Graph deep learning models have been popular in graph based applications such as node classification, link prediction, community detection, etc. The expressive power of deep learning models combined with the increasing amount of graph data enables researchers to solve graph tasks that are traditionally done via algorithmic approaches. For example, neural network based models have shown state-of-the-art performance on the graph classification task, outperforming kernel based methods. However, how to perform graph matching related tasks such as Graph Edit Distance (GED) computation, Maximum Common Subgraph (MCS) detection, still requires careful design and utilization of graph deep learning models, due to the unique challenges posed by these NP-hard problems. Besides, in terms of the expressive power of neural network models, few research works have explored the usage of inter-graph information such as the graph-graph proximity to learn a better graph-level representation in an unsupervised way. In this prospectus, three recent published and ongoing researches on neural network-based graph-level operator learning are introduced. Extensive experiments show that the proposed models gain extraordinary improvements compared to the baseline appraoches.