用图神经网络和物理约束模型替代有限元分析，加速结构设计优化。
A Comprehensive Evaluation of Graph Neural Networks and Physics Informed Learning for Surrogate Modelling of Finite Element Analysis
- 采用图神经网络与物理信息神经网络结合，学习参数化工字梁的力学行为。
- 图变压器模型误差低至2.6%，比最优U-Net降低4.4个百分点，且物理约束提升泛化能力。
- 物理约束+课程学习策略稳定训练，MPNN-PINN是兼顾精度与推理速度的最佳选择。
尽管有限元分析（FEA）是产品设计流程的核心环节，但其计算成本高昂，难以应用于多数设计优化问题。深度学习可作为有效替代方案，但选择能高精度模拟FEA的模型架构仍具挑战。本文系统评估了图神经网络（GNNs）与3D U-Net作为参数化工字梁FEA代理模型的表现。提出基于Navier-Cauchy方程的物理信息神经网络（PINN）框架，以强制施加物理定律。关键发现：课程学习策略（先数据预训练，再物理微调）对训练稳定性至关重要。结果表明，GNN整体优于U-Net：最差的GCN相对L2误差为8.7%，而最优的带注意力机制的高分辨率U-Net为13.0%。在图模型中，消息传递神经网络（MPNN）与图变压器分别达到3.5%与2.6%的相对L2误差。引入物理规律后，高信号任务误差最高降低11.3%。虽图变压器精度最高，但推理速度慢37.5%于次优模型MPNN-PINN。综合性能、模型规模与推理速度，增强型MPNN-PINN为最实用方案。
原文摘要 · Abstract (English)
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.
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