arXiv 2407.045162024-07cs.LGcs.NAmath.NA

G-Adaptivity：面向有限元方法的优化图基网格重定位

G-Adaptivity: optimised graph-based mesh relocation for finite element methods

Department of Applied Mathematics；Theoretical Physics, University of Cambridge, UK；Department of Mathematical Sciences, University of Bath, UK；Department of Computer Science, University of Bath, UK；School of Computing；Mathematical Sciences, University of Leicester, UK；Laboratoire Jacques-Louis Lions, Sorbonne Universit\'e, France；Department of Computer Science；Technology, University of Cambridge, UK

James Rowbottom, Georg Maierhofer, Teo Deveney, Eike Mueller, Alberto Paganini, Katharina Schratz, Pietro Liò, Carola-Bibiane Schönlieb, Chris Budd

发表信息：: Proceedings of the 42nd International Conference on Machine Learning, 2025

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AI总结：

提出G-Adaptivity，一种基于图神经网络直接最小化有限元求解误差的网格重定位方法，替代经典非线性PDE求解，实现更优精度与显著加速。

摘要：

我们提出了一种新颖且有效的方法，用于在有限元方法（FEMs）中实现最优网格重定位。FEM的成本和精度关键取决于网格点的选择。网格重定位（r-自适应）旨在优化网格几何形状，以在给定的计算预算下获得最佳的求解精度。经典的r-自适应方法依赖于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。这在重新网格化时会产生显著成本，并且依赖于将插值误差与FEM误差相关联的估计。最近的机器学习方法侧重于为这类经典方法构建快速代理模型。相反，我们的新方法训练一个图神经网络（GNN），通过直接最小化来自PDE系统Firedrake的有限元求解误差来确定网格点位置，以实现更高的求解精度。我们的GNN架构将网格解空间与经典网格生成方法紧密对齐，从而用可学习的策略取代了经典的最优性估计。这使得训练快速且稳健，并产生了一种极其高效且有效的用于在线r-自适应的GNN方法。我们的方法在r-自适应网格生成方面优于经典方法和先前的机器学习方法。特别是，它在获得更低的有限元求解误差的同时，保留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。

英文摘要：

We present a novel, and effective, approach to achieve optimal mesh relocation in finite element methods (FEMs). The cost and accuracy of FEMs is critically dependent on the choice of mesh points. Mesh relocation (r-adaptivity) seeks to optimise the mesh geometry to obtain the best solution accuracy at given computational budget. Classical r-adaptivity relies on the solution of a separate nonlinear "meshing" PDE to determine mesh point locations. This incurs significant cost at remeshing, and relies on estimates that relate interpolation- and FEM-error. Recent machine learning approaches have focused on the construction of fast surrogates for such classical methods. Instead, our new approach trains a graph neural network (GNN) to determine mesh point locations by directly minimising the FE solution error from the PDE system Firedrake to achieve higher solution accuracy. Our GNN architecture closely aligns the mesh solution space to that of classical meshing methodologies, thus replacing classical estimates for optimality with a learnable strategy. This allows for rapid and robust training and results in an extremely efficient and effective GNN approach to online r-adaptivity. Our method outperforms both classical, and prior ML, approaches to r-adaptive meshing. In particular, it achieves lower FE solution error, whilst retaining the significant speed-up over classical methods observed in prior ML work.

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