Patch-Based Deep Kernel Learning for Data Assimilation and Bias Correction with Extremely Sparse Observations

Emanuele Mele; Angelo Coluccia; Massimo Cafaro; Italo Epicoco; Paola Nassisi

cs.AIOct 30, 2026

Patch-Based Deep Kernel Learning for Data Assimilation and Bias Correction with Extremely Sparse Observations

Authors: Emanuele Mele, Angelo Coluccia, Massimo Cafaro, Italo Epicoco, Paola Nassisi

Organizations: Department of Engineering for Innovation, University of Salento, 73100 Lecce, Italy · Euro-Mediterranean Center on Climate Change, Foundation (CMCC), 73100 Lecce, Italy

Abstract

Data Assimilation (DA) reconstructs the state of a spatio-temporal system by combining sensor observations with the output of a numerical model. Since model states are high-dimensional while observations cover only a small fraction of the domain, DA is a challenging large-scale estimation problem, central to weather forecasting and oceanography. Classical variational methods such as 3D-Var rely on a hand-crafted background error covariance that assumes unbiased errors, and therefore cannot capture state-dependent correlations or systematic model biases. Moreover, their cost grows with the size of the domain. Deep learning (DL) methods can perform DA and bias correction (BC) non-linearly, but typically offer limited interpretability and no calibrated uncertainty estimates. We propose a hybrid framework that couples a convolutional neural network (CNN) with a learnable-kernel Gaussian Process (GP) to jointly perform DA and BC from extremely sparse observations. The model is trained end-to-end by maximizing the marginal log-likelihood, learns a data-driven covariance over the whole grid, and provides both the Maximum A Posteriori analysis and its uncertainty by design. A patch-based strategy reduces the $O(N^3)$ training cost of exact GPs over $N$ grid points to $O(N)$ for a fixed patch size. Inference remains exact and only requires solving a linear system whose $O(m^3)$ cost depends on the number of observations $m \ll N$ , not on the domain. The learned covariance can also replace the hand-crafted one in 3D-Var, improving its accuracy without changing the variational formulation. Experiments on sea surface temperature and salinity, with both synthetic and real, extremely sparse in-situ observations, show that the framework outperforms 3D-Var, a classical GP and a competing DL model, while reducing inference time by more than 98% with respect to 3D-Var.

Figures & tables

Fig. 1 : Proposed data assimilation framework combining a neural network and a Gaussian process. Top: During training, the neural network learns an embedding $\mathbf{Z}$ of the input $\mathbf{x}$ , which is fed to the learnable kernel $k_{\bm{\gamma}}$ to compute the covariance matrix over the grid and optimize the marginal log-likelihood. Bottom: At inference, the embeddings $\mathbf{Z}^{*}$ of the background state are used to compute the covariance matrices between observed and unobserved locations; conditioned on the residuals at the observed positions, the model predicts the background error corrections $\hat{\bm{\mu}}_{\mathbf{Z}^{*}}$ and their uncertainties $\hat{\bm{\Sigma}}_{\mathbf{Z}^{*}}$ , which are added to the background state to obtain the assimilated field.

Fig. 2 : Mediterranean Sea domain selected for the numerical experiments. The dashed box delimits the $128\times 128$ computational domain; the shaded areas outside it are excluded. Yellow markers indicate the positions of in-situ mooring buoys.

Fig. 3 : Training cost per day and RMSE improvement over the unassimilated simulation as a function of the patch size.

Fig. 4 : Mean daily RMSE of SST and SSS over the test year as a function of the density of synthetic observations. The dashed line is the unassimilated simulation.

Fig. 5 : Spatial distribution of the SST and SSS assimilation error and of the error reduction with respect to the unassimilated baseline. Black markers indicate the observation locations.

Fig. 6 : Posterior standard deviation estimated by the proposed model over the test year for SST and SSS. Black markers indicate the observation locations.

Fig. 7 : Mean daily RMSE of the proposed method as a function of

\alpha

for SST and SSS, at different densities of synthetic observations. Stars mark the best value of

\alpha

for each density on the test year.

Fig. 8 : Daily mean RMSE of SST and SSS over the test year with real buoy observations.

Obs. Density	3D-Var [s]	GP [s]	FiLM CNN [s]	Proposed [s]
$1.0\%$	17.9385	0.0595	0.0743	0.1617
$1.4\%$	16.1644	0.0614	0.0764	0.2776
$1.9\%$	17.7259	0.0628	0.0850	0.3038
$2.3\%$	16.9607	0.0763	0.0858	0.2836
$2.8\%$	18.3246	0.0697	0.0810	0.3082
$3.2\%$	16.7395	0.0734	0.0820	0.3156

TABLE I : Per-day inference time of the four assimilation approaches as a function of the density of synthetic observations.

Fig. 9 : Learned SST and SSS correlations between the location marked by the star and every other point.

Fig. 10 : Paired comparison of the daily RMSE obtained by 3D-Var with the classical Gaussian background error covariance (x-axis) and with the covariance learned by the proposed method (y-axis), over the test year for SST and SSS.

Data Assimilation

Data assimilation combines observations with dynamical models to estimate hidden system states and their uncertainty over time.

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Data Assimilation

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Data assimilation (DA) estimates the state of an evolving dynamical system from noisy, partial observations, and is widely used in scientific simulation as well as weather and climate science. In practice, filtering methods rely on frame-to-frame transition models. However, these models are fragile when observations are non-Markovian (when they form only a partial slice of a higher-dimensional latent state as in real-world weather data): they tend to accumulate errors over long horizons. At the same time, learned DA methods typically commit to a single regime, either filtering (nowcasting, real-time forecasting) or smoothing (retrospective reanalysis), which splits what should be a shared prior across application-specific pipelines. To address both issues, we introduce ForcingDAS, a unified and robust DA framework. Built on Diffusion Forcing with an independent noise level assigned to each frame, ForcingDAS learns a joint-trajectory prior instead of frame-to-frame transitions. This allows it to capture long-horizon temporal dependencies and reduce error accumulation. In addition, the same trained model spans the full filtering to smoothing spectrum at inference time. Specifically, nowcasting, fixed-lag smoothing, and batch reanalysis are selected through the inference schedule alone, without retraining. We evaluate ForcingDAS on 2D Navier-Stokes vorticity, precipitation nowcasting, and global atmospheric state estimation. Across all settings, a single model is competitive with or outperforms both learned and classical baselines that are specialized for individual regimes, with the largest gains observed on real-world weather benchmarks.

Abstract

Data Assimilation

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