用神经细胞自动机学习偏微分方程的时间步进
Learning PDE Time-Stepping with Neural Cellular Automata
浏览论文内容
中文总结 AI 辅助
该研究针对偏微分方程经典数值求解器重复计算成本高的问题，提出基于神经细胞自动机的代理模型，在5类偏微分方程实验中多数场景下长时程相对误差低于PDE-Net、PINN、FNO等基准模型。
中文摘要 AI 辅助
经典的偏微分方程（PDE）数值求解器在不同初始条件下重复求解时计算成本高昂，这催生了对学习型代理模型的需求。本文提出一种可训练的基于神经细胞自动机（NCA）的代理模型，用于学习偏微分方程的长时间动力学。该模型并非一次性将整个初始场映射为完整轨迹，而是学习一个小型、局部、齐次的更新规则，该规则在每个网格单元上相同且重复应用，契合微分算子的局部性。我们在5种经典偏微分方程（热传导、平流、伯格斯、艾伦-蔡恩、费希尔-KPP）上，将该框架与3种基准模型（PDE-Net、改进的物理信息神经网络（PINN）、傅里叶神经算子（FNO））进行对比，评估的时域超出训练时域2倍。所提模型在多数实验中实现了最低的长时程相对误差。
英文摘要
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.
发表机构
- University of Alberta（阿尔伯塔大学）
- Interdisciplinary Lab for Mathematical Ecology and Epidemiology (ILMEE)（数学生态学与流行病学跨学科实验室）
机构由 AI 辅助整理，请以论文原文为准。
