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.

Machine Learning (cs.LG); Machine Learning (stat.ML)

arXiv:2608.30328 [cs.LG]

(or arXiv:2608.30328v1 [cs.LG] for this version)

https://doi.org/10.48550/arXiv.2608.30328 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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