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.
翻译:在不同尺度上代表物理信号是工程领域最具挑战性的问题之一。已经开发了几种多尺度的模型工具来描述由 emph{Partpartdial Equations} (PDE) (PDE) 所规范的物理系统。这些工具处于有原则的物理模型和数字模型的交叉路口。最近,引入了数据驱动模型来加快PDE解决方案与数字求解器的近似。在这些最近的数据驱动方法中,神经集成操作器是一个学习功能空间间绘图的班级。这些功能在图表(meshes)上分离,适合在物理现象中进行模拟互动。在这项工作中,我们研究三种多分辨率模型,与集成的内核操作器(MPGNNNs)相近。为了验证我们的研究,我们用考虑稳定不稳定的和不稳定的 PDEs 的精选指标进行广泛的MPGNNS实验。