Efficient mapping of application communication patterns to the network topology is a critical problem for optimizing the performance of communication bound applications on parallel computing systems. The problem has been extensively studied in the past, but they mostly formulate the problem as finding an isomorphic mapping between two static graphs with edges annotated by traffic volume and network bandwidth. But in practice, the network performance is difficult to be accurately estimated, and communication patterns are often changing over time and not easily obtained. Therefore, this work proposes a deep reinforcement learning (DRL) approach to explore better task mappings by utilizing the performance prediction and runtime communication behaviors provided from a simulator tolearn an efficient task mapping algorithm. We extensively evaluated our approach using both synthetic and real applications with varied communication patterns on Torus and Dragonfly networks. Compared with several existing approaches from literature and software library, our proposed approach found task mappings that consistently achieved comparable or better application performance. Especially for a real application, the average improvement of our approach on Torus and Dragonfly networks are 11% and 16%, respectively. In comparison, the average improvements of other approaches are all less than 6%.