Multiscale analysis of nonlinear systems using a hierarchy of deep neural networks

The application of multiscale methods that are based on computational homogenization, such as the well established FE2, remains in most cases a computationally challenging task. In a system that involves more than two length scales, the task of a FEN (N>2) calculation becomes practically intractable if applied in the frame of conventional computational setups. This paper addresses this issue by introducing a novel surrogate model designed to accelerate the solution procedure of hierarchically formulated multiscale problems. The idea is to employ a sequence of deep feedforward neural networks (DNNs) that represent the hierarchy of the separate scales in the multiscale problem. Each DNN is trained to learn the constitutive law of a corresponding length scale of the problem. In a similar manner to the original problem, where each finer scale is contained in a coarser scale, DNNs representing fine scales are contained in the DNNs that represent coarser scales. At the end of the training process, a single deep network is produced which emulates the macroscopic behaviour by incorporating all physical mechanisms arising at each of the problem's finer scales. This approach takes full advantage of the accuracy and modelling capabilities of the FEN schemes, while at the same time handles the immense computational requirements associated with their implementation. By virtue of the elaborated surrogate modelling setup, a four-scale structural system is solved using a FE4 scheme at reasonable computational times. In turn, this allows us to perform laborious sensitivity analyses in order to assess the uncertainty in the material parameters and its propagation to the macroscopic structural response.