Stress prediction in porous materials and structures is challenging due to the high computational cost associated with direct numerical simulations. Convolutional Neural Network (CNN) based architectures have recently been proposed as surrogates to approximate and extrapolate the solution of such multiscale simulations. These methodologies are usually limited to 2D problems due to the high computational cost of 3D voxel based CNNs. We propose a novel geometric learning approach based on a Graph Neural Network (GNN) that efficiently deals with three-dimensional problems by performing convolutions over 2D surfaces only. Following our previous developments using pixel-based CNN, we train the GNN to automatically add local fine-scale stress corrections to an inexpensively computed coarse stress prediction in the porous structure of interest. Our method is Bayesian and generates densities of stress fields, from which credible intervals may be extracted. As a second scientific contribution, we propose to improve the extrapolation ability of our network by deploying a strategy of online physics-based corrections. Specifically, we condition the posterior predictions of our probabilistic predictions to satisfy partial equilibrium at the microscale, at the inference stage. This is done using an Ensemble Kalman algorithm, to ensure tractability of the Bayesian conditioning operation. We show that this innovative methodology allows us to alleviate the effect of undesirable biases observed in the outputs of the uncorrected GNN, and improves the accuracy of the predictions in general.

翻译：多孔材料与结构中的应力预测极具挑战性，这源于直接数值模拟所需的高昂计算成本。基于卷积神经网络（CNN）的架构近期被提出作为替代模型，用于近似和推断这类多尺度仿真的解。由于三维体素型CNN的高计算开销，这些方法通常局限于二维问题。我们提出一种基于图神经网络（GNN）的新型几何学习方法，该方法通过仅在二维表面上执行卷积操作，高效处理三维问题。借鉴我们此前基于像素型CNN的研究，我们训练GNN为感兴趣的多孔结构中廉价计算的粗略应力预测自动添加局部精细尺度应力修正。该方法具有贝叶斯特性，可生成应力场密度分布，并从中提取可信区间。作为第二项科学贡献，我们提出通过部署在线物理修正策略来提升网络的推断能力。具体而言，我们在推理阶段对概率预测的后验预测施加条件，使其满足微观尺度的局部平衡。该过程采用集合卡尔曼算法实现，以确保贝叶斯条件操作的可处理性。研究表明，这种创新方法能够减轻未修正GNN输出中观测到的不良偏差影响，并普遍提升预测精度。