Cellular Automaton With CNN

TL;DR

This work addresses the challenge of identifying hidden jump parameters in a two-dimensional cellular automaton by training a custom convolutional neural network on CA-generated state images that combine multiple spatial resolutions and temporal iterations. The CNN learns to classify the jump parameter $\sigma$ across varied domain sizes and evolution stages, achieving an overall test accuracy of $89.31\%$ while offering substantially faster inference than conventional architectures. Key findings include the positive impact of larger domain sizes, diminishing returns after about 25 CA iterations, and the advantage of multi-resolution inputs over single-resolution training. The results demonstrate the practical potential of CNN-based parameter estimation in CA models for real-time analysis and pave the way for extending to multiple hidden parameters and higher-dimensional CA.

Abstract

Cellular automata (CA) models are widely used to simulate complex systems with emergent behaviors, but identifying hidden parameters that govern their dynamics remains a significant challenge. This study explores the use of Convolutional Neural Networks (CNN) to identify jump parameters in a two-dimensional CA model. We propose a custom CNN architecture trained on CA-generated data to classify jump parameters, which dictates the neighborhood size and movement rules of cells within the CA. Experiments were conducted across varying domain sizes (25 x 25 to 150 x 150) and CA iterations (0 to 50), demonstrating that the accuracy improves with larger domain sizes, as they provide more spatial information for parameter estimation. Interestingly, while initial CA iterations enhance the performance, increasing the number of iterations beyond a certain threshold does not significantly improve accuracy, suggesting that only specific temporal information is relevant for parameter identification. The proposed CNN achieves competitive accuracy (89.31) compared to established architectures like LeNet-5 and AlexNet, while offering significantly faster inference times, making it suitable for real-time applications. This study highlights the potential of CNNs as a powerful tool for fast and accurate parameter estimation in CA models, paving the way for their use in more complex systems and higher-dimensional domains. Future work will explore the identification of multiple hidden parameters and extend the approach to three-dimensional CA models.

Figures (6)

• Figure 1: CA with grids demonstrating the Vonn Neumann Neighborhood. Cell O is the center and the neighborhood increases from 1 (blue cells) to 3 (yellow cells).
• Figure 2: Cell movement within one time step to form an agglomerate. Initially, Ⓑ moves downwards and then the single cells move to favorable positions
• Figure 3: Visualization of structural changes in the CA model using varying porosities and jump parameters. Larger porosities and jump parameters result in sparse and dispersed patterns.
• Figure 4: The illustration of the CNN model used for jump parameter prediction.
• Figure 5: Confusion matrix demonstrating the accuracy of the model. The horizontal axis is the predicted label while the vertical axis is the true label. The correct classes belong to the diagonal positions.