Discovering Partial Differential Equations With Neural Cellular Automata

Abstract Neural cellular automata (NCA) are a class of cellular automata where the update rule is parameterized by a neural network that can be trained using gradient descent. In this article, 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 spatiotemporal domain is discretized, and Euler integration is used to numerically simulate the dynamics. Crucially, it is unclear whether a ground-truth PDE even exists for the task, and NCA training only supervises the final steady state without any trajectory supervision, leaving it an open question whether a trained NCA truly learns continuous dynamics or merely overfits the discretization used during training. 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 a “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 spatiotemporal granularities. We further show that the resulting model is robust to a stochastic updating scheme and modest additive Gaussian noise. 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 avenues to studying NCA as a class of PDEs and from a dynamical system’s perspective.