Identifying internal patterns in (1+1)-dimensional directed percolation using neural networks

TL;DR

This work tackles phase transitions and hidden percolation structures in a $(1+1)$-dimensional directed percolation replication process. It introduces a neural architecture that combines a spatial mixing component, a temporal encoder, and a GRU into a multi-head classifier trained on raw space–time configurations, coupled with calibration strategies to produce reliable phase maps. The approach reproduces the DP phase diagram, identifies multiple internal patterns (A, PL, Q+, D+, Q, D), and demonstrates robustness to sequence length and sampling, including indirect critical-point estimation via calibrated probability sweeps. The findings show that deep architectures can extract hierarchical, pattern-based information from raw percolation data, offering fast, feature-free phase mapping and potential applicability to complex connectivity patterns in other networks.

Abstract

In this paper we present a neural network-based method for the automatic detection of phase transitions and classification of hidden percolation patterns in a (1+1)-dimensional replication process. The proposed network model is based on the combination of CNN, TCN and GRU networks, which are trained directly on raw configurations without any manual feature extraction. The network reproduces the phase diagram and assigns phase labels to configurations. It shows that deep architectures are capable of extracting hierarchical structures from the raw data of numerical experiments.

Figures (10)

• Figure 1: Space--time lattice of binary states with axes $i$ (vertical) and $t$ (horizontal); filled blue cells denote $n_{i,t}=1$, empty cells $n_{i,t}=0$. Local stochastic update rules: survival of the center with probability $p$, activation from each neighbor with probability $q$; two active neighbors yield $q(2-q)$.
• Figure 2: Schematic percolation patterns considered in this work. Active (occupied) sites are shown in blue, empty sites in green. (a) Percolation on the original lattice sites. (b) Dipolar percolation: phase $D$ accounts for nearest neighbors (solid red lines), while phase $D^{+}$ includes also next-nearest neighbors (dashed black lines). (c) Similarly for quadrupole patterns $Q$ and $Q^{+}$. (d) Plaquette pattern $PL$ includes either nearest fully occupied $2\times2$ blocks or blocks with one empty site.
• Figure 3: Zoomed view of a representative $PL$ configuration at $p=0.51$, $q=0.90$.
• Figure 4: Base model architecture for the BDP task.
• Figure 5: One model—multiple sequence lengths. A TCN--GRU trained at $N=50$, $T=1000$ applies directly to $T/2$ and $2T$; boundaries sharpen with $T$.