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 different time-scales still remains a challenge. In this work, we make a comprehensive comparison between various FNO and U-Net like approaches on 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 performance. Finally, we show promising results on generalization to different PDE parameters and time-scales with a single surrogate model.
翻译:局部差异方程式(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对深神经网络替代机器人的兴趣增加。然而,培训这种替代机器人的实际效用取决于他们是否有能力模拟复杂的多尺度的时空现象。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),它们通过不同Fourier模式的参数化,以及通过下取样和上取样路径处理当地和全球信息的U-Net,自然处理全球空间信息。然而,推广不同等式参数或不同时间尺度的通用化仍然是一项挑战。在这项工作中,我们对各种FNO和U-Net(U-Net)之间关于流动性机械问题的方法进行了全面比较,例如,在变色流和速度功能表上。对于U-Net,我们从计算机的视野,特别是对象分解和基因化模型,将最近的建筑改进从计算机图像转换为设计考虑因素,我们进一步分析了使用FNO层次来改进U-DE网络结构模型的性能和不同时间尺度的性能最终显示我们有前景的模型。