Timely detection and identification of faults in railway track circuits are crucial for the safety and availability of railway networks. In this paper, the use of the long-short-term memory (LSTM) recurrent neural network is proposed to accomplish these tasks based on the commonly available measurement signals. By considering the signals from multiple track circuits in a geographic area, faults are diagnosed from their spatial and temporal dependences. A generative model is used to show that the LSTM network can learn these dependences directly from the data. The network correctly classifies 99.7% of the test input sequences, with no false positive fault detections. In addition, the t-Distributed Stochastic Neighbor Embedding (t-SNE) method is used to examine the resulting network, further showing that it has learned the relevant dependences in the data. Finally, we compare our LSTM network with a convolutional network trained on the same task. From this comparison, we conclude that the LSTM network architecture is better suited for the railway track circuit fault detection and identification tasks than the convolutional network.