We explore emergent properties of Isotropic Neural Cellular Automata models (IsoNCA) trained on two simple problems, and evaluate them on static image outputs. The first task involves pixel-perfect reconstruction of fixed target images starting from a fixed initial state. The other task requires NCA to synthesize a pattern that, when fed into a separate Imagenet-pretrained convolutional network, triggers a strong response in the feature channels. In order to satisfy the given objectives and model constraints, optimization arrives at solutions that display intriguing life-like dynamics and mimic well known phenomena, such as swarming of Boids, Voronoi cell formation and network-like structures. Finally we discuss the multi-scale hierarchical nature of the solutions that arise in NCA.

Cellular Automata have intrigued curious minds for the better part of the last century, with significant contributions to their field from the likes of Von Neumann et al. (1966), John Conway (Gardner (1970)), and Wolfram and Gad-el Hak (2003). They can simulate and model phenomena in biology, chemistry, and physics (Chopard and Droz (1998)). Recently, Neural Cellular Automata (NCA) have demonstrated a capacity to learn complex behaviour, including constructing a target morphology (Mordvintsev et al. (2020)), classifying the shape they occupy (Randazzo et al. (2020)), or segmentation of images (Sandler et al. (2020)). As a computational model, NCA have appealing properties. They are parallelisable, fault tolerant and partially robust to operating on manifolds other than those used during training. A strong parallel exists between training NCA and system identification of a partial differential equation (PDE) satisfying certain boundary value conditions. In the original work by Mordvintsev et al. (2020), asynchronicity in cell updates is justified by a desire to have purely local communication between cells. We demonstrate that asynchronicity is not just an ideological feature of the model and is in fact necessary to learn a well-behaved PDE and to allow the model to be used in arbitrary integrators.

Reaction-Diffusion (RD) systems provide a computational framework that governs many pattern formation processes in nature. Current RD system design practices boil down to trial-and-error parameter search. We propose a differentiable optimization method for learning the RD system parameters to perform example-based texture synthesis on a 2D plane. We do this by representing the RD system as a variant of Neural Cellular Automata and using task-specific differentiable loss functions. RD systems generated by our method exhibit robust, non-trivial “life-like” behavior.