Growing Isotropic Neural Cellular Automata

TL;DR

This paper introduces an isotropic neural cellular automata model that overcomes anisotropy limitations of previous models, enabling the growth of accurate asymmetrical patterns through invariant training methods and symmetry-breaking techniques.

Contribution

The paper proposes a novel Isotropic NCA model that is rotation-invariant, allowing for more flexible pattern growth and addressing limitations of the original anisotropic NCA.

Findings

IsoNCA can grow asymmetrical patterns accurately.

Training with symmetry-breaking methods improves pattern diversity.

Invariant training objectives enable orientation-independent pattern growth.

Abstract

Modeling the ability of multicellular organisms to build and maintain their bodies through local interactions between individual cells (morphogenesis) is a long-standing challenge of developmental biology. Recently, the Neural Cellular Automata (NCA) model was proposed as a way to find local system rules that produce a desired global behaviour, such as growing and persisting a predefined target pattern, by repeatedly applying the same rule over a grid starting from a single cell. In this work, we argue that the original Growing NCA model has an important limitation: anisotropy of the learned update rule. This implies the presence of an external factor that orients the cells in a particular direction. In other words, "physical" rules of the underlying system are not invariant to rotation, thus prohibiting the existence of differently oriented instances of the target pattern on the same grid. We propose a modified Isotropic NCA (IsoNCA) model that does not have this limitation. We demonstrate that such cell systems can be trained to grow accurate asymmetrical patterns through either of two methods: (1) by breaking symmetries using structured seeds or (2) by introducing a rotation-reflection invariant training objective and relying on symmetry-breaking caused by asynchronous cell updates.