Growing Steerable Neural Cellular Automata

Neural Cellular Automata (NCA) models have shown remarkable capacity for pattern formation and complex global behaviors stemming from local coordination. However, in the original implementation of NCA, cells are incapable of adjusting their own orientation, and it is the responsibility of the model designer to orient them externally. A recent isotropic variant of NCA (Growing Isotropic Neural Cellular Automata) makes the model orientation-independent - cells can no longer tell up from down, nor left from right - by removing its dependency on perceiving the gradient of spatial states in its neighborhood. In this work, we revisit NCA with a different approach: we make each cell responsible for its own orientation by allowing it to "turn" as determined by an adjustable internal state. The resulting Steerable NCA contains cells of varying orientation embedded in the same pattern. We observe how, while Isotropic NCA are orientation-agnostic, Steerable NCA have chirality: they have a predetermined left-right symmetry. We therefore show that we can train Steerable NCA in similar but simpler ways than their Isotropic variant by: (1) breaking symmetries using only two seeds, or (2) introducing a rotation-invariant training objective and relying on asynchronous cell updates to break the up-down symmetry of the system.