Diffusion-Weighted Magnetic Resonance Imaging is the only non-invasive technique available to infer the underlying brain tissue microstructure. Currently, one of the promising methods for microstructure imaging is signal modelling using convex formulation, e.g. using the COMMIT framework. Despite the benefits introduced with such a framework, an important limitation is the long convergence time, making the method unappealing for clinical applications. In order to address this limitation, we propose to use a neural network to learn the sparse encode representation of the data and performed an end-to-end reconstruction of the microstructure estimates directly from the diffusion-weighted data. Our results show that the neural network can accurately estimate the microstructure maps, 4 orders of magnitude faster than the convex formulation.