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

TL;DR

This paper introduces a graph neural network-based method for mesh relocation in finite element methods, significantly improving accuracy and efficiency over classical and prior machine learning approaches.

Contribution

It presents a novel GNN-based approach that directly minimises FEM solution error for mesh adaptation, replacing classical error estimates with a learnable strategy.

Findings

Achieves lower FEM solution error compared to classical methods.

Provides faster mesh adaptation with significant speed-up.

Outperforms prior ML approaches in accuracy and efficiency.

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.