## 6 Conclusions  
In this work, we addressed neural-network-based models for the spatio-temporal interpolation of satellite-derived SST fields. We introduced a novel probabilistic NN-based representation of geophysical dynamics. This representation, which relies on a patch-level and EOF-based decomposition, allows us to propagate in time a mean component and the covariance of the SST field. It makes direct the derivation of an associated Kalman filter for the spatio-temporal interpolation of SST dynamics. The relevance of the proposed framework is demonstrated in our numerical experiments with respect to the state-of-the-art approaches. Our method clearly outperforms the optimal interpolation and DINEOF based schemes which fail retrieving fine scale structures due to the high missing data rate in our observations. Comparing our data-driven data assimilation scheme to the analog data assimilation framework reveals the importance of investigating such filtering representations. From our numerical experiments, an important gain is stressed with respect to analog forecasting based schemes which is principally due to the formulation of our stochastic dynamical model. The patch based identification procedure allows to significantly reduce the identification complexity while still giving good priors. The recollection of the patches to form the global output allows getting ride of fine tuning post-processing step that can decrease the results as illustrated in our experiments. Finally the stochastic formulation of our dynamical model allows the propagation of a parametric PDF of our transition function in a Kalman like assimilation scheme. This stochastic formulation is completely learnt from data and allows getting ride of the ensemble formulation that may cause limitations in terms of numerical complexity. We believe that this study opens a new research avenue for the design of stochastic dynamical representations for spatio-temporal fields. The application of the proposed framework to other sea surface geophysical tracers, including multi-source and multi-modal interpolation issues is considered as our first priority. SLA (Sea Level Anomaly) fields could provide an interesting case-study as the associated space-time sampling is particularly scarce and multi-source strategies are of key interest [40]. Improving the formulation and training of the covariance model is also an important issue. Learning our covariance model based on one step ahead ensemble forecasting is most likely to fail in sequential assimilation frameworks when provided with observations with highly irregular  
Figure 5: Radially averaged power spectral density of the interpolated SST fields with respect to the reference SST. temporal sampling. Optimizing our covariance model based on the spatio-temporal sampling of our observations seems to be an interesting path to investigate as one of our further works. The use of the RMSE for training our data-driven models and as a diagnosis tool raises the question of the relevance of the proposed criterion. Although from the qualitative analysis based on the visual analysis of our reconstructed fields proved the relevance of the proposed technique. The development of more rigorous diagnosis and training criterions based on structures matching is an appalling research avenue. Exploiting stability analysis tools such as Lyapunov exponents is an interesting approach that may increase the modeling capabilities of our data-driven framework. Finally, the interpretation of the parametrization of the reconstruction network is an open issue. In our work, our reconstruction network was tuned to give the best forecasting performances with a low computational complexity. However, defining a relationship between the reconstruction network parameters (e.g., number of filters, kernel size, activation function) and the physical system (e.g., fine scale structures identification, patch boundaries) is an open research topic that might be answered in the next years due to the advances of deep learning interpretability.