The mineral content of shale plays an important role in the study of reservoir characteristics. Obtaining high-precision mineral content data from a simple and straightforward method is helpful for the study of reservoir properties. Due to the complex and varied elemental combination of minerals and the lithological diversity in different formations, there are limitations in the traditional statistical methods for predicting mineral content. In this study, a multi-source fusion dataset is constructed first using a detailed workflow to process multi-source data from elemental logging and conventional well logging from the Funing shale formations in China. A hybrid neural network model consisting of a back-propagation artificial neural network (BP-ANN) and gated recurrent unit (GRU) structure, with a custom constraint loss function, is developed to predict the mineral contents. The results show that the predictive performance of this hybrid multi-source model performs statistically significantly better than that of the BP-ANN and GRU models using single source data. In addition, the accuracy of the hybrid model is further improved by using the custom constraint loss function. This method provides high-resolution distributions of mineral content compared to the X-ray powder diffraction testing method. It will help enhance the understanding of the formation.