Learning PDE Time-Stepping with Neural Cellular Automata

Abstract

Classical numerical solvers for partial differential equations (PDEs) are computationally expensive to solve repeatedly across varying initial conditions, motivating the need for learned surrogates. In this paper, we propose a trainable Neural Cellular Automata (NCA) based surrogate model for learning long time PDE dynamics. Rather than mapping an entire initial field to a full trajectory in one shot, our proposed model learns a small, local, homogeneous update rule that is applied identically and repeatedly at every grid cell, mirroring the locality of differential operators. We benchmark this framework against three baselines: PDE - Net, a modified physics-informed neural network (PINN), and a Fourier Neural Operator (FNO), on five canonical PDEs (heat, advection, Burgers, Allen - Cahn, and Fisher - KPP), evaluated at temporal domain two times beyond the training temporal domain. The proposed model achieves the lowest long-horizon relative errors on the majority of the experiments.