Non-intrusive reduced order modeling of dynamical systems with large stochastic dimensions

Abstract

The work done in this thesis falls under the broad category of computational mechanics. This thesis addresses the computational challenge of high-dimensional regression and interpolation within the framework of a proper orthogonal decomposition (POD)-based non-intrusive reduced order model (ROM) of stochastic dynamical systems. The novelty of this work lies in developing new non-intrusive ROMs to handle large stochastic dimensions originating from random excitations. These excitations upon discretization lead to a large number of random variables — hundreds or thousands, handling of which is computationally expensive. While existing non-intrusive ROMs work well for low stochastic dimensions, they are practically infeasible for large stochastic systems due to the requirement of substantial computational resources. To the best of the author’s knowledge, no non-intrusive ROMs exist that address the large stochastic system.

To address this issue, novel non-intrusive ROMs are proposed in this thesis that directly work on random excitations, altogether avoiding the need for discretization. The proposed ROMs address the issue of high dimensionality by adopting artificial neural network-based regression models in the reduced space. Although the proposed ROMs are general, their accuracy and efficiency are elucidated through uncertainty quantification — specifically reliability estimation — of various dynamical systems throughout the thesis. Two novel ROMs are proposed: one for linear and another for nonlinear systems. For linear systems, the thesis proposes a ROM that utilizes feed-forward neural networks (FFNNs) for regression between compressed excitations --- achieved through principal component analysis --- and reduced-order solutions. This ROM demonstrates high accuracy and a substantial speed-up of sixty in estimating the failure probability of a linear soil-structure interaction (SSI) problem. However, an FFNN cannot sequentially process the data, which is crucial in analyzing nonlinear dynamical systems, especially those that show hysteretic behavior. To address this issue, a long short-term memory (LSTM) network is adopted. Accordingly, an LSTM-integrated non-intrusive ROM is proposed for nonlinear systems. The numerical studies show that the proposed LSTM-integrated ROM is accurate and efficient, gaining a speed-up of two-order magnitude for stationary and non-stationary random excitations.

In addition to the above novelties, the thesis proposes a novel approach for computing more efficient POD bases. The proposed ROMs mentioned above focus on forming efficient regression models within the framework of non-intrusive ROMs. These ROMs are based on POD bases, computed using the snapshots of the displacement of the original system. However, several researchers have shown that such POD bases may yield inaccurate results. Consequently, energy-preserving POD bases are proposed, which significantly improve the efficiency of the resulting ROM compared to traditional ROMs.