# Cellular automata as convolutional neural networks

arXiv: https://arxiv.org/abs/1809.02942 · HTML: https://mlsanity.com/paper/1809.02942/

## Key facts

- Cited by 45 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 9 Sep 2018.

## Built on by (most cited first, 30 of 45)

- [Chaos as an interpretable benchmark for forecasting and data-driven modelling](https://mlsanity.com/paper/2110.05266/) (2021)
- [Learning Graph Cellular Automata](https://mlsanity.com/paper/2110.14237/) (2021)
- [Attention-based Neural Cellular Automata](https://mlsanity.com/paper/2211.01233/) (2022)
- [Med-NCA: Robust and Lightweight Segmentation with Neural Cellular Automata](https://mlsanity.com/paper/2302.03473/) (2023)
- [Generative learning for nonlinear dynamics](https://mlsanity.com/paper/2311.04128/) (2023)
- [HyperNCA: Growing Developmental Networks with Neural Cellular Automata](https://mlsanity.com/paper/2204.11674/) (2022)
- [DyNCA: Real-time Dynamic Texture Synthesis Using Neural Cellular Automata](https://mlsanity.com/paper/2211.11417/) (2022)
- [M3D-NCA: Robust 3D Segmentation with Built-in Quality Control](https://mlsanity.com/paper/2309.02954/) (2023)
- [ARC-NCA: Towards Developmental Solutions to the Abstraction and Reasoning Corpus](https://mlsanity.com/paper/2505.08778/) (2025)
- [Neural Cellular Automata for ARC-AGI](https://mlsanity.com/paper/2506.15746/) (2025)
- [E(n)-equivariant Graph Neural Cellular Automata](https://mlsanity.com/paper/2301.10497/) (2023)
- [Frequency-Time Diffusion with Neural Cellular Automata](https://mlsanity.com/paper/2401.06291/) (2024)
- [Discovering Sensorimotor Agency in Cellular Automata using Diversity Search](https://mlsanity.com/paper/2402.10236/) (2024)
- [Neural cellular automata: applications to biology and beyond classical AI](https://mlsanity.com/paper/2509.11131/) (2025)
- [Learning spatio-temporal patterns with Neural Cellular Automata](https://mlsanity.com/paper/2310.14809/) (2023)
- [CAX: Cellular Automata Accelerated in JAX](https://mlsanity.com/paper/2410.02651/) (2024)
- [Physical Neural Cellular Automata for 2D Shape Classification](https://mlsanity.com/paper/2203.07548/) (2022)
- [Generalization over different cellular automata rules learned by a deep feed-forward neural network](https://mlsanity.com/paper/2103.14886/) (2021)
- [Learning stochastic dynamics and predicting emergent behavior using transformers](https://mlsanity.com/paper/2202.08708/) (2022)
- [Learning Locally Interacting Discrete Dynamical Systems: Towards Data-Efficient and Scalable Prediction](https://mlsanity.com/paper/2404.06460/) (2024)
- [Neural Cellular Automata: From Cells to Pixels](https://mlsanity.com/paper/2506.22899/) (2025)
- [Severe Damage Recovery in Evolving Soft Robots through Differentiable Programming](https://mlsanity.com/paper/2206.06674/) (2022)
- [AdaNCA: Neural Cellular Automata As Adaptors For More Robust Vision Transformer](https://mlsanity.com/paper/2406.08298/) (2024)
- [Unsupervised Training of Neural Cellular Automata on Edge Devices](https://mlsanity.com/paper/2407.18114/) (2024)
- [Learning Elementary Cellular Automata with Transformers](https://mlsanity.com/paper/2412.01417/) (2024)
- [Beyond Memorization: Extending Reasoning Depth with Recurrence, Memory and Test-Time Compute Scaling](https://mlsanity.com/paper/2508.16745/) (2025)
- [Synaptic motor adaptation: A three-factor learning rule for adaptive robotic control in spiking neural networks](https://mlsanity.com/paper/2306.01906/) (2023)
- [On the Spatiotemporal Dynamics of Generalization in Neural Networks](https://mlsanity.com/paper/2602.01651/) (2026)
- [A New Kind of Network? Review and Reference Implementation of Neural Cellular Automata](https://mlsanity.com/paper/2604.24990/) (2026)
- [NCAdapt: Dynamic adaptation with domain-specific Neural Cellular Automata for continual hippocampus segmentation](https://mlsanity.com/paper/2410.23368/) (2024)

## Abstract (arXiv)

Deep learning techniques have recently demonstrated broad success in predicting complex dynamical systems ranging from turbulence to human speech, motivating broader questions about how neural networks encode and represent dynamical rules. We explore this problem in the context of cellular automata (CA), simple dynamical systems that are intrinsically discrete and thus difficult to analyze using standard tools from dynamical systems theory. We show that any CA may readily be represented using a convolutional neural network with a network-in-network architecture. This motivates our development of a general convolutional multilayer perceptron architecture, which we find can learn the dynamical rules for arbitrary CA when given videos of the CA as training data. In the limit of large network widths, we find that training dynamics are nearly identical across replicates, and that common patterns emerge in the structure of networks trained on different CA rulesets. We train ensembles of networks on randomly-sampled CA, and we probe how the trained networks internally represent the CA rules using an information-theoretic technique based on distributions of layer activation patterns. We find that CA with simpler rule tables produce trained networks with hierarchical structure and layer specialization, while more complex CA produce shallower representations---illustrating how the underlying complexity of the CA's rules influences the specificity of these internal representations. Our results suggest how the entropy of a physical process can affect its representation when learned by neural networks.