Software-defined networking (SDN) separates the control layer from the data layer, and decisions to manage the network are issued through a controller. The distributed SDN architecture is an effective solution addressing modern WAN SDN architectures and allows multiple controllers to manage different parts of the network to ensure efficient and stable operation. To solve the problems of high switch migration cost, load imbalance, and inefficient load balancing in SDN multi-controller environments, we propose a deep learning-based controller load prediction switch migration (LPSM) strategy that uses a migration switch selection algorithm, target controller selection algorithm, and switch migration decision algorithm. Then, we propose a load balancing algorithm based on this decision algorithm. The final experimental results show that the LPSM reduces the migration cost by 16% and 8%, respectively, compared with time-sharing switch migration (TSSM) and distributed decision migration (DDM) strategies, reduces load variance from 0.02 to 0.004 compared with the DDM strategy, and improves load balancing efficiency by 27.6% compared with the TSSM strategy.