Accelerating multigrid solver with generative super-resolution

TL;DR

This work investigates accelerating geometric multigrid solvers for the pressure-Poisson equation by replacing the traditional prolongation operator with a generative adversarial network-based super-resolution (GAN-SR) operator. By training on fluid-pressure data and applying the SR interpolation within a two-level V-cycle, the approach injects finer-scale structure earlier in the iteration, achieving convergence comparable to spline-based interpolation and, in some cases, faster with alternating operators. The study reveals a rich hybrid-parameter landscape: mixed strategies (e.g., alternating GAN and spline interpolations) often outperform purely spline or purely SR methods, and transferability across Reynolds numbers and grid scales is observed, albeit with caveats at very high resolutions. Overall, GAN-SR prolongation offers a viable, interpretable accelerator for multigrid Poisson solves in fluid mechanics, with practical guidance on when and how to mix interpolation operators for best performance.

Abstract

The geometric 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 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 deep learning super-resolution, commonly used to increase the resolution of images. We (i) propose the 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 can improve the convergence properties of multigrid in comparison to cubic spline interpolation on a class of multiscale PDEs typically solved in fluid mechanics and engineering simulations. We also highlight the importance of characterizing hybrid (machine learning/traditional) algorithm parameters.

Figures (9)

• Figure 1: Diagrams for two-level multigrid and GAN interpolation.
• Figure 2: Schematic of ghost cell-like splitting of the domain. The blue dotted line shows the input grid of size $n_s \times n_s$ for the SR GAN. Only the upscaled data from the inner $2\times 2$ grid is kept for the final grid reconstruction. The next input grid is denoted by red dotted line, which is shifted 2 grid elements to the right, so that the inner $2\times 2$ grid data (marked by 'x') that is kept is next to the previous upscaled data. The processes continues with the next grid in green, and so on until. The algorithm proceeds similarly in downwards.
• Figure 3: Test of super-resolution GAN model on pressure data generated with a sinusoidal function at a fixed length scale. The sinusoidal function is $p(x,y) = \rm{cos}(nx) + \rm{cos}(ny)$, where ($x,y$) are the spatial coordinates in the range $[0,2\pi]$ and $n$ is an integer. The first and second rows correspond to $n=2$ and $n=4$ respectively. The first column shows generated data at $N=48$. The second and third columns show the interpolated high-resolution grid at $N=192$ using a spline and SR GAN method, respectively. The fourth column shows the power spectra of spatial scales for all three grids, each normalized to their peak value. Please note that the slope of the SR GAN spectra is not zero, and appears small only due to the y-axis range.
• Figure 4: Examination of super-resolution GAN model on grids from the test set. The predicted power spectra are smoothed with $\rm{N}_{\rm{smooth}}=5$ via the Gauss-Seidel iterations. Each line is the power spectra relative to the true fine solution, averaged from 50 test grids. The dashed line shows the line value of one.
• Figure 5: Averaged results of multigrid algorithm solving $100$ different grids. Norm of the difference of grid between iterations as a function of iteration, shown for two choices of prolongation operator, two-dimensional spline and GAN super resolution interpolation. The multigrid parameters used are $N_{\rm{smooth, pre}}$ = 10, $N_{\rm{smooth}}$ = 20, $N_{\rm{step}}$ = 4, and $r_{\rm{min}}$ = 12. Top: The GAN operator is used for interpolation in every iteration. Bottom: The GAN and spline operators are alternated every other iteration.