The paper presents for the first time a methodology for solving supervised learning problems, such as classification and regression, based on deep Gaussian mixture models (DGMMs). We use a self-supervised approach to construct a classifier as well as a semi-supervised one for a regressor. More than 20 public UCI datasets with various parameters were used for testing. It has been demonstrated that the greatest increase in classification accuracy of \(37.69\%\) is achieved by using the ensemble of DGMM and extreme gradient boosting (XGBoost). The accuracy of this method exceeds that of the combination of GMM and SVM by \(14.51\%\) . The DGMM regression (DGMMR) analogue of the Gaussian mixture model regression (GMMR) is introduced as a semi-supervised learning algorithm. On the test data, the best results were shown by the ensemble of DGMMR and XGBoost regression. The accuracy of this method exceeded the combination with support vector machines regression (SVR), as well as variants of GMMR with SVR and linear regression with SVR by \(3.58\%\) , \(11.63\%\) and \(32.78\%\) , respectively.