In this work, we investigate the existence and effect of percolation in training deep Neural Networks (NNs) with dropout. Dropout methods are regularisation techniques for training NNs, first introduced by G. Hinton et al. (2012). These methods temporarily remove connections in the NN, randomly at each stage of training, and update the remaining subnetwork with Stochastic Gradient Descent (SGD). The process of removing connections from a network at random is similar to percolation, a paradigm model of statistical physics. If dropout were to remove enough connections such that there is no path between the input and output of the NN, then the NN could not make predictions informed by the data. We study new percolation models that mimic dropout in NNs and characterise the relationship between network topology and this path problem. The theory shows the existence of a percolative effect in dropout. We also show that this percolative effect can cause a breakdown when training NNs without biases with dropout; and we argue heuristically that this breakdown extends to NNs with biases.

Percolation theory serves as a cornerstone for studying phase transitions and critical phenomena, with broad implications in statistical physics, materials science, and complex networks. However, most machine learning frameworks for percolation analysis have focused on two-dimensional systems, oversimplifying the spatial correlations and morphological complexity of real-world three-dimensional materials. To bridge this gap and improve label efficiency and scalability in 3D systems, we propose a Siamese Neural Network (SNN) that leverages features of the largest cluster as discriminative input. Our method achieves high predictive accuracy for both site and bond percolation thresholds and critical exponents in three dimensions, with sub-1% error margins using significantly fewer labeled samples than traditional approaches. This work establishes a robust and data-efficient framework for modeling high-dimensional critical phenomena, with potential applications in materials discovery and complex network analysis.

Modern machine learning, grounded in the Universal Approximation Theorem, has achieved significant success in the study of phase transitions in both equilibrium and non-equilibrium systems. However, identifying the critical points of percolation models using raw configurations remains a challenging and intriguing problem. This paper proposes the use of the Kolmogorov-Arnold Network, which is based on the Kolmogorov-Arnold Representation Theorem, to input raw configurations into a learning model. The results demonstrate that the KAN can indeed predict the critical points of percolation models. Further observation reveals that, apart from models associated with the density of occupied points, KAN is also capable of effectively achieving phase classification for models where the sole alteration pertains to the orientation of spins, resulting in an order parameter that manifests as an external magnetic flux, such as the Ising model.

In the field of statistical physics, machine learning has gained significant popularity and has achieved remarkable results in recent studies on phase transitions.In this paper, we apply Principal Component Analysis (PCA) and Autoencoder(AE) based on Unsupervised learning to study the various configurations of the percolation model in equilibrium phase transition. In certain phase transition models, such as the DP model in non-equilibrium phase transitions, the order parameter is particle density. However, in some other phase transition models, such as the percolation model, it is not. This study involved randomizing and selecting percolation graphs to be used as input for a neural network, and analyzed the obtained results, indicating that the outputs of the single latent variable of AE and the first principal component of PCA are signals related to particle density.

Percolation is an important topic in climate, physics, materials science, epidemiology, finance, and so on. Prediction of percolation thresholds with machine learning methods remains challenging. In this paper, we build a powerful graph convolutional neural network to study the percolation in both supervised and unsupervised ways. From a supervised learning perspective, the graph convolutional neural network simultaneously and correctly trains data of different lattice types, such as the square and triangular lattices. For the unsupervised perspective, combining the graph convolutional neural network and the confusion method, the percolation threshold can be obtained by the "W" shaped performance. The finding of this work opens up the possibility of building a more general framework that can probe the percolation-related phenomenon.