Online social media have significantly boosted the creation and transmission of information, accelerating the dissemination and interaction of vast amounts of data, thereby making the prediction of information cascades increasingly important. In recent years, deep learning has been extensively applied in the domain of information cascade prediction. This paper primarily classifies, organizes, and summarizes the current research status and classic algorithms of information cascade prediction methods based on deep learning. According to the different focuses on characterizing information cascade features, studies on deep learning-based information cascade prediction are classified from two perspectives, i.e., prediction targets and prediction methods. Each category is explained in detail, along with its principles, advantages, and disadvantages, and the commonly used datasets and evaluation metrics in this field are introduced. Additionally, this paper explores the role of symmetry in the structural patterns of information diffusion networks, analyzing how symmetry impacts the pathways and efficiency of information dissemination. Finally, this paper summarizes the potential future research directions and development trends in this domain.