Hyperspectral unmixing is an essential tool for analyzing hyperspectral data, especially in remote sensing. Many approaches have been developed for this problem, ranging from model-based to deep learning-based, and (hybrid) unrolled methods. However, the development of supervisedly trained deep learning-based unmixing methods is hindered by the lack of available labeled training datasets. In this paper, to enable the supervised training of neural networks for hyperspectral unmixing, we propose a methodology to construct a synthetic training database directly from the hyperspectral image to unmix. We use this data generation approach to train an unrolled unmixing method LPALM. The trained LPALM is assessed on two real hyperspectral datasets and shows the best performances compared to other classical, unrolled, and autoencoder-based unmixing methods. The code of this work will be available at https://github.com/rhadjeres/Synthetic-