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.

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