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Efficient data transfers over high-speed, long-distance shared networks require proper utilization of available network bandwidth. Using parallel TCP streams enables an application to utilize network parallelism and can improve transfer throughput; however, finding the optimum number of parallel TCP streams is challenging due to nondeterministic background traffic sharing the same network. Additionally, the non-stationary, multi-objectiveness, and partially-observable nature of network signals in the host systems add extra complexity in finding the current network condition. In this work, we present a novel approach to finding the optimum number of parallel TCP streams using deep reinforcement learning (RL). We devise a learning-based algorithm capable of generalizing different network conditions and utilizing the available network bandwidth intelligently. Contrary to rule-based heuristics that do not generalize well in unknown network scenarios, our RL-based solution can dynamically discover and adapt the parallel TCP stream numbers to maximize the network bandwidth utilization without congesting the network and ensure fairness among contending transfers. We extensively evaluated our RL-based algorithm's performance, comparing it with several state-of-the-art online optimization algorithms. The results show that our RL-based algorithm can find near-optimal solutions 40% faster while achieving up to 15% higher throughput. We also show that, unlike a greedy algorithm, our devised RL-based algorithm can avoid network congestion and fairly share the available network resources among contending transfers.