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Accumulating evidence indicates that microRNAs (miRNAs) can control and coordinate various biological processes. Consequently, abnormal expressions of miRNAs have been linked to various complex diseases. Recognizable proof of miRNA-disease associations (MDAs) will contribute to the diagnosis and treatment of human diseases. Nevertheless, traditional experimental verification of MDAs is laborious and limited to small-scale. Therefore, it is necessary to develop reliable and effective computational methods to predict novel MDAs. In this work, a multi-kernel graph attention deep autoencoder (MGADAE) method is proposed to predict potential MDAs. In detail, MGADAE first employs the multiple kernel learning (MKL) algorithm to construct an integrated miRNA similarity and disease similarity, providing more biological information for further feature learning. Second, MGADAE combines the known MDAs, disease similarity, and miRNA similarity into a heterogeneous network, then learns the representations of miRNAs and diseases through graph convolution operation. After that, an attention mechanism is introduced into MGADAE to integrate the representations from multiple graph convolutional network (GCN) layers. Lastly, the integrated representations of miRNAs and diseases are input into the bilinear decoder to obtain the final predicted association scores. Corresponding experiments prove that the proposed method outperforms existing advanced approaches in MDA prediction. Furthermore, case studies related to two human cancers provide further confirmation of the reliability of MGADAE in practice.