Mesh-based Graph Neural Networks (GNNs) have recently shown capabilities to simulate complex multiphysics problems with accelerated performance times. However, mesh-based GNNs require a large number of message-passing (MP) steps and suffer from over-smoothing for problems involving very fine mesh. In this work, we develop a multiscale mesh-based GNN framework mimicking a conventional iterative multigrid solver, coupled with adaptive mesh refinement (AMR), to mitigate challenges with conventional mesh-based GNNs. We use the framework to accelerate phase field (PF) fracture problems involving coupled partial differential equations with a near-singular operator due to near-zero modulus inside the crack. We define the initial graph representation using all mesh resolution levels. We perform a series of downsampling steps using Transformer MP GNNs to reach the coarsest graph followed by upsampling steps to reach the original graph. We use skip connectors from the generated embedding during coarsening to prevent over-smoothing. We use Transfer Learning (TL) to significantly reduce the size of training datasets needed to simulate different crack configurations and loading conditions. The trained framework showed accelerated simulation times, while maintaining high accuracy for all cases compared to physics-based PF fracture model. Finally, this work provides a new approach to accelerate a variety of mesh-based engineering multiphysics problems

翻译：基于网格的图神经网络（GNN）近期展现出模拟复杂多物理场问题并加速计算性能的能力。然而，针对涉及极细网格的问题，基于网格的GNN需要大量消息传递步骤，并面临过平滑问题。本研究开发了一种模拟传统迭代多重网格求解器的多尺度网格GNN框架，结合自适应网格细化，以缓解传统网格GNN面临的挑战。我们利用该框架加速涉及近奇异算子（由裂纹内部接近零的模量导致）的耦合偏微分方程相位场断裂问题。初始图表示采用所有网格分辨率层级定义，通过Transformer消息传递GNN执行一系列下采样步骤直至最粗图，再通过上采样步骤恢复原始图。在下采样过程中，通过跳跃连接保留生成的嵌入表示以防止过平滑。采用迁移学习显著缩减模拟不同裂纹构型和加载条件所需训练数据集规模。相较于基于物理的相位场断裂模型，训练后的框架在所有案例中均保持高精度，同时实现仿真时间加速。本研究为加速各类基于网格的工程多物理场问题提供了新途径。