Operator Inference Aware Quadratic Manifolds with Isotropic Reduced Coordinates for Nonintrusive Model Reduction

Learning reduced models from data in a nonintrusive fashion is an important problem in science and engineering [1, 2, 3]. A typical approach is to first learn an encoder-decoder pair, embed the training snapshot trajectories with the learned encoder, and then fit a reduced dynamical-system model to the embedded trajectories. However, the decomposition of the training process into first learning an encoder-decoder pair for the embedding and only sub-sequentially learning a model of the dynamics typically means that the encoder-decoder pair are trained with the objective of accurately approximating the training data, rather than taking the reduced-model prediction error into account. Thus, the encoder-decoder pair can overfit to achieving a low reconstruction error on the training data by learning embeddings of the snapshot trajectories that are non-smooth, which means that learning a reduced model can become challenging. Correspondingly, it has been observed that learning embeddings and models together can be beneficial; see, e.g., [4, 5, 6, 7]. In the context of intrusive model reduction with linear approximations, there is work that optimizes the reduced basis with respect to the model prediction error [8], quantities of interest [9], and to achieve stability [10]; however, we focus here on the setting of nonintrusive model reduction and nonlinear approximations.