The residence time Ti of water inside axons is an important biomarker for white matter pathologies of the human central nervous system, as myelin damage is hypothesised to increase axonal permeability, and thus reduce Ti. Diffusion-weighted (DW) MRI is potentially able to measure Ti as it is sensitive to the average displacement of water molecules in tissue. However, previous work addressing this has been hampered by a lack of both sensitive data and accurate mathematical models. We address the latter problem by constructing a computational model using Monte Carlo simulations and machine learning in order to learn a mapping between features derived from DW MR signals and ground truth microstructure parameters. We test our method using simulated and in vivo human brain data. Simulation results show that our approach provides a marked improvement over the most widely used mathematical model. The trained model also predicts sensible microstructure parameters from in vivo human brain data, matching values of Ti found in the literature.