Learning PDE Time-Stepping with Neural Cellular Automata

arXiv:2608.30328v1 Announce Type: new 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.

Related

• MD-PNOP: Equation-Recast Neural Operators for Minimal-Data Extrapolation and PDE Solver Acceleration
• MiNO: Cotangent-bundle propagator learning for PDEs
• Inertial Manifold Neural Operator for Dissipative Time-Dependent Partial Differential Equations
• Active Learning with Selective Time-Step Acquisition for PDEs

Source: arXiv cs.LG | 2026-09-01