Routing services in next generation networks not only need to provide good transmission quality in heterogeneous network environments, but also need to meet the differentiated performance requirements of different applications. For example, real-time applications require low latency performance guarantees, while low-cost applications pay more attention to strict cost guarantees. Recently, deep learning has been widely applied in the field of network. With the aid of the powerful deep neural networks, the communication network can perform the routing operation intelligently to avoid the possible failure and congestion. However, existing deep learning-based network routing algorithms cannot satisfy the specific performance requirements of users, because this kind of algorithm is an unconstrained feature learning method essentially, while the routing requirements of different applications are really a constrained problem. In order to solve the above problems, we propose a deep learning-based constrained intelligent routing method, which combines the advantages of Lagrange multiplier method for solving constrained problems and the learning ability of deep learning methods, making the routing service can not only learn complex features to adapt to network environments, but also can meet differentiated requirement of users on the performance. To the best of our knowledge, this is the first work to solve the constrained routing problem by using deep learning system. Experimental results prove the effectiveness of the proposed method and show it is a method suitable for providing high-quality routing services for the next generation network.