The hybrid electromagnetic and elastic foil gas bearing is explored based on the radial basis function (RBF) neural network in this study so as to improve its stabilization in work. The related principles and structure of hybrid electromagnetic and elastic foil gas bearings is introduced firstly. Then, the proportional, integral, and derivative (PID) bearing controller is introduced and improved into two controllers: IPD and CPID. The controllers and hybrid bearing system are controlled based on the RBF neural network based on deep learning. The characteristics of the hybrid bearing system are explored at the end of this study, and the control simulation research is developed based on the Simulink simulation platform. The effects of the PID, IPD, and CIPD controllers based on the RBF neural network are compared, and they are also compared based on the traditional particle swarm optimization (PSO). The results show that the thickness, spread angle, and rotation speed of the elastic foil have great impacts on the bearing system. The proposed CIPD bearing control method based on RBF neural network has the shortest response time and the best control effect. The controller parameter tuning optimization starts to converge after one generation, which is the fastest iteration. It proves that RBF neural network control based on deep learning has high feasibility in hybrid bearing system. Therefore, the results provide an important reference for the application of deep learning in rotating machinery.