A Graph-based probabilistic geometric deep learning framework with online physics-based corrections to predict the criticality of defects in porous materials

Abstract

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.

Convolution, aggregation and attention based deep neural networks for accelerating simulations in mechanics

“…Graph-based approaches leverage the topological information of the input to perform local operations in the respective neighborhood only, and can learn efficiently on generally structured data. Recently, GDL methods have shown promising performance for their applications as well in the field of mechanics, (Battaglia et al, 2018;Vlassis et al, 2020;Pfaff et al, 2021;Krokos et al, 2022a;Strönisch et al, 2022). More recently (Deshpande et al, 2022b), proposed MAgNET, a novel graph U-Net framework for efficiently learning on mesh-based data.…”