Mesh-based Super-Resolution of Fluid Flows with Multiscale Graph Neural Networks

TL;DR

This work addresses the challenge of producing high-resolution 3D fluid-flow fields from coarse mesh data by introducing a mesh-based, element-local, multiscale SRGNN. The architecture combines a coarse-scale processor, a graph unpooling layer, and a fine-scale processor to map $\mathbf{Y}_1(t)$ to $\mathbf{Y}_7(t)$, with node/edge features encoded into a shared latent space and synchronized message passing to enforce spatial consistency. Key contributions include the novel synchronized MP layer, KNN-based unpooling, and a residual, one-shot training objective tested on Taylor-Green Vortex, backward-facing step, and cavity flows, including Reynolds-number and geometry extrapolations. The results show substantial improvements over spectral-element interpolation, reveal how neighborhood size and multiscale processing impact fidelity, and demonstrate potential for cross-geometry transfer and targeted fine-tuning, enabling efficient mesh-aware super-resolution for complex 3D flows.

Abstract

A graph neural network (GNN) approach is introduced in this work which enables mesh-based three-dimensional super-resolution of fluid flows. In this framework, the GNN is designed to operate not on the full mesh-based field at once, but on localized meshes of elements (or cells) directly. To facilitate mesh-based GNN representations in a manner similar to spectral (or finite) element discretizations, a baseline GNN layer (termed a message passing layer, which updates local node properties) is modified to account for synchronization of coincident graph nodes, rendering compatibility with commonly used element-based mesh connectivities. The architecture is multiscale in nature, and is comprised of a combination of coarse-scale and fine-scale message passing layer sequences (termed processors) separated by a graph unpooling layer. The coarse-scale processor embeds a query element (alongside a set number of neighboring coarse elements) into a single latent graph representation using coarse-scale synchronized message passing over the element neighborhood, and the fine-scale processor leverages additional message passing operations on this latent graph to correct for interpolation errors. Demonstration studies are performed using hexahedral mesh-based data from Taylor-Green Vortex and backward-facing step flow simulations at Reynolds numbers of 1600 and 3200. Through analysis of both global and local errors, the results ultimately show how the GNN is able to produce accurate super-resolved fields compared to targets in both coarse-scale and multiscale model configurations. Reconstruction errors for fixed architectures were found to increase in proportion to the Reynolds number. Geometry extrapolation studies on a separate cavity flow configuration show promising cross-mesh capabilities of the super-resolution strategy.

Figures (22)

• Figure 1: (a) Visualization of the TGV initial condition using contours of z-component of vorticity ($w_z$), with $w_z=1$ in red and $w_z=-1$ in blue. (b) Computational mesh ($36^3$ elements) used in NekRS simulations. (c) Visualization of GLL nodes in a single coarse (P=1, top) and fine (P=7, bottom) element.
• Figure 2: (a) Visualization of Re=1600 flow-field ${\bf Y}_7$ at $t=9$. Top plot shows vorticity contours, middle shows vorticity in the 2D x-y plane at z=$\pi/2$, and bottom shows velocity magnitude in the same plane. (b) Same as (a), but for coarsened flow-field ${\bf Y}_1$. (c) Turbulent kinetic energy versus wavenumber (energy spectrum) for ${\bf Y}_7$ (DNS, black) and ${\bf Y}_1$ (projected DNS, blue) at $t=9$. Solid and dashed curves correspond to Re=1600 and Re=3200, respectively.
• Figure 3: Meshes for backward-facing step (left) and cavity (right) configurations. Bottom schematics show 2D cross-sections.
• Figure 4: (a) Coarse-fine training snapshot pairs for BFS configuration at Re=1600, showing z-component vorticity contours (top), x-y plane slices of vorticity (z-component) taken at z=1 (middle), and x-y plane slices of velocity magnitude (bottom). (b) Same as (a), but for BFS at Re=3200. (c) Same as (a), but for cavity at Re=1600. Vorticity contours show $w_z=-5$ in red and $w_z=5$ in blue. Domains are cropped in x- and y-directions for near-step visualization.
• Figure 5: (Left) Illustration of the SRGNN modeling scope, as described in Sec. \ref{['sec:modeling_scope']}. Shown are the three different sizes of coarse element neighborhoods considered here (0, 6, and 26), with central query element nodes marked in red. Coarse element neighborhood (whose graph node features contain velocity fields) is passed as input into the SRGNN, which outputs the super-resolved velocity field in the same query element. (Right) Illustration of element-local graph generation procedure for P=1 and P=7 discretizations, as described in Sec. \ref{['sec:gnn_architecture']}.