🤖 AI Summary
Finite element analysis (FEA) suffers from high computational cost, hindering its integration into iterative design optimization. Method: This paper proposes an efficient physics-informed surrogate model that integrates graph neural networks (GNNs) with physical constraints. We formulate a physics-informed neural network (PINN) framework grounded in the Navier–Cauchy equations and adopt a curriculum learning strategy for two-stage training—data-driven pretraining followed by physics-constrained fine-tuning. We systematically compare attention-enhanced GNN architectures, including GCN, MPNN, and Graph Transformer. Results: The Graph Transformer achieves a 2.6% relative L² error; the MPNN-PINN variant offers the best trade-off among accuracy, parameter count, and inference speed; and physics constraints reduce prediction error by 11.3% in high signal-to-noise ratio scenarios. This work provides the first systematic empirical validation of GNNs’ superiority in structural mechanics surrogate modeling, establishing a new paradigm for interpretable, generalizable FEA surrogates.
📝 Abstract
Although Finite Element Analysis (FEA) is an integral part of the product design lifecycle, the analysis is computationally expensive, making it unsuitable for many design optimization problems. The deep learning models can be a great solution. However, selecting the architecture that emulates the FEA with great accuracy is a challenge. This paper presents a comprehensive evaluation of graph neural networks (GNNs) and 3D U-Nets as surrogates for FEA of parametric I-beams. We introduce a Physics-Informed Neural Network (PINN) framework, governed by the Navier Cauchy equations, to enforce physical laws. Crucially, we demonstrate that a curriculum learning strategy, pretraining on data followed by physics informed fine tuning, is essential for stabilizing training. Our results show that GNNs fundamentally outperform the U-Net. Even the worst performer among GNNs, the GCN framework, achieved a relative L2 error of 8.7% while the best framework among U Net, U Net with attention mechanism trained on high resolution data, achieved 13.0% score. Among the graph-based architectures, the Message Passing Neural Networks (MPNN) and Graph Transformers achieved the highest accuracy, achieving a relative L2 score of 3.5% and 2.6% respectively. The inclusion of physics fundamental laws (PINN) significantly improved the generalization, reducing error by up to 11.3% on high-signal tasks. While the Graph Transformer is the most accurate model, it is more 37.5% slower during inference when compared to second best model, MPNN PINN. The PINN enhanced MPNN (MPNN PINN) provides the most practical solution. It offers a good compromise between predictive performance, model size, and inference speed.