Deep Neural Networks (DNNs) have gained widespread adoption in diverse fields, including image classification, object detection, and natural language processing. However, training large-scale DNN models often encounters significant memory bottlenecks, which ask for efficient management of extensive tensors. Heterogeneous memory system, which combines persistent memory (PM) modules with traditional DRAM, offers an economically viable solution to address tensor management challenges during DNN training. However, existing memory management methods on heterogeneous memory systems often lead to low PM access efficiency, low bandwidth utilization, and incomplete analysis of model characteristics. To overcome these hurdles, we introduce an efficient tensor management approach, DeepTM, tailored for heterogeneous memory to alleviate memory bottlenecks during DNN training. DeepTM employs page-level tensor aggregation to enhance PM read and write performance and executes contiguous page migration to increase memory bandwidth. Through an analysis of tensor access patterns and model characteristics, we quantify the overall performance and transform the performance optimization problem into the framework of Integer Linear Programming. Additionally, we achieve tensor heat recognition by dynamically adjusting the weights of four key tensor characteristics and develop a global optimization strategy using Deep Reinforcement Learning. To validate the efficacy of our approach, we implement and evaluate DeepTM, utilizing the TensorFlow framework running on a PM-based heterogeneous memory system. The experimental results demonstrate that DeepTM achieves performance improvements of up to 36% and 49% compared to the current state-of-the-art memory management strategies AutoTM and Sentinel, respectively. Furthermore, our solution reduces the overhead by 18 times and achieves up to 29% cost reduction compared to AutoTM.