# Learning PDE Time-Stepping with Neural Cellular Automata

arXiv: https://arxiv.org/abs/2608.30328 · HTML: https://mlsanity.com/paper/2608.30328/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 14 of its 25 references that are in the map.
- First posted on arXiv 31 Aug 2026.

## Builds on (most cited first, 14 of 14)

- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Physics-informed neural networks (PINNs) for fluid mechanics: A review](https://mlsanity.com/paper/2105.09506/) (2021)
- [Multipole Graph Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2006.09535/) (2020)
- [PDE-Net: Learning PDEs from Data](https://mlsanity.com/paper/1710.09668/) (2017)
- [PDE-Net 2.0: Learning PDEs from Data with A Numeric-Symbolic Hybrid Deep Network](https://mlsanity.com/paper/1812.04426/) (2018)
- [Physics-informed learning of governing equations from scarce data](https://mlsanity.com/paper/2005.03448/) (2020)
- [PDE-GCN: Novel Architectures for Graph Neural Networks Motivated by Partial Differential Equations](https://mlsanity.com/paper/2108.01938/) (2021)
- [Learned Coarse Models for Efficient Turbulence Simulation](https://mlsanity.com/paper/2112.15275/) (2021)
- [Variational Neural Cellular Automata](https://mlsanity.com/paper/2201.12360/) (2022)
- [Neural cellular automata: applications to biology and beyond classical AI](https://mlsanity.com/paper/2509.11131/) (2025)
- [Learning spatio-temporal patterns with Neural Cellular Automata](https://mlsanity.com/paper/2310.14809/) (2023)
- [Dispersion based Recurrent Neural Network Model for Methane Monitoring in Albertan Tailings Ponds](https://mlsanity.com/paper/2411.06741/) (2024)
- [Learning Coupled System Dynamics under Incomplete Physical Constraints and Missing Data](https://mlsanity.com/paper/2512.23761/) (2025)

## Abstract (arXiv)

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.