Computation of the critical point of complex fluid mixtures is an important part of thermodynamic phase behaviour computations. While algorithms for these calculations are well established, they are often slow when the number of constituting components is large. In this work, we propose a new procedure to significantly accelerate critical point calculations by leveraging deep neural network (DNN) models. A DNN model for critical point predictions of a given mixture is first trained based on the critical points of such a mixture with various compositions. Such a DNN model is then used to initialize both of the commonly used algorithms for mixture critical point calculations: root finding and global minimization. We demonstrate that when using the DNN-based predictions to initialize the root-finding-based and optimization-based algorithms, we can achieve 50-90% and 80-90% reductions in the number of required iterations, respectively.