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：多孔材料及结构中的应力预测因直接数值模拟计算成本高昂而极具挑战性。基于卷积神经网络（CNN）的架构近期被提出作为近似和外推此类多尺度模拟结果的替代方法。由于三维体素CNN计算成本过高，这类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，仅通过二维曲面卷积即可高效处理三维问题。延续先前基于像素CNN的研究，我们训练GNN自动为多孔结构中廉价计算的粗尺度应力预测补充局部细尺度应力修正。该方法采用贝叶斯框架，可生成应力场密度分布并提取置信区间。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泛化外推能力：在推理阶段，将概率预测的后验结果约束为满足微尺度局部平衡条件。我们采用集合卡尔曼算法确保贝叶斯条件运算的可追溯性。研究表明，这种创新方法能够有效缓解未修正GNN输出中存在的非期望偏差效应，并普遍提升预测精度。