Due to the stiffness and damping controllability of magnetorheological elastomer under external magnetic field, it has attracted great attention in the field of vibration control. In order to realize the vibration control system, accurate modeling of magnetorheological elastomer isolator is necessary. However, for the highly nonlinearity and inherently hysteresis of isolator, the existing parametric modeling methods are required to identify too many parameters and the corresponding inverse model cannot be derived from it. Therefore, this paper proposes a nonparametric neural network to approximate the dynamic behaviors of the magnetorheological elastomer isolator. Firstly, the dynamic characteristics of the isolator with shear-compression mixed mode are experimentally tested under different loading conditions. Next, based on these experimental data, the forward model of the isolator is established by using recurrent neural network. Finally, the effectiveness of the network model is validated by comparing the experimental and predicted responses of the isolator, and the results show that the performance of recurrent neural network outperforms that of BP network.