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.

翻译：神经细胞自动机（NCA）是机器学习与机理建模的强大结合。我们训练NCA从图像时间序列和偏微分方程轨迹中学习复杂动力学。该方法旨在识别支配大规模动态涌现行为的底层局部规则。此前关于NCA的研究主要关注学习能生成静态涌现结构的规则。我们将NCA扩展至能够同时捕获同一系统中的瞬态结构与稳定结构，并学习能捕获非线性偏微分方程（PDE）中图灵斑图形成动力学的规则。我们证明NCA能够在其PDE训练数据之外实现良好的泛化能力，展示了如何约束NCA使其尊重给定对称性，并探索了相关超参数对模型性能与稳定性的影响。由于能够学习任意动力学，NCA作为数据驱动的建模框架具有巨大潜力，尤其适用于生物斑图形成建模。