Data Assimilation is the process in which we improve the representation of the state of a physical system by combining information coming from a numerical model, real-world observations, and some prior modelling. It is widely used to model and to improve forecast systems in Earth science fields such as meteorology, oceanography and environmental sciences. One key aspect of Data assimilation is the analysis step, where the output of the numerical model is adjusted in order to account for the observational data. In Variational Data Assimilation and under Gaussian assumptions, the analysis step comes down to solving a high-dimensional non-linear least-square problem. In practice, this minimization involves successive inversions of large, and possibly ill-conditioned matrices constructed using linearizations of the forward model. In order to improve the convergence rate of these methods, and thus reduce the computational burden, preconditioning techniques are often used to get better-conditioned matrices, but require either the sparsity pattern of the matrix to inverse, or some spectral information. We propose to use Deep Neural Networks in order to construct a preconditioner. This surrogate is trained using some properties of the singular value decomposition, and is based on a dataset which can be constructed online to reduce the storage requirements.