MicroRNAs (miRNAs) have been proved to play critical roles in diverse biological processes, including the human disease development process. Exploring the potential associations between miRNAs and diseases can help us better understand complex disease mechanisms. Given that traditional biological experiments are expensive and time-consuming, computational models can serve as efficient means to uncover potential miRNA-disease associations. This study presents a new computational model based on variational graph auto-encoder with matrix factorization (VGAMF) for miRNA-disease association prediction. More specifically, VGAMF first integrates four different types of information about miRNAs into an miRNA comprehensive similarity network and two types of information about diseases into a disease comprehensive similarity network, respectively. Then, VGAMF gets the non-linear representations of miRNAs and diseases, respectively, from those two comprehensive similarity networks with variational graph auto-encoders. Simultaneously, a non-negative matrix factorization is conducted on the miRNA-disease association matrix to get the linear representations of miRNAs and diseases. Finally, a fully connected neural network combines linear and non-linear representations of miRNAs and diseases to get the final predicted association score for all miRNA-disease pairs. In the 10-fold cross-validation experiments, VGAMF achieves an average AUC of 0.9280 on HMDD v2.0 and 0.9470 on HMDD v3.2, which outperforms other competing methods. Besides, the case studies on colon cancer and esophageal cancer further demonstrate the effectiveness of VGAMF in predicting novel miRNA-disease associations.