This paper proposes a graph neural network (GNN)-enabled coordinated beamforming design, termed HSTGNN, for high speed railway (HSR) communication systems, where a high-speed train (HST) and low-speed users (LUs) coexist in multi-cell scenarios. Two transmission schemes with the goal of maximizing quality-of-service (QoS)-constrained sum rate and rate of the HST are formulated and then reformulated using a hybrid maximum ratio transmission and zero-forcing strategy. The HSR communication system is abstracted into a heterogeneous graph, and HSTGNN consists of complex heterogeneous graph attention layers and fully-connected layers. To meet QoS requirements and power budget constraints, we employ constraint-based penalty terms and a numerical scaling operation. HSTGNN is trained via unsupervised learning to solve the two schemes in a unified framework. Numerical results demonstrate that HSTGNN achieves millisecond-level inference speed with an average optimality gap of only 4% relative to traditional optimization algorithm across various scenarios. Moreover, HSTGNN exhibits strong scalability to unseen configurations of both cells and LUs.

This paper investigates the graph neural network (GNN)-enabled beamforming design for interference channels. We propose a model termed interference channel GNN (ICGNN) to solve a quality-of-service constrained energy efficiency maximization problem. The ICGNN is two-stage, where the direction and power parts of beamforming vectors are learned separately but trained jointly via unsupervised learning. By formulating the dimensionality of features independent of the transceiver pairs, the ICGNN is scalable with the number of transceiver pairs. Besides, to improve the performance of the ICGNN, the hybrid maximum ratio transmission and zero-forcing scheme reduces the output ports, the feature enhancement module unifies the two types of links into one type, the subgraph representation enhances the message passing efficiency, and the multi-head attention and residual connection facilitate the feature extracting. Furthermore, we present the over-the-air distributed implementation of the ICGNN. Ablation studies validate the effectiveness of key components in the ICGNN. Numerical results also demonstrate the capability of ICGNN in achieving near-optimal performance with an average inference time less than 0.1 ms. The scalability of ICGNN for unseen problem sizes is evaluated and enhanced by transfer learning with limited fine-tuning cost. The results of the centralized and distributed implementations of ICGNN are illustrated.