With the emerging trends of virtualization of cloud computing and big data applications, network management has become a challenging problem for optimizing the network state while satisfying the applications' Quality of Service (QoS) requirements. This paper proposes a policy-based management framework over Software-Defined Networks (SDN) for QoS provisioning. The proposed approach monitors the QoS parameters of the active flows and dynamically enforces new decisions on the underlying SDN switches to adapt the network state to the current demanded high-level policies. Moreover, the proposed solution makes use of Neural Networks to identify the violating flows causing the network congestion. Upon detection of a policy violation two route management techniques are implemented, such as: rerouting and rate limiting. The proposed framework was implemented and evaluated within an experimental testbed setup. The results indicate that the proposed PBNM-based SDN framework enables QoS provisioning and outperforms the default SDN in terms of throughput, packet loss rate and latency.