Neural Renormalization Group Flow for Percolation

Anaclara Alvez; Luca Camagna; Sergio Chibbaro; Cyril Furtlehner; François Landes; Gianluca Manzan; Lorenzo Mensi

cond-mat.dis-nnAug 27, 2026

Neural Renormalization Group Flow for Percolation

Authors: Anaclara Alvez, Luca Camagna, Sergio Chibbaro, Cyril Furtlehner, François Landes, Gianluca Manzan, Lorenzo Mensi

Organizations: LISN, AO team, Bˆat 660 Universit´e Paris-Saclay, Orsay Cedex 91405 · Inria Saclay - Tau team, Bˆat 660 Universit´e Paris-Saclay, Orsay Cedex 91405

Abstract

Machine learning offers a possible route to data-driven real-space renormalization when the relevant observables are nonlocal and difficult to prescribe explicitly. We explore this idea for two-dimensional site percolation developping a supervised, scale-shared neural architecture. The model recursively applies the same learned coarse-graining rule across scales, producing a latent field from which the crossing probability is predicted, while a corresponding fine-graining decoder reconstructs the largest-cluster mask. Trained only on small lattices, the model extrapolates to substantially larger systems, recovers the spanning cluster with high fidelity, and produces observables obeying the expected finite-size scaling near the critical point. We observe that to get such performance it is key that the learned latent representation exhibits critical fluctuations and scale-dependent flows consistent with the renormalization-group structure of percolation.

Figures & tables

Figure 1: Site-percolation configuration at criticality (left) , mask of the spanning cluster (center) , and prediction of the neural network (right) . The lattice size is $L=1024$ , which lies outside the training range and therefore probes finite-size extrapolation.

Figure 2: Scale-shared RG-inspired encoder-decoder architecture. The encoder applies the same learned coarse-graining map $f_{\theta}$ at every scale until the lattice is represented by a single latent vector, from which a readout $g_{\theta}$ predicts the percolation probability. The decoder applies the same learned fine-graining map $f^{\prime}_{\theta}$ at every scale, combining coarse information with encoder features to reconstruct the largest-cluster mask.

Figure 3: Finite-size scaling of the learned percolation observables. Top Panel: predicted crossing probability

\widehat{\Pi}(p,L)

as a function of

p

, with inset showing the collapse as a function of

x=(p-p_{c})L^{1/\nu}

. Bottom Panel: predicted order parameter

\hat{P}_{\infty}(p,L)

obtained from the decoder output, compared with the target cluster mass fraction. The inset shows the behavior above the critical point and the corresponding estimate of

\beta

Figure 4: Classification and reconstruction performance as a function of the rescaled occupation probability

x=(p-p_{c})L^{1/\nu}

. Top: encoder accuracy for the percolation label. Bottom: decoder accuracy for the largest-cluster mask. The largest size used during training is

L=64

, while the test sizes extend up to

L=1024

. For each value of

x

, the reported value is the median over

N=5000

test samples. Error bars indicate bootstrap uncertainty over test samples. Insets show the performance at criticality as a function of

L

Panel of Figure 5 — Figure 5: Top:Projection of the coarsest latent representation $\Phi$ onto principal components computed from critical samples at the largest training size. Points show critical samples. Colors correspond to distinct connectivity patterns, including non-percolating configurations (light blue), configurations spanning in both directions (orange), and other percolating patterns related by rotations (green). Six clusters are clearly apparent. Arrows indicate the motion of the distribution $\Phi(p,L)$ ’s centroid as the lattice size increases, for values of $p$ below and above $p_{c}$ , providing an empirical visualization of the learned scale flow. Bottom: Finite-size scaling of the normalized latent susceptibility $\bar{\Sigma}(p,L)$ as function of $p$ with its collapse inset as function of $x=(p-pc)L^{1/\nu}$

Phase Transitions

Phase transitions are abrupt changes in collective behavior as temperature, density, noise, or other control parameters vary.

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

Shanshan Wang, Dian Xu, Jianmin Shen, Feng Gao+2

Key Laboratory of Quark and Lepton Physics (MOE) and Institute of Particle Physics, Central China Normal University, , Wuhan, 430079, Hubei, China · School of Engineering and Technology, Baoshan University, , Baoshan, 678000, Yunnan, China · Ecole Superieure d’Informatique Electronique Automatique, , Ivry-sur-Seine, 94200, , France

Phase Transitions Siamese Neural Networks Scientific ML Data-Efficient Learning

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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.

Abstract

Phase Transitions

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