Partial differential equations (PDEs) are central to describing complex physical system simulations. Their expensive solution techniques have led to an increased interest in deep neural network based surrogates. However, the practical utility of training such surrogates is contingent on their ability to model complex multi-scale spatio-temporal phenomena. Various neural network architectures have been proposed to target such phenomena, most notably Fourier Neural Operators (FNOs), which give a natural handle over local & global spatial information via parameterization of different Fourier modes, and U-Nets which treat local and global information via downsampling and upsampling paths. However, generalizing across different equation parameters or time-scales still remains a challenge. In this work, we make a comprehensive comparison between various FNO, ResNet, and U-Net like approaches to fluid mechanics problems in both vorticity-stream and velocity function form. For U-Nets, we transfer recent architectural improvements from computer vision, most notably from object segmentation and generative modeling. We further analyze the design considerations for using FNO layers to improve performance of U-Net architectures without major degradation of computational cost. Finally, we show promising results on generalization to different PDE parameters and time-scales with a single surrogate model. Source code for our PyTorch benchmark framework is available at https://github.com/microsoft/pdearena.
翻译:局部偏差方程式(PDEs)是描述复杂的物理系统模拟的核心。其昂贵的解决方案技术已导致人们对深神经网络以代孕器为基础的代孕器的兴趣增加。然而,培训这种代孕器的实际效用取决于它们是否有能力模拟复杂的多尺度超标准时空现象。提出了各种神经网络结构来针对这些现象,其中最主要的是四级神经操作员(FNOs),它们通过对不同的Fourier模式和通过下取样和上层采样路径处理当地和全球信息的U-Net,自然处理当地和全球空间信息。然而,推广不同等式参数或时间尺度的通用化仍然是一项挑战。在这项工作中,我们全面比较了各种FNO、ResNet和U-Net(U-Net)方法,以对付这些现象。对于U-Nets来说,我们将最新的建筑改进从计算机视野,特别是对象分割和基因化模型,我们进一步分析了在使用FNO的层和上层对当地和全球信息进行处理的设计考虑,从而改进了不同等式等式的等式等式平面的公式结构的绩效。我们最后将展示了U-del-del-DE标准的U-delisal-delisal 的U-delvial 基础结构结构模型,没有显示我们总成本基础化,我们总基数级的模型。