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

arXiv: https://arxiv.org/abs/2510.15294 · HTML: https://mlsanity.com/paper/2510.15294/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 5 of its 46 references that are in the map.
- First posted on arXiv 17 Oct 2025.

## Builds on (most cited first, 5 of 5)

- [On Calibration of Modern Neural Networks](https://mlsanity.com/paper/1706.04599/) (2017)
- [Empirical Evaluation of Gated Recurrent Neural Networks on Sequence Modeling](https://mlsanity.com/paper/1412.3555/) (2014)
- [An Empirical Evaluation of Generic Convolutional and Recurrent Networks for Sequence Modeling](https://mlsanity.com/paper/1803.01271/) (2018)
- [Transfer learning of phase transitions in percolation and directed percolation](https://mlsanity.com/paper/2112.15516/) (2021)
- [Determination of the critical points for systems of directed percolation class using machine learning](https://mlsanity.com/paper/2307.10456/) (2023)

## Abstract (arXiv)

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.