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

arXiv: https://arxiv.org/abs/2507.14159 · HTML: https://mlsanity.com/paper/2507.14159/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 12 of its 62 references that are in the map.
- First posted on arXiv 5 Jul 2025.

## Builds on (most cited first, 12 of 12)

- [Safe, Multi-Agent, Reinforcement Learning for Autonomous Driving](https://mlsanity.com/paper/1610.03295/) (2016)
- [A high-bias, low-variance introduction to Machine Learning for physicists](https://mlsanity.com/paper/1803.08823/) (2018)
- [Scientific Machine Learning Benchmarks](https://mlsanity.com/paper/2110.12773/) (2021)
- [Discovering Phase Transitions with Unsupervised Learning](https://mlsanity.com/paper/1606.00318/) (2016)
- [Unsupervised learning of phase transitions: from principal component analysis to variational autoencoders](https://mlsanity.com/paper/1703.02435/) (2017)
- [Mutual Information, Neural Networks and the Renormalization Group](https://mlsanity.com/paper/1704.06279/) (2017)
- [Bayesian stochastic blockmodeling](https://mlsanity.com/paper/1705.10225/) (2017)
- [Deep-Learning based Inverse Modeling Approaches: A Subsurface Flow Example](https://mlsanity.com/paper/2007.15580/) (2020)
- [Unsupervised Learning of Rydberg Atom Array Phase Diagram with Siamese Neural Networks](https://mlsanity.com/paper/2205.04051/) (2022)
- [Content-based Representations of audio using Siamese neural networks](https://mlsanity.com/paper/1710.10974/) (2017)
- [Transfer learning of phase transitions in percolation and directed percolation](https://mlsanity.com/paper/2112.15516/) (2021)
- [Identifying percolation phase transitions with unsupervised learning based on largest clusters](https://mlsanity.com/paper/2311.14725/) (2023)

## Abstract (arXiv)

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 nu 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 Smax/L3 (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.