Some microstructure parameters, such as permeability, remain elusive because mathematical models that express their relationship to the MR signal accurately are intractable. Here, we propose to use computational models learned from simulations to estimate these parameters. We demonstrate the approach in an example which estimates water residence time in brain white matter. The residence time τi of water inside axons is a potentially important biomarker for white matter pathologies of the human central nervous system, as myelin damage is hypothesised to affect axonal permeability, and thus τi. We construct a computational model using Monte Carlo simulations and machine learning (specifically here a random forest regressor) in order to learn a mapping between features derived from diffusion weighted MR signals and ground truth microstructure parameters, including τi. We test our numerical model using simulated and in vivo human brain data. Simulation results show that estimated parameters have strong correlations with the ground truth parameters (R2={0.88,0.95,0.82,0.99}) for volume fraction, residence time, axon radius and diffusivity respectively), and provide a marked improvement over the most widely used Kärger model (R2={0.75,0.60,0.11,0.99}). The trained model also estimates sensible microstructure parameters from in vivo human brain data acquired from healthy controls, matching values found in literature, and provides better reproducibility than the Kärger model on both the voxel and ROI level. Finally, we acquire data from two Multiple Sclerosis (MS) patients and compare to the values in healthy subjects. We find that in the splenium of corpus callosum (CC-S) the estimate of the residence time is 0.57±0.05 s for the healthy subjects, while in the MS patient with a lesion in CC-S it is 0.33±0.12 s in the normal appearing white matter (NAWM) and 0.19±0.11 s in the lesion. In the corticospinal tracts (CST) the estimate of the residence time is 0.52±0.09 s for the healthy subjects, while in the MS patient with a lesion in CST it is 0.56±0.05 s in the NAWM and 0.13±0.09 s in the lesion. These results agree with our expectations that the residence time in lesions would be lower than in NAWM because the loss of myelin should increase permeability. Overall, we find parameter estimates in the two MS patients consistent with expectations from the pathology of MS lesions demonstrating the clinical potential of this new technique.