  Neural Cellular Automata (NCA) is a class of Cellular Automata where the
update rule is parameterized by a neural network that can be trained using
gradient descent. In this paper, we focus on NCA models used for texture
synthesis, where the update rule is inspired by partial differential equations
(PDEs) describing reaction-diffusion systems. To train the NCA model, the
spatio-temporal domain is discretized, and Euler integration is used to
numerically simulate the PDE. However, whether a trained NCA truly learns the
continuous dynamic described by the corresponding PDE or merely overfits the
discretization used in training remains an open question. We study NCA models
at the limit where space-time discretization approaches continuity. We find
that existing NCA models tend to overfit the training discretization,
especially in the proximity of the initial condition, also called "seed". To
address this, we propose a solution that utilizes uniform noise as the initial
condition. We demonstrate the effectiveness of our approach in preserving the
consistency of NCA dynamics across a wide range of spatio-temporal
granularities. Our improved NCA model enables two new test-time interactions by
allowing continuous control over the speed of pattern formation and the scale
of the synthesized patterns. We demonstrate this new NCA feature in our
interactive online demo. Our work reveals that NCA models can learn continuous
dynamics and opens new venues for NCA research from a dynamical system's
perspective.
