Learning phase transitions by siamese neural network

The wide application of machine learning (ML) techniques in statistics physics has presented new avenues for research in this field. In this paper, we introduce a semi-supervised learning method based on Siamese Neural Networks (SNN), trying to explore the potential of neural network (NN) in the study of critical behaviors beyond the approaches of supervised and unsupervised learning. By focusing on the (1+1) dimensional bond directed percolation (DP) model of nonequilibrium phase transition and the 2 dimensional Ising model of equilibrium phase transition, we use the SNN to predict the critical values and critical exponents of the systems. Different from traditional ML methods, the input of SNN is a set of configuration data pairs and the output prediction is similarity, which prompts to find an anchor point of data for pair comparison during the test. In our study, during test we set different bond probability $p$ or temperature $T$ as anchors, and discuss the impact of the configurations at this anchors on predictions. In addition, we use an iterative method to find the optimal training interval to make the algorithm more efficient, and the prediction results are comparable to other ML methods.