Siamese Neural Network for Label-Efficient Critical Phenomena Prediction in 3D Percolation Models

Predicting critical phenomena from limited labeled data remains a challenging task in statistical physics. As percolation theory provides a canonical model for phase transitions with well-established critical exponents, it serves as an ideal benchmark for validating new machine learning frameworks. Here, we introduce a label-efficient learning framework based on a Siamese Neural Network (SNN) to identify phase transitions in three-dimensional site and bond percolation models. Using only 22 labeled probability points drawn entirely from non-critical regions, the method locates percolation thresholds with percent-level accuracy and yields estimates of the critical exponent $ν$ consistent with literature values within statistical uncertainty. Analysis of the learned representations clarifies what the network learns: although trained solely on binary similarity labels, the network autonomously converges to a statistic that coincides quantitatively with the normalized largest-cluster size $S_{max}/L^3$ ($r > 0.99$), the finite-size order parameter of percolation. This underlies the framework's most distinctive capability -- a model trained solely on simple cubic lattices identifies the phase transition in face-centered cubic lattices without retraining. The framework thus offers a complementary route to criticality detection in settings where no quantitative order parameter is explicitly defined or labeled data is scarce.