Towards Universal Mesh Movement Networks

TL;DR

The paper tackles the challenge of delivering accurate, efficient mesh movement for PDE solvers across varying equations and geometries without retraining. It introduces UM2N, a two-stage graph-based architecture (Graph Transformer encoder and Graph Attention Network decoder) trained on a PDE-independent dataset, with an element-volume loss to prevent inverted elements. By decoupling movement from the underlying PDE through Monge-Ampère-based supervision and employing a PDE-agnostic monitor function, UM2N achieves zero-shot generalization and strong performance on advection, Navier–Stokes, and tsunami simulations, often matching or exceeding traditional MA methods while being far faster. The work demonstrates robustness to complex boundaries, highlights failure modes in extreme geometries, and provides thorough ablations to justify architectural and loss choices, offering a practical, scalable path toward universal mesh adaptation in scientific computing.

Abstract

Solving complex Partial Differential Equations (PDEs) accurately and efficiently is an essential and challenging problem in all scientific and engineering disciplines. Mesh movement methods provide the capability to improve the accuracy of the numerical solution without increasing the overall mesh degree of freedom count. Conventional sophisticated mesh movement methods are extremely expensive and struggle to handle scenarios with complex boundary geometries. However, existing learning-based methods require re-training from scratch given a different PDE type or boundary geometry, which limits their applicability, and also often suffer from robustness issues in the form of inverted elements. In this paper, we introduce the Universal Mesh Movement Network (UM2N), which -- once trained -- can be applied in a non-intrusive, zero-shot manner to move meshes with different size distributions and structures, for solvers applicable to different PDE types and boundary geometries. UM2N consists of a Graph Transformer (GT) encoder for extracting features and a Graph Attention Network (GAT) based decoder for moving the mesh. We evaluate our method on advection and Navier-Stokes based examples, as well as a real-world tsunami simulation case. Our method outperforms existing learning-based mesh movement methods in terms of the benchmarks described above. In comparison to the conventional sophisticated Monge-Ampère PDE-solver based method, our approach not only significantly accelerates mesh movement, but also proves effective in scenarios where the conventional method fails. Our project page is at https://erizmr.github.io/UM2N/.

Figures (18)

• Figure 1: Overview of Universal Mesh Movement Network.
• Figure 2: Results for swirl case. The plot shows the end state of the swirl simulation (bottom row) and corresponding meshes (top row). As the process is reversible, the end state should be consistent to the initial state. It can be observed that the solution obtained on high resolution almost recover the initial state. The proposed UM2N outperform M2N and achieve comparable performance to MA method. Quantitatively in the rightmost plot, the orange line (UM2N) and blue line (MA) best suppress error accumulation along timesteps.
• Figure 3: Results for the flow past cylinder case in time. In the upper part, we visualize $5$ snapshots of adapted meshes and vorticity intensity from $4.00s$ to $4.20s$. Please refer to the full video in the supplementary materials. It shows that our UM2N can output meshes which adapt to the dynamics around the cylinder. It can be observed from the zoom-in mini-plots at the top-left that UM2N reduces errors/noise in vorticity intensity compared to the original mesh. In the lower part, the blue, orange and green lines show the drag $C_D$ coefficients obtained on a high resolution fixed mesh, UM2N adapted mesh and the original coarse mesh for the first 6 seconds (i.e., 6000 steps). It can be observed that the UM2N output mesh improves the accuracy of $C_D$ in magnitude and periodicity compared to the original mesh.
• Figure 4: Analysis on wake flow vorticity for the cylinder case. The adapted mesh output by UM2N (see top-left part of the plot). Qualitatively, the error maps at the top-right part indicate that our UM2N reduce the vorticity difference to the high resolution mesh result in general. The quantitative comparison results along the probes are shown in the lower part of the figure. There are four cross-section probes place on $x=[0.5, 1.0, 1.5, 2.0]$ marked by the black dash line in the middle left plot. The green line (our UM2N) is much more consistent to the blue dash line (high resolution) compared to the orange line (original mesh), which further verifies that our method can reduce the PDE errors.
• Figure 5: Qualitative results for the Tōhoku tsunami simulation. The boundary of the simulation is generated based on the real coastline data (see the mesh overlaid on the satellite map). We show 9 snapshots of the wave elevation from 80 steps of the tsunami simulation enhanced by our UM2N. In these visualizations, red and blue hues indicate wave elevations that are, respectively, above or below mean sea level. Regions exhibiting significant elevation magnitudes necessitate increased resolution to accurately resolve the underlying dynamics. Our method dynamically and robustly adjusts the mesh to increase resolution at the wave front, thereby effectively tracking its propagation.