The multigrid algorithm is an efficient numerical method for solving a variety of elliptic partial differential equations (PDEs). The method damps errors at progressively finer grid scales, resulting in faster convergence compared to standard iterative methods such as Gauss-Seidel. The prolongation, or coarse-to-fine interpolation operator within the multigrid algorithm lends itself to a data-driven treatment with ML super resolution, commonly used to increase the resolution of images. We (i) propose the novel integration of a super resolution generative adversarial network (GAN) model with the multigrid algorithm as the prolongation operator and (ii) show that the GAN-interpolation improves the convergence properties of the multigrid in comparison to cubic spline interpolation on a class of multiscale PDEs typically solved in physics and engineering simulations.
翻译:多电离层算法是解决各种椭圆部分差异方程式(PDEs)的有效数字方法。方法在逐渐细化的网格尺度上断裂错误,与标准迭代法(如Gauss-Seidel)相比,导致更快的趋同。多电格算法中的延长,或粗线到线的内插操作器,可以使用数据驱动处理法,通常用于增加图像分辨率的超分辨率超分辨率超分辨率超分辨率超分辨率。我们提议将超级分辨率对抗网络(GAN)模型与作为延长操作员的多格网算法(GAN)模型进行新颖的整合,并(二)表明GAN的内插法提高了多电网的趋同性,与在物理学和工程模拟中通常解析的多尺度PDE类的三次螺旋内插。