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-Amp\`ere 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/.

翻译：精确高效地求解复杂偏微分方程（PDE）是所有科学与工程领域至关重要且具有挑战性的问题。网格移动方法能够在无需增加整体网格自由度数量的情况下，提高数值解的精度。传统的复杂网格移动方法计算成本极高，且难以处理具有复杂边界几何形状的场景。然而，现有的基于学习的方法在面对不同的PDE类型或边界几何形状时，需要从头开始重新训练，这限制了其适用性，并且常常因出现单元反转问题而遭受鲁棒性困扰。本文提出了通用网格移动网络（UM2N），该网络一旦训练完成，便能够以非侵入式、零样本的方式应用于移动具有不同尺寸分布和结构的网格，适用于求解不同类型PDE和不同边界几何形状的求解器。UM2N由一个用于提取特征的图Transformer（GT）编码器和一个基于图注意力网络（GAT）的解码器组成，用于移动网格。我们在平流方程和Navier-Stokes方程相关的算例以及一个真实世界的海啸模拟案例上评估了我们的方法。我们的方法在上述描述的基准测试中优于现有的基于学习的网格移动方法。与传统的基于复杂Monge-Ampère PDE求解器的方法相比，我们的方法不仅显著加速了网格移动，而且在传统方法失效的场景中也证明了其有效性。我们的项目页面位于 https://erizmr.github.io/UM2N/。