用神经元胞自动机学习长期偏微分方程动态，误差更低。
Learning PDE Time-Stepping with Neural Cellular Automata
- 基于局部同质更新规则，重复应用在每个网格点上。
- 在五类典型PDE上，长期预测相对误差最低。
- 适合需要高效长时模拟的科学计算场景。
经典数值求解器在不同初值下反复求解偏微分方程（PDE）计算成本高，亟需可学习的代理模型。本文提出一种可训练的神经元胞自动机（NCA）代理模型，用于学习长期PDE动力学。与直接将整个初值场映射为完整轨迹不同，该模型学习一个小型、局部、均匀的更新规则，该规则在每个网格单元上一致重复应用，模拟微分算子的局部性。我们在五个典型PDE（热方程、对流方程、Burgers方程、Allen-Cahn方程和Fisher-KPP方程）上进行基准测试，评估时间域超出训练范围两倍的情况。相比三种基线模型（PDE-Net、改进的物理信息神经网络PINN、傅里叶神经算子FNO），所提模型在多数实验中实现了最低的长期相对误差。
原文摘要 · Abstract (English)
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.
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