Recovering the T(2) distribution from multi-echo T(2) magnetic resonance (MR) signals is challenging but has high potential as it provides biomarkers characterizing the tissue micro-structure, such as the myelin water fraction (MWF). In this work, we propose to combine machine learning and aspects of parametric (fitting from the MRI signal using biophysical models) and non-parametric (model-free fitting of the T(2) distribution from the signal) approaches to T(2) relaxometry in brain tissue by using a multi-layer perceptron (MLP) for the distribution reconstruction. For training our network, we construct an extensive synthetic dataset derived from biophysical models in order to constrain the outputs with a priori knowledge of in vivo distributions. The proposed approach, called Model-Informed Machine Learning (MIML), takes as input the MR signal and directly outputs the associated T(2) distribution. We evaluate MIML in comparison to a Gaussian Mixture Fitting (parametric) and Regularized Non-Negative Least Squares algorithms (non-parametric) on synthetic data, an ex vivo scan, and high-resolution scans of healthy subjects and a subject with Multiple Sclerosis. In synthetic data, MIML provides more accurate and noise-robust distributions. In real data, MWF maps derived from MIML exhibit the greatest conformity to anatomical scans, have the highest correlation to a histological map of myelin volume, and the best unambiguous lesion visualization and localization, with superior contrast between lesions and normal appearing tissue. In whole-brain analysis, MIML is 22 to 4980 times faster than the non-parametric and parametric methods, respectively.