Graph convolutional network-based unsupervised learning of percolation transition

In this study, we address the challenge of detecting percolation phase transitions using unsupervised machine learning methods. Unlike the Ising model, where machine learning has shown success, percolation problems have proven more difficult due to the lack of direct interactions between sites. We propose a Graph Convolutional Network (GCN) approach combined with the confusion method to identify percolation transitions and estimate percolation thresholds. Various pooling methods, including state-of-the-art differentiable pooling, are evaluated for their effectiveness in aggregating global information. Our experiments on 10 x 10 lattices demonstrate that the physical-intuition-based coarse-graining method significantly outperforms other pooling techniques, providing more accurate and reliable predictions. We further validate the model's performance on larger systems, confirming its robustness and applicability. Our findings highlight the potential of GCNs with appropriate pooling methods in studying percolation transitions and suggest promising direction for future applications in condensed matter physics.