Physics-based deep learning frameworks have shown to be effective in accurately modeling the dynamics of complex physical systems with generalization capability across problem inputs. However, time-independent problems pose the challenge of requiring long-range exchange of information across the computational domain for obtaining accurate predictions. In the context of graph neural networks (GNNs), this calls for deeper networks, which, in turn, may compromise or slow down the training process. In this work, we present two GNN architectures to overcome this challenge - the Edge Augmented GNN and the Multi-GNN. We show that both these networks perform significantly better (by a factor of 1.5 to 2) than baseline methods when applied to time-independent solid mechanics problems. Furthermore, the proposed architectures generalize well to unseen domains, boundary conditions, and materials. Here, the treatment of variable domains is facilitated by a novel coordinate transformation that enables rotation and translation invariance. By broadening the range of problems that neural operators based on graph neural networks can tackle, this paper provides the groundwork for their application to complex scientific and industrial settings.

翻译：基于物理的深度学习框架已在精确建模复杂物理系统动力学方面展现出有效性，且具备跨问题输入的泛化能力。然而，时间无关问题对跨计算域的长程信息交换提出了挑战，需要网络具备获取精确预测的能力。在图神经网络（GNN）背景下，这要求构建更深层次的网络结构，而这可能削弱或减缓训练过程。本文提出两种克服该挑战的GNN架构——边增强图神经网络（Edge Augmented GNN）与多图神经网络（Multi-GNN）。研究表明，当应用于时间无关固体力学问题时，这两种网络性能显著优于基线方法（提升幅度达1.5至2倍）。此外，所提出的架构对未见域、边界条件及材料展现出良好的泛化能力。其中，一种新型坐标变换处理可变域，实现了旋转和平移不变性。通过拓展基于图神经网络的神经算子可解决的问题范围，本文为其在复杂科学与工业场景中的应用奠定了基础。