GDEA: Global-Weight Deep Equilibrium Attention for Finite Element Systems

Graph neural networks have been explored as surrogate models for the finite element method due to the computational cost. However, many models fail to effectively capture the influence of distant nodes. This is critical in surrogate models because boundary conditions at a distance can significantly influence the results. To mitigate these issues, we propose iterative message passing using a global-weight matrix assembled via the direct stiffness method. This method is implemented through a deep equilibrium model with Anderson acceleration to ensure fast convergence and low memory cost. Our model is evaluated on published FEM datasets as well as a new dataset, Deformed ABC, featuring diverse geometries, materials, and boundary conditions. Our model outperforms others across all criteria, including a 12% lower average RMSE and a 5% higher average R ² on a complex new dataset, but differences are marginal due to saturation of the benchmark across all models in simpler datasets.