Physics-informed coarsening for multigrid graph neural networks surrogates

Learning-based surrogates for partial differential equations have recently matched the accuracy of classical solvers while achieving orders-of-magnitude speedups, predominantly in fluid settings and structured geometries. In contrast, robust surrogates for deformable solids remain underexplored, despite the presence of nonlinear elasticity, plasticity, and transient behavior that challenge standard architectures. We introduce a multigrid graph neural network for solid mechanics that couples an encoder-processor-decoder backbone with a physics-informed coarsening strategy. Instead of downsampling via geometric heuristics, our method scores nodes using a residual-based measure of local physical activity and preferentially retains regions of high strain or stress concentration, allocating multiscale capacity where it is most needed. This preserves long-range interactions through hierarchical message passing while improving stability over long rollouts. We evaluate on multiple datasets covering linear, nonlinear, and transient regimes, and observe consistent gains in accuracy and rollout stability compared to standard sampling baselines. Our results highlight the importance of physics-informed coarsening for scalable surrogate modeling in solid mechanics.

Modern engineering design often relies on computer simulations to predict how solid objects bend, deform, or change shape under forces. These simulations are accurate but can be very slow, especially for complex materials and industrial processes. In this work, we develop a machine learning model that learns to approximate these simulations more efficiently on 3D meshes. The key idea is to simplify the mesh in a smart way: instead of keeping points only based on geometry, the model gives more attention to regions where the physical behavior is most difficult, such as areas with large deformation, contact, or stress concentration. This allows the model to focus its computational effort where it matters most. We test the approach on several solid mechanics problems, including elastic deformation and industrial metal forming, and show that it improves prediction accuracy compared with existing learning-based simulation methods.