Concurrent multiscale simulations of nonlinear random materials using probabilistic learning

This work is concerned with the construction of statistical surrogates for concurrent multiscale modeling in structures comprising nonlinear random materials. The development of surrogates approximating a homogenization operator is a fairly classical topic that has been addressed through various methods, including polynomial- and deep-learning-based models. Such approaches, and their extensions to probabilistic settings, remain expensive and hard to deploy when the nonlinear upscaled quantities of interest exhibit large statistical variations (in the case of non-separated scales, for instance) and potential non-locality. The aim of this paper is to present a methodology that addresses this particular setting from the point of view of probabilistic learning. More specifically, we formulate the approximation problem using conditional statistics, and use probabilistic learning on manifolds to draw samples of the nonlinear constitutive model at mesoscale. Two applications, relevant to inverse problem solving and forward propagation, are presented in the context of nonlinear elasticity. We show that the framework enables accurate predictions (in probability law), despite the small amount of training data and the very high levels of nonlinearity and stochasticity in the considered system.