This study introduces a novel framework for estimating permeability from diffusion-weighted MRI data using deep learning. Recent work introduced a random forest (RF) regressor model that outperforms approximate mathematical models (Kärger model). Motivated by recent developments in machine learning, we propose a deep neural network (NN) approach to estimate the permeability associated with the water residence time. We show in simulations and in in-vivo mouse brain data that the NN outperforms the RF method. We further show that the performance of either ML method is unaffected by the choice of training data, i.e. raw diffusion signals or signal-derived features yield the same results.