## 1 Introduction
The implementation of this dynamical prior in an ensemble filtering scheme also limits the representativity of the model as a trad-off between the method's parameters and the ensemble size need to be addressed carefully to decrease the computational complexity of the assimilation method. From this point of view, several works [23; 24] tried to formulate stochastic representations of dynamical operators for their optimal use in sequential filtering schemes. Methods based on prior knowledge of the variability of dynamical models have already been addressed to infer probabilistic representations. However, such techniques are limited to systems with available dynamical priors. Complex dynamical models in the other hand may require complex priors which may be unavailable or hard to derive. In the last years, Neural Networks have enriched the state-of-the-art in probabilistic modelling. This is principally due to the advances in deep learning models which allow better understanding of complicated systems. Probabilistic representations such as structured inference [25] and deep Gaussian processes [26] have rapidly became very popular in applications such as generative modeling and dynamical inference. From this point of view the stochastic modelization of spatio-temporal fields is an interesting open challenge that may benefit from these advances and can allow the representation of complex stochastic dynamics without any prior knowledge regarding our system. In this paper, we investigate data-driven interpolation approaches within a statistical data assimilation framework. We aim to derive stochastic data-driven representations of complex geophysical tracers. Among other representations [27] Neural networks are particularly appealing due to their efficient trad-off between modeling abilities and interpretability of the learnt models. This models have rapidly become the state-of-the-art in machine learning for a wide range of applications, including inverse imaging issues [28]. Recent applications to the assimilation of low-dimensional dynamical systems [27] and to the forecasting of geophysical dynamics [29; 30] have been developed. However, to our knowledge, the design of neural-network-based assimilation models for the spatio-temporal interpolation of geophysical dynamics remain an open challenge, which may greatly benefit from the ability of deep learning models to capture computationally-efficient representations from available ocean observation and simulation datasets. In this study, we address this challenge and propose a novel NN-based Kalman filtering scheme applied to the spatio-temporal interpolation of satellite-derived sea surface temperature. We aim to propose a parametric data driven framework that embed a stochastic representation of spatio-temporal dynamics. this architecture conveys a probabilistic representation through the prediction of a mean component and a covariance pattern. The latter may be regarded as a NN-based representation of the covariance patterns issued from Monte Carlo approximations in ensemble assimilation schemes [31]. Our model may then be directly exploited in sequential filtering schemes which allows us to overcome both issues encountered in analog data assimilation and parametric stochastic representations based on prior knowledge in terms of numerical complexity and availability of dynamical priors. Overall, the methodological contributions of this work are two-fold: (i) we propose a new probabilistic NN-based representation of 2D geophysical dynamics, (ii) we derive the associated NN-based Kalman filtering scheme for spatio-temporal interpolation issues. We demonstrate the relevance of these contributions with respect to state-of-the-art approaches [5; 11; 22] for the spatio-temporal interpolation of satellite-derived SST fields in a case study region off South Africa. This region involves complex fine-scale SST dynamics (e.g., fronts, filaments) which can't be retrieved using classical state-of-the-art techniques. This paper is organized as follows.