged CRPS (b) of the IEnKF-G as a function of Ng , applied to the large-scale GL15 model. The shaded areas around the curves indicate plus or minus twice the standard deviation across replicates.
Original model
Nx
Δt
Netr
Blocks
Channels
Core
Ntp
L63
3
0.10 , 0.25 , 0.50
3
16
8
dense, 6→8→3
2899
GL15
40
0.05
8
8
40
conv., kin=5 , kb=3
79201
Table 3: Architectures used for the stochastic transition model. Here Netr is the number of ensemble members generated per training pair, kin and kb are the input and residual-block kernel widths, respectively, and Ntp denotes the number of trainable parameters.
Long-range geophysical forecasts are fundamentally limited by chaotic dynamics and numerical errors. While data assimilation can mitigate these issues, classical variational smoothers require computationally expensive tangent-linear and adjoint models. Conversely, recent efficient latent filtering methods often enforce weak trajectory-level constraints and assume fixed observation grids. To bridge this gap, we propose Latent Ensemble Variational Data Assimilation (LEVDA), an ensemble-space variational smoother that operates in the low-dimensional latent space of a pretrained differentiable neural dynamics surrogate. By performing four-dimensional ensemble-variational (4DEnVar) optimization within an ensemble subspace, LEVDA jointly assimilates states and unknown parameters without the need for adjoint code or auxiliary observation-to-latent encoders. Leveraging the fully differentiable, continuous-in-time-and-space nature of the surrogate, LEVDA naturally accommodates highly irregular sampling at arbitrary spatiotemporal locations. Across three challenging geophysical benchmarks, LEVDA matches or outperforms state-of-the-art latent filtering baselines under severe observational sparsity while providing more reliable uncertainty quantification. Simultaneously, it achieves substantially improved assimilation accuracy and computational efficiency compared to full-state 4DEnVar.
Phillip Si, Peng Chen
School of Computational Sciences and Engineering, Georgia Institute of Technology
Data Assimilation (DA) aims to recover the full state of a dynamical system that is only partially observed. A solution is to use Score-based models to generate physically consistent trajectories that agree with the observations. These Autoregressive Diffusion models are trained by conditioning on the previous state; however, they do not take into account the uncertainty of their past predictions. We propose a new diffusion-based assimilation algorithm that dynamically balances the confidence in the current state and the new observations. Crucially, we choose to learn the distribution of the joint state containing both the past and future. This allows us to use a modified version of En4DVar, a classical DA algorithm that relies on the covariance of an ensemble of particles. Experiments on fluid and traffic flow simulations show improved reconstruction performance, especially in situations where observations are sparse and non-homogeneous.
Julien Moreau, Marc Lelarge
Inria Paris, DI ENS PSL Research University Renault Group
Hybrid forecasting systems combining machine learning and numerical weather prediction offer a promising route toward improved forecast skill while preserving physical consistency. In this study, we investigate online learning of state-dependent model-error corrections within the Integrated Forecasting System using 4D-Var. Building on previous work, we represent model error using a neural network embedded within the forecast model and trained from observations through the data assimilation cycle. First, we introduce a targeted variant of neural-network 4D-Var in which only the neural-network parameters are estimated, while the atmospheric state is provided by a precomputed analysis. We then propose a batched extension that processes assimilation windows in random order and averages parameter increments over multiple windows, drawing inspiration from mini-batch stochastic optimization. Experiments are conducted with the Integrated Forecasting System of the European Centre for Medium-Range Weather Forecasts at an intermediate resolution. Sequential online learning substantially improves forecasts within the training period, with error reductions exceeding 10%in some cases. However, these improvements fail to generalize to unseen data. In contrast, the proposed batched formulation produces hybrid models that consistently improve forecast skill in out-of-sample experiments. Larger batch sizes accelerate convergence and improve robustness by reducing the noise of parameter updates. Diagnostics further show that the online-trained corrections preserve realistic forecast variability and do not improve skill through artificial smoothing. These results demonstrate that the formulation of the online learning problem is critical for generalisation and establish batched online training as a promising framework for hybrid machine-learning/data-assimilation systems.
Alban Farchi, Marcin Chrust, Massimo Bonavita +3
ECMWF Bonn, Germany · ECMWF Reading, United Kingdom · CEREA, ENPC, EDF R&D Institut Polytechnique de Paris Île–de–France, France