Physics-informed two-tier neural network for non-linear model order reduction

In recent years, machine learning (ML) has had a great impact in the area of non-intrusive, non-linear model order reduction (MOR). However, the offline training phase often still entails high computational costs since it requires numerous, expensive, full-order solutions as the training data. Furthermore, in state-of-the-art methods, neural networks trained by a small amount of training data cannot be expected to generalize sufficiently well, and the training phase often ignores the underlying physical information when it is applied with MOR. Moreover, state-of-the-art MOR techniques that ensure an efficient online stage, such as hyper reduction techniques, are either intrusive or entail high offline computational costs. To resolve these challenges, inspired by recent developments in physics-informed and physics-reinforced neural networks, we propose a non-intrusive, physics-informed, two-tier deep network (TTDN) method. The proposed network, in which the first tier achieves the regression of the unknown quantity of interest and the second tier rebuilds the physical constitutive law between the unknown quantities of interest and derived quantities, is trained using pretraining and semi-supervised learning strategies. To illustrate the efficiency of the proposed approach, we perform numerical experiments on challenging non-linear and non-affine problems, including multi-scale mechanics problems.