Executive Summary : | Physics-based modeling and simulation has become ubiquitous in many engineering applications, ranging from critical system monitoring to aircraft design. Thrust areas like control, optimization, data assimilation, and uncertainty quantification demand fine spatiotemporal resolutions on the computational models under study, leading to an overwhelming burden on the computational resources. This computational bottleneck precludes the integration of such systems in many-query and real-time applications such as in model predictive control or uncertainty propagation. Work during my Ph.D. revealed how system-theoretic methods can be used to construct low-dimensional approximations of large-scale and complex nonlinear systems pervasive in various engineering domains. Notably, we demonstrated that a connection between the linear and nonlinear enhancements of moment-matching allows extracting reduced models in a snapshot-free architecture that drastically reduces the complexity bottleneck in nonlinear MOR. However, an underlying limitation of these methods is the assumption of a linear subspace approximation of the original high-dimensional nonlinear state-space in which the solution evolves. This imposes a fundamental limit on the accuracy of the reduced-order models thus obtained. This project aims to approach this problem by exploiting machine learning techniques to generate low-dimensional surrogate models in a non-intrusive manner. After collecting large-scale data either from numerical simulations of benchmark nonlinear systems or from data repositories, we intend to use custom deep-autoencoder networks for discovering an efficient coordinate system followed by reduced-space time evolution using recurrent neural networks to overcome the linear subspace limitation. The goal will be to address the fundamental n-width deficiency of linear trial subspace by pursuing an approach that is both general and only requires the snapshot data as in typical POD-based methods. Another objective will be to discover global coordinates on which the dynamics are globally linear by using a modified autoencoder in an operator-theoretic framework. To achieve this goal, we intend to use the Koopman operator to obtain the best-fit linear dynamical system governing the actual dynamics. This linear representation of nonlinear dynamics will have a tremendous potential to enable analysis, prediction, and control via textbook methods developed for linear methods. Finally, we address the challenge of obtaining parsimonious models of large-scale systems by combining autoencoder networks with sparse-regression techniques to obtain interpretable models. To accomplish this objective, we will use the SINDy method to determine the fewest active terms in the dynamic governing equations required to represent the data accurately. The project is novel in the sense that it is discovery-based and fundamental to the overarching goal of understanding complex nonlinear dynamical systems. |