We present a novel, and effective, approach to the long-standing problem of mesh adaptivity in finite element methods (FEM). FE solvers are powerful tools for solving partial differential equations (PDEs), but their cost and accuracy are critically dependent on the choice of mesh points. To keep computational costs low, mesh relocation (r-adaptivity) seeks to optimise the position of a fixed number of mesh points to obtain the best FE solution accuracy. Classical approaches to this problem require the solution of a separate nonlinear "meshing" PDE to find the mesh point locations. This incurs significant cost at remeshing and relies on certain a-priori assumptions and guiding heuristics for optimal mesh point location. Recent machine learning approaches to r-adaptivity have mainly focused on the construction of fast surrogates for such classical methods. Our new approach combines a graph neural network (GNN) powered architecture, with training based on direct minimisation of the FE solution error with respect to the mesh point locations. The GNN employs graph neural diffusion (GRAND), closely aligning the mesh solution space to that of classical meshing methodologies, thus replacing heuristics with a learnable strategy, and providing a strong inductive bias. This allows for rapid and robust training and results in an extremely efficient and effective GNN approach to online r-adaptivity. This method outperforms classical and prior ML approaches to r-adaptive meshing on the test problems we consider, in particular achieving lower FE solution error, whilst retaining the significant speed-up over classical methods observed in prior ML work.

翻译：本文针对有限元方法（FEM）中由来已久的网格自适应问题，提出了一种新颖且有效的解决方案。有限元求解器是求解偏微分方程（PDEs）的强大工具，但其计算成本与精度关键取决于网格点的选取。为降低计算成本，网格重定位（r-自适应）旨在优化固定数量网格点的位置，以获得最佳的有限元求解精度。该问题的经典方法需要求解一个独立的非线性“网格生成”偏微分方程来确定网格点位置，这会在重新网格化时产生显著计算开销，并且依赖于关于最优网格点位置的某些先验假设与启发式准则。近期针对r-自适应的机器学习方法主要聚焦于为这类经典方法构建快速代理模型。我们提出的新方法结合了基于图神经网络（GNN）的架构，其训练直接以最小化有限元解关于网格点位置的误差为目标。该GNN采用图神经扩散（GRAND）机制，使网格解空间与经典网格生成方法的空间紧密对齐，从而用可学习策略替代启发式规则，并提供强归纳偏置。这使得训练过程快速稳健，最终形成一种高效且有效的在线r-自适应GNN方法。在我们考察的测试问题上，该方法在r-自适应网格生成方面超越了经典方法与先前的机器学习方法，尤其实现了更低的有限元求解误差，同时保持了先前机器学习工作中观察到的相对于经典方法的显著加速优势。