# Learning spatio-temporal patterns with Neural Cellular Automata

arXiv: https://arxiv.org/abs/2310.14809 · HTML: https://mlsanity.com/paper/2310.14809/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 56 references that are in the map.
- First posted on arXiv 23 Oct 2023.

## Builds on (most cited first, 13 of 13)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Learning Transferable Visual Models From Natural Language Supervision](https://mlsanity.com/paper/2103.00020/) (2021)
- [Xception: Deep Learning with Depthwise Separable Convolutions](https://mlsanity.com/paper/1610.02357/) (2016)
- [Sharpness-Aware Minimization for Efficiently Improving Generalization](https://mlsanity.com/paper/2010.01412/) (2020)
- [What is the State of Neural Network Pruning?](https://mlsanity.com/paper/2003.03033/) (2020)
- [Recurrent Residual Convolutional Neural Network based on U-Net (R2U-Net) for Medical Image Segmentation](https://mlsanity.com/paper/1802.06955/) (2018)
- [Adan: Adaptive Nesterov Momentum Algorithm for Faster Optimizing Deep Models](https://mlsanity.com/paper/2208.06677/) (2022)
- [Revisiting Small Batch Training for Deep Neural Networks](https://mlsanity.com/paper/1804.07612/) (2018)
- [Learning Low-rank Deep Neural Networks via Singular Vector Orthogonality Regularization and Singular Value Sparsification](https://mlsanity.com/paper/2004.09031/) (2020)
- [Cellular automata as convolutional neural networks](https://mlsanity.com/paper/1809.02942/) (2018)
- [Growing Isotropic Neural Cellular Automata](https://mlsanity.com/paper/2205.01681/) (2022)
- [Extended Dynamic Mode Decomposition with Learned Koopman Eigenfunctions for Prediction and Control](https://mlsanity.com/paper/1911.08751/) (2019)
- [μNCA: Texture Generation with Ultra-Compact Neural Cellular Automata](https://mlsanity.com/paper/2111.13545/) (2021)

## Built on by (most cited first, 8 of 8)

- [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)
- [AutomataGPT: Forecasting and Ruleset Inference for Two-Dimensional Cellular Automata](https://mlsanity.com/paper/2506.17333/) (2025)
- [A New Kind of Network? Review and Reference Implementation of Neural Cellular Automata](https://mlsanity.com/paper/2604.24990/) (2026)
- [Mixtures of Neural Cellular Automata: A Stochastic Framework for Growth Modelling and Self-Organization](https://mlsanity.com/paper/2506.20486/) (2025)
- [AugRelNet: Relation-Augmented Dynamics for Structure Discovery and Forecasting from Limited Data](https://mlsanity.com/paper/2606.11251/) (2026)
- [Learning PDE Time-Stepping with Neural Cellular Automata](https://mlsanity.com/paper/2608.30328/) (2026)
- [Applications of Neural Cellular Automata: State of the Art, Challenges and Opportunities](https://mlsanity.com/paper/2609.24595/) (2026)

## Abstract (arXiv)

Neural Cellular Automata (NCA) are a powerful combination of machine learning and mechanistic modelling. We train NCA to learn complex dynamics from time series of images and PDE trajectories. Our method is designed to identify underlying local rules that govern large scale dynamic emergent behaviours. Previous work on NCA focuses on learning rules that give stationary emergent structures. We extend NCA to capture both transient and stable structures within the same system, as well as learning rules that capture the dynamics of Turing pattern formation in nonlinear Partial Differential Equations (PDEs). We demonstrate that NCA can generalise very well beyond their PDE training data, we show how to constrain NCA to respect given symmetries, and we explore the effects of associated hyperparameters on model performance and stability. Being able to learn arbitrary dynamics gives NCA great potential as a data driven modelling framework, especially for modelling biological pattern formation.