Model Predictive Control (MPC) is an advanced control method broadly applied to chemical processes. However, the prohibitive online computation time limits its application to nonlinear systems. Although the approximation of the MPC control law via deep neural networks (DNNs) has been studied in these recent years, this approach cannot be applied to nonlinear systems if the optimal control problems have multiple optima. When the MPC control law follows one-to-many mappings, it cannot be effectively approximated via DNNs, which provide one-to-one mappings. In this paper, we propose a mixture density network(MDN)-based approximation method for nonlinear MPC. MDNs approximate the MPC control law through conditional probabilities by mixing several estimated Gaussians and then generate several control inputs with the highest probabilities, which means that the network can realize the one-to-many mappings. We also investigate a case study of a nonlinear benchmark process, which demonstrates that our proposed scheme exhibits better control performance than the DNN-based approximation method.