Abstract In geosciences, data assimilation (DA) addresses the reconstruction of a hidden dynamical process given some observation data. DA is at the core of operational systems such as weather forecasting, operational oceanography and climate studies. Beyond the reconstruction of the mean or most likely state, the inference of the state posterior distribution remains a key challenge, that is, quantify uncertainties as well as to inform intrinsical stochastic variabilities. Indeed, DA schemes, such as variational DA and Kalman methods, can have difficulty in dealing with complex non‐linear processes. A growing literature investigates the cross‐fertilization of DA and machine learning. This study proposes an end‐to‐end neural scheme based on a variational Bayes inference formulation to jointly address DA and uncertainty quantification. It combines an Evidence Lower BOund variational cost to a trainable gradient‐based solver to infer the state posterior probability distribution function given observation data. The inference of the posterior and the trainable solver are learnt jointly. We demonstrate the relevance of the proposed scheme for a Gaussian parameterization of the posterior and different case‐study experiments, including Lorenz 63 dynamics and river flow measurements. A benchmark with respect to state‐of‐the‐art schemes is provided.