Understanding the phases of water molecules based on local structure is essential for understanding their anomalous properties. However, due to complicated structural motifs formed via hydrogen bonds, conventional order parameters represent water molecules incompletely. In this paper, we develop GCIceNet, which automatically generates machine-based order parameters for classifying the phases of water molecules via supervised and unsupervised learning. The multiple graph convolutional layers in GCIceNet can learn topological information on the complex hydrogen bond networks. It shows a substantial improvement in accuracy for predicting the phase of water molecules in a bulk system and an ice/vapor interface system. A relative importance analysis shows that GCIceNet can capture the structural features of the given system hidden in the input data. Augmented with the vast amount of data provided by molecular dynamics simulations, GCIceNet is expected to serve as a powerful tool for the fields of glassy liquids and hydration layers around biomolecules.