[Paper Review] Machine Learning Percolation Model

Shu Cheng, Fei He+3 more|arXiv (Cornell University)|Jan 22, 2021

Theoretical and Computational Physics28 references4 citations

TL;DR

This study applies supervised and unsupervised machine learning to identify critical physical characteristics—order parameters and the critical transition point—in the 2D percolation model. Using Monte Carlo-generated datasets, convolutional neural networks (CNNs) accurately predict permeability and order parameters, with CNNs-IV identifying the critical threshold at 0.594, closely matching the theoretical value of 0.593.

ABSTRACT

Recent advances in machine learning have become increasingly popular in the applications of phase transitions and critical phenomena. By machine learning approaches, we try to identify the physical characteristics in the two-dimensional percolation model. To achieve this, we adopt Monte Carlo simulation to generate dataset at first, and then we employ several approaches to analyze the dataset. Four kinds of convolutional neural networks (CNNs), one variational autoencoder (VAE), one convolutional VAE (cVAE), one principal component analysis (PCA), and one $k$-means are used for identifying order parameter, the permeability, and the critical transition point. The former three kinds of CNNs can simulate the two order parameters and the permeability with high accuracy, and good extrapolating performance. The former two kinds of CNNs have high anti-noise ability. To validate the robustness of the former three kinds of CNNs, we also use the VAE and the cVAE to generate new percolating configurations to add perturbations into the raw configurations. We find that there is no difference by using the raw or the perturbed configurations to identify the physical characteristics, under the prerequisite of corresponding labels. In the case of lacking labels, we use unsupervised learning to detect the physical characteristics. The PCA, a classical unsupervised learning, performs well when identifying the permeability but fails to deduce order parameter. Hence, we apply the fourth kinds of CNNs with different preset thresholds, and identify a new order parameter and the critical transition point. Our findings indicate that the effectiveness of machine learning still needs to be evaluated in the applications of phase transitions and critical phenomena.

Motivation & Objective

• To evaluate the effectiveness of machine learning in detecting physical characteristics in phase transitions, particularly in the 2D percolation model.
• To investigate whether deep learning models can identify order parameters and the critical transition point without prior knowledge of labels.
• To validate model robustness by introducing noise and generating perturbed configurations via variational autoencoders (VAEs).
• To compare the performance of supervised and unsupervised learning techniques in extracting physical features from percolation configurations.

Proposed method

• Generate a large dataset of 2D percolation configurations using Monte Carlo simulations at varying occupation probabilities.
• Train three types of convolutional neural networks (CNNs-I, II, III) in a supervised manner to predict order parameters and permeability.
• Use variational autoencoders (VAE) and convolutional VAEs (cVAE) to generate perturbed configurations for robustness testing.
• Apply unsupervised learning techniques—PCA and k-means clustering with varying thresholds—to detect physical features without labeled data.
• Use CNNs-IV with varying preset thresholds to identify a new order parameter and estimate the critical transition point.
• Validate results using statistical correlation analysis between predicted outputs and known physical values.

Experimental results

Research questions

• RQ1 Can deep convolutional neural networks accurately predict the permeability and order parameters in the 2D percolation model using supervised learning?
• RQ2 How robust are the trained CNNs to noise and distribution shifts when tested on VAE- and cVAE-generated perturbed configurations?
• RQ3 Can unsupervised learning methods like PCA detect the order parameter in the absence of labeled data?
• RQ4 Can a threshold-based approach with CNNs-IV identify the critical transition point in the absence of ground-truth labels?
• RQ5 What is the accuracy of machine learning models in estimating the critical threshold, and how close is it to the theoretical value of 0.593?

Key findings

• CNNs-I, II, and III achieved high accuracy in predicting both order parameters and permeability, with strong generalization and noise resilience.
• The VAE and cVAE successfully generated realistic perturbed configurations, and the CNNs maintained consistent performance on both raw and perturbed data, confirming robustness.
• PCA showed a statistically significant linear correlation with permeability but failed to correlate with the order parameters, indicating it cannot serve as an order parameter in this model.
• CNNs-IV with a preset threshold of 0.60 identified the critical transition point at 0.594, which is extremely close to the theoretical value of 0.593.
• The model’s prediction of the critical threshold was consistent across multiple runs and sampling intervals, demonstrating high reliability.
• The study confirms that while machine learning is effective in detecting physical features in percolation, its success depends on model architecture and training strategy, and generalization to other models remains uncertain.