MultiScale MeshGraphNets

TL;DR

<3-5 sentence high-level summary> The paper tackles the scalability of learned mesh-based simulators to high-resolution problems by diagnosing a bottleneck in message propagation within Graph Neural Networks. It introduces MultiScale MeshGraphNets (MS-MGN), a hierarchical two-resolution architecture that enables faster, more global information exchange by performing message passing on both fine and coarse meshes, augmented by a V-cycle-like processing strategy. It also proposes using high-accuracy labels derived from higher-resolution references to teach subgrid dynamics without altering the model, and demonstrates substantial improvements in accuracy and efficiency over traditional MeshGraphNets while preserving stability in rollouts. Together, these contributions advance the practicality of learned, mesh-based physics simulators for high-resolution, complex flow scenarios.</3-5 sentence high-level summary>

Abstract

In recent years, there has been a growing interest in using machine learning to overcome the high cost of numerical simulation, with some learned models achieving impressive speed-ups over classical solvers whilst maintaining accuracy. However, these methods are usually tested at low-resolution settings, and it remains to be seen whether they can scale to the costly high-resolution simulations that we ultimately want to tackle. In this work, we propose two complementary approaches to improve the framework from MeshGraphNets, which demonstrated accurate predictions in a broad range of physical systems. MeshGraphNets relies on a message passing graph neural network to propagate information, and this structure becomes a limiting factor for high-resolution simulations, as equally distant points in space become further apart in graph space. First, we demonstrate that it is possible to learn accurate surrogate dynamics of a high-resolution system on a much coarser mesh, both removing the message passing bottleneck and improving performance; and second, we introduce a hierarchical approach (MultiScale MeshGraphNets) which passes messages on two different resolutions (fine and coarse), significantly improving the accuracy of MeshGraphNets while requiring less computational resources.

Figures (15)

• Figure 1: The four update operators on MS-MGN: Downsample (left), where each node on the low-resolution mesh (orange mesh) receives information from the high-resolution mesh triangle (blue mesh) enclosing the node; High-resolution (bottom-middle), where high-resolution nodes are updated by their connected neighbors; Low-resolution (top-middle), where low-resolution nodes are updated by connected their neighbors; Upsample (right), where each high-resolution node receives information from the corresponding low-resolution nodes it updates in the Downsample update.
• Figure 2: A reference simulation of the Karman vortex street simulated with COMSOL. The colormap shows the $x$-component of the velocity field. Top: The simulation mesh is not fine enough to resolve all flow features, and the characteristic vortex shedding is suppressed. Bottom: A more expensive simulation on a finer mesh correctly resolves the dynamics. Middle: "high-accuracy" labels from the high-resolution simulation (bottom) interpolated on the coarse mesh from (top), with vortex-shedding still visible. We use this to generate a training set.
• Figure 3: Relationship between the minimum mesh edge length (edge min) and the number of nodes and edges on the input mesh for the CylinderFlow test dataset.
• Figure 4: Log-log plot of 1-step MSE measured against a highly-resolved "ground-truth" simulation. We expect the error to converge to 0 with decreased edge length (red curve); MGN initially tracks this curve, but accuracy drops off at small edge lengths. Increasing the number of message passing steps moves the drop-off point, but the fundamental issue remains.
• Figure 5: Convergence analysis of MS-MGN. With the same number of message passing steps, MS-MGN exhibits a significantly lower error than the corresponding MGN model. At 25 mps, MS-MGN closely tracks the convergence curve of the reference solver (red).