GraphMesh: Geometrically Generalized Mesh Refinement Using GNNs

Optimal mesh refinement is important for finite element simulations, facilitating the generation of non-uniform meshes. While existing neural network-based approaches have successfully generated high quality meshes, they can only handle a fixed number of vertices seen during training. We introduce GraphMesh, a novel mesh refinement method designed for geometric generalization across meshes with varying vertex counts. Our method employs a two-step process, initially learning a unified embedding for each node within an input coarse mesh, and subsequently propagating this embedding based on mesh connectivity to predict error distributions. By learning a node-wise embedding, our method achieves superior simulation accuracy with reduced computational costs compared to current state-of-the-art methods. Through experimentation and comparisons, we showcase the effectiveness of our approach across various scenarios, including geometries with different vertex counts. We validated our approach by predicting the local error estimates for the solution of Poisson’s equation.