Error-Driven Graph Augmentation for Mesh-Based PDE Surrogates

Graph Neural Networks (GNNs) on meshes have emerged as promising surrogates for computational mechanics, but standard local message passing struggles to propagate information across unstructured meshes, leading to large errors in regions with complex physics (e.g., shocks, wakes, boundary layers). Existing approaches enlarge connectivity with long-range edges chosen a priori via geometric heuristics or random sampling, which lack a mechanism to prioritize high-error regions and often introduce redundant communication. We propose MiSe-GNN, a dual-head architecture that adaptively augments graph connectivity using model-predicted a posteriori errors. MiSe-GNN jointly predicts physical fields and a node-wise error indicator; the predicted error is periodically converted into a hierarchy of augmentation edges via an adaptive tree that links high-error nodes to spatial pivots at multiple scales. This error-guided connectivity concentrates message passing where the surrogate is uncertain while keeping the graph sparse elsewhere, yielding a transparent and physically interpretable graph-space analogue of adaptive mesh refinement. Across industrial CFD and CSD benchmarks, MiSe-GNN consistently improves accuracy and accuracy-compute trade-offs over strong baselines, and qualitative analyses show that it routes communication toward physically challenging regions. These results establish error-guided edge augmentation as a robust and general design principle for long-range message passing in physics-aware GNNs.