With the continuous popularization of Global Navigation Satellite System (GNSS) in various applications, the performance requirement for integrity is also increasing, especially in the field of safety-of-life. Although the existing Receiver Autonomous Integrity Monitoring (RAIM) algorithm has been embedded in the GNSS receiver as a standard method, it might still suffer from small fault detection and delay alarm problem for time series fault models. In an effort to solve this problem, a Deep Neural Network (DNN) for RAIM, named RAIM-NET, is investigated in this paper. The main idea of RAIM-NET is to propose a combination of feature vector extraction and DNN model to improve the performance of integrity monitoring, with a problem specifically designed for loss function, obtaining the model parameters. Inspired by the powerful advantages of Recurrent Neural Network (RNN) in time series data processing, a multilayer RNN is applied to build the DNN model structure and improve the detection rate for small faults and reduce the alarm delay for the time series fault event. Finally, real GNSS data experiments are designed to verify the performance of RAIM-NET in fault detection and time delay for integrity monitoring.