Diffusion MRI (DW-MRI) allows for the detailed exploration of the brain white matter microstructure, with applications in both research and the clinic. However, state-of-the-art methods for microstructure estimation suffer from known limitations, such as the overestimation of the mean axon diameter, and the infeasibility of fitting diameter distributions. In this study, we propose to eschew current modeling based approaches in favor of a novel, simulation-assisted machine learning approach. In particular, we train machine learning (ML) algorithms on a large dataset of simulated diffusion MRI signals from white matter regions with different axon diameter distributions and packing densities. We show, on synthetic data, that the trained models in-silico and from DW-MRI data with moderately high b-values (4000 s/mm2). Further, we show, on in-vivo data, that the estimators trained from simulations can provide parameter estimates which are close to the values expected from histology.