Improving the predictive capability of molecular properties in ab-initio simulations is essential for advanced material discovery. Despite recent progress making use of machine learning, utilizing deep neural networks to improve quantum chemistry modelling remains severely limited by the scarcity and heterogeneity of appropriate experimental data. Here we show how training a neural network to replace the exchange-correlation functional within a fully-differentiable three-dimensional Kohn-Sham density functional theory (DFT) simulation can greatly improve its accuracy and generalizability. Using only eight experimental data points on diatomic molecules, our trained exchange-correlation networks provided improved predictions of atomization energies across a collection of 104 molecules containing new bonds and atoms that are not present in the training.