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有力结合。我们训练NCA从图像时间序列和偏微分方程（PDE）轨迹中学习复杂动力学。该方法旨在识别控制大规模动态涌现行为的基本局部规则。先前关于NCA的研究主要集中在学习能产生静止涌现结构的规则。我们将NCA扩展至能同时捕捉同一系统中的瞬态与稳定结构，并学习捕捉非线性偏微分方程中图灵斑图形成动力学的规则。我们证明了NCA能很好地在PDE训练数据范围之外实现泛化，展示了如何约束NCA以尊重给定对称性，并探索了相关超参数对模型性能与稳定性的影响。能够学习任意动力学特性使NCA作为数据驱动建模框架具有巨大潜力，尤其在模拟生物斑图形成方面。