In this paper, we proposed a novel approach for proactive edge caching in wireless small cell networks. Specifically, we propose using a recurrent neural network for predicting the content popularity with low computational complexity. The mean estimation error of the adopted recurrent neural network could be very close to that of the optimal linear prediction filter utilizing all past history. Based on the predicted content popularity, we formulate and solve a minimum cost flow problem in order to optimally place content files at edge caches. Since the computational complexity of the adopted recurrent neural network is relatively low and the minimum cost flow problem can be solved in polynomial time, the proposed approach is feasible in practice. Simulation results show that the proposed approach outperforms a greedy approach and can significantly reduce the bandwidth consumption of the backhaul network.