Abstract: Although many graph-based clustering methods attempt to model the stationary diffusion state in their objectives, their performance limits to using a predefined graph. We argue that the estimation of the stationary diffusion state can be achieved by gradient descent over neural networks. We specifically design the Stationary Diffusion State Neural Estimation (SDSNE) to exploit multiview structural graph information for co-supervised learning. We explore how to design a graph neural network specially for unsupervised multiview learning and integrate multiple graphs into an unified consensus graph by a shared self-attentional module. The view-shared self-attentional module utilizes the graph structure to learn a view-consistent global graph. Meanwhile, instead of using auto-encoder in most unsupervised learning graph neural networks, SDSNE uses a co-supervised strategy with structure information to supervise the model learning. The co-supervised strategy as the loss function guides SDSNE in achieving the stationary state. With the help of the loss and the self-attentional module, we learn to obtain a graph in which nodes in each connected component fully connect by the same weight. Experiments on several multiview datasets demonstrate effectiveness of SDSNE in terms of six clustering evaluation metrics.