Variational models are among the state-of-the-art formulations for the resolution of ill-posed inverse problems. Following recent advances in learning-based variational settings, we investigate the end-to-end learning of variational models, more precisely of the regularization term given some observation model, jointly to the associated solver, so that we can optimize the reconstruction performance. In the proposed end-to-end setting, both the variational cost and the gradient-based solver are stated as neural networks using automatic differentiation for the latter. We consider an application to inverse problems with incomplete datasets (image inpainting and multivariate time series interpolation). We experimentally illustrate that this framework can lead to a significant gain in terms of reconstruction performance, including w.r.t. the direct minimization of the variational formulation derived from the known generative model.