论文标题
使用图神经操作员近似PDE解决方案的多尺度物理表示
Multi-scale Physical Representations for Approximating PDE Solutions with Graph Neural Operators
论文作者
论文摘要
在不同尺度上代表物理信号是工程中最具挑战性的问题之一。已经开发了几种多尺度建模工具来描述由\ emph {部分微分方程}（PDES）控制的物理系统。这些工具位于原则性物理模型和数值模式的十字路口。最近，与数值求解器相比，已经引入了数据驱动的模型来加快PDE溶液的近似值。在这些最新数据驱动的方法中，神经积分运算符是一个学习函数空间之间映射的类。这些功能在图形（网格）上离散化，适用于在物理现象中建模相互作用。在这项工作中，我们使用\ emph {消息传递图神经网络}（mpgnns）近似的整体内核操作员研究了三个多分辨率架构。为了验证我们的研究，我们通过考虑稳定且不稳定的PDE进行了精心选择的指标进行广泛的MPGNN实验。
Representing physical signals at different scales is among the most challenging problems in engineering. Several multi-scale modeling tools have been developed to describe physical systems governed by \emph{Partial Differential Equations} (PDEs). These tools are at the crossroad of principled physical models and numerical schema. Recently, data-driven models have been introduced to speed-up the approximation of PDE solutions compared to numerical solvers. Among these recent data-driven methods, neural integral operators are a class that learn a mapping between function spaces. These functions are discretized on graphs (meshes) which are appropriate for modeling interactions in physical phenomena. In this work, we study three multi-resolution schema with integral kernel operators that can be approximated with \emph{Message Passing Graph Neural Networks} (MPGNNs). To validate our study, we make extensive MPGNNs experiments with well-chosen metrics considering steady and unsteady PDEs.
