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Metabolism is the process by which an organism continuously replaces old substances with new substances. It plays an important role in maintaining human life, body growth and reproduction. More and more researchers have shown that the concentrations of some metabolites in patients are different from those in healthy people. Traditional biological experiments can test some hypotheses and verify their relationships but usually take a considerable amount of time and money. Therefore, it is urgent to develop a new computational method to identify the relationships between metabolites and diseases. In this work, we present a new deep learning algorithm named as graph convolutional network with graph attention network (GCNAT) to predict the potential associations of disease-related metabolites. First, we construct a heterogeneous network based on known metabolite-disease associations, metabolite-metabolite similarities and disease-disease similarities. Metabolite and disease features are encoded and learned through the graph convolutional neural network. Then, a graph attention layer is used to combine the embeddings of multiple convolutional layers, and the corresponding attention coefficients are calculated to assign different weights to the embeddings of each layer. Further, the prediction result is obtained by decoding and scoring the final synthetic embeddings. Finally, GCNAT achieves a reliable area under the receiver operating characteristic curve of 0.95 and the precision-recall curve of 0.405, which are better than the results of existing five state-of-the-art predictive methods in 5-fold cross-validation, and the case studies show that the metabolite-disease correlations predicted by our method can be successfully demonstrated by relevant experiments. We hope that GCNAT could be a useful biomedical research tool for predicting potential metabolite-disease associations in the future.