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

2024-07-05

AI Summary

To address the accuracy-efficiency trade-off in r-adaptive mesh relocation within the finite element method (FEM), this paper proposes an end-to-end, error-driven optimization framework based on graph neural networks (GNNs). We introduce the first GNN-integrated online r-adaptivity pipeline for FEM, wherein a learnable policy directly minimizes the numerical error of partial differential equation (PDE) solutions—replacing conventional paradigms reliant on nonlinear PDE solves and hand-crafted error estimators. A geometry-aware graph representation ensures alignment between mesh and solution spaces, and the framework is implemented and trained end-to-end within the Firedrake environment. Experiments demonstrate that our method achieves several-fold computational speedup over traditional approaches while significantly reducing numerical error, outperforming both classical r-adaptive schemes and state-of-the-art machine learning–based adaptive mesh methods in overall accuracy-efficiency balance.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Search and Optimization: Learning to Search
• Intelligent Robots: Learning & Optimization for ROB

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• User Modeling, Personalization and Recommendation: Federated recommendation systems and personalization

Abstract

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.

Problem

• Optimise mesh relocation
• Minimise FE solution error
• Enhance computational efficiency

Innovation

• Graph neural network optimizes mesh
• Direct minimization of FE error
• Replaces classical estimates with learning