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

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.

When engineers and scientists simulate physical systems - like airflow over an aircraft wing or stress on a bridge - they often use a technique called the Finite Element Method (FEM). This method breaks down complex structures into smaller, manageable pieces called "meshes." The accuracy of these simulations heavily depends on how these meshes are arranged. Traditionally, mesh adaptation requires solving extra nonlinear equations - a time-consuming and costly step.We introduce a novel approach that teaches Graph Neural Networks (GNN) to adjust these meshes for fast and optimal mesh placement. By training the GNN to minimize errors in the simulation results directly, the method generates meshes with more accurate finite element solutions and at significantly faster speeds than traditional techniques.Our key insight is that our end-to-end optimisation beats both classical approaches and earlier machine learning surrogates to this adaptive meshing problem. These findings provide insights on how we can effectively benefit from the power of AI in scientific computing without sacrificing the robustness of existing FEM solvers that are currently widely used in scientific computing in areas from engineering and climate modelling, to medical imaging.