We investigate the effect of spatial filtering on the recently proposed data-driven correction reduced order model (DDC-ROM). We compare two filters: the ROM projection, which was originally used to develop the DDC-ROM, and the ROM differential filter, which uses a Helmholtz operator to attenuate the small scales in the input signal. We focus on the following questions: ``Do filtering and differentiation with respect to space variable commute, when filtering is applied to the diffusion term?'' or in other words ``Do we have commutation error (CE) in the diffusion term?" and ``If so, is the commutation error data-driven correction ROM (CE-DDC-ROM) more accurate than the original DDC-ROM?'' If the CE exists, the DDC-ROM has two different correction terms: one comes from the diffusion term and the other from the nonlinear convection term. We investigate the DDC-ROM and the CE-DDC-ROM equipped with the two ROM spatial filters in the numerical simulation of the Burgers equation with different diffusion coefficients and two different initial conditions (smooth and non-smooth). / M.S. / We propose reduced order models (ROMs) for an efficient and relatively accurate numerical simulation of nonlinear systems. We use the ROM projection and the ROM differential filters to construct a novel data-driven correction ROM (DDC-ROM). We show that the ROM spatial filtering and differentiation do not commute for the diffusion operator. Furthermore, we show that the resulting commutation error has an important effect on the ROM, especially for low viscosity values. As a mathematical model for our numerical study, we use the one-dimensional Burgers equations with smooth and non-smooth initial conditions.

Accelerating the simulation of turbulence to stationarity is a critical challenge in various engineering applications. This study presents an innovative equation-free multiscale approach combined with a machine learning technique to address this challenge in the context of the one-dimensional stochastic Burgers' equation, a widely used toy model for turbulence. We employ an encoder-decoder recurrent neural network to perform super-resolution reconstruction of the velocity field from lower-dimensional energy spectrum data, enabling seamless transitions between fine and coarse levels of description. The proposed multiscale-machine learning framework significantly accelerates the computation of the statistically stationary turbulent Burgers' velocity field, achieving up to 442 times faster wall clock time compared to direct numerical simulation, while maintaining three-digit accuracy in the velocity field. Our findings demonstrate the potential of integrating equation-free multiscale methods with machine learning methods to efficiently simulate stochastic partial differential equations and highlight the possibility of using this approach to simulate stochastic systems in other engineering domains. / Master of Science / In many practical engineering problems, simulating turbulence can be computationally expensive and time-consuming. This research explores an innovative method to accelerate these simulations using a combination of equation-free multiscale techniques and deep learning. Multiscale methods allow researchers to simulate the behavior of a system at a coarser scale, even when the specific equations describing its evolution are only available for a finer scale. This can be particularly helpful when there is a notable difference in the time scales between the coarser and finer scales of a system. The ``equation-free approach multiscale method coarse projective integration" can then be used to speed up the simulations of the system's evolution. Turbulence is an ideal candidate for this approach since it can be argued that it evolves to a statistically steady state on two different time scales. Over the course of evolution, the shape of the energy spectrum (the coarse scale) changes slowly, while the velocity field (the fine scale) fluctuates rapidly. However, applying this multiscale framework to turbulence simulations has been challenging due to the lack of a method for reconstructing the velocity field from the lower-dimensional energy spectrum data. This is necessary for moving between the two levels of description in the multiscale simulation framework. In this study, we tackled this challenge by employing a deep neural network model called an encoder-decoder sequence-to-sequence architecture. The model was used to capture and learn the conversions between the structure of the velocity field and the energy spectrum for the one-dimensional stochastic Burgers' equation, a simplified model of turbulence. By combining multiscale techniques with deep learning, we were able to achieve a much faster and more efficient simulation of the turbulent Burgers' velocity field. The findings of this study demonstrated that this novel approach could recover the final steady-state turbulent Burgers' velocity field up to 442 times faster than the traditional direct numerical simulations, while maintaining a high level of accuracy. This breakthrough has the potential to significantly improve the efficiency of turbulence simulations in a variety of engineering applications, making it easier to study and understand these complex phenomena.

In this dissertation, we consider two different strategies for improving the projection-based reduced order model (ROM) accuracy: (I) adding closure terms to the standard ROM; (II) using Lagrangian data to improve the ROM basis. Following strategy (I), we propose a new data-driven reduced order model (ROM) framework that centers around the hierarchical structure of the variational multiscale (VMS) methodology and utilizes data to increase the ROM accuracy at a modest computational cost. The VMS methodology is a natural fit for the hierarchical structure of the ROM basis: In the first step, we use the ROM projection to separate the scales into three categories: (i) resolved large scales, (ii) resolved small scales, and (iii) unresolved scales. In the second step, we explicitly identify the VMS-ROM closure terms, i.e., the terms representing the interactions among the three types of scales. In the third step, we use available data to model the VMS-ROM closure terms. Thus, instead of phenomenological models used in VMS for standard numerical discretizations (e.g., eddy viscosity models), we utilize available data to construct new structural VMS-ROM closure models. Specifically, we build ROM operators (vectors, matrices, and tensors) that are closest to the true ROM closure terms evaluated with the available data. We test the new data-driven VMS-ROM in the numerical simulation of four test cases: (i) the 1D Burgers equation with viscosity coefficient $nu = 10^{-3}$; (ii) a 2D flow past a circular cylinder at Reynolds numbers $Re=100$, $Re=500$, and $Re=1000$; (iii) the quasi-geostrophic equations at Reynolds number $Re=450$ and Rossby number $Ro=0.0036$; and (iv) a 2D flow over a backward facing step at Reynolds number $Re=1000$. The numerical results show that the data-driven VMS-ROM is significantly more accurate than standard ROMs. Furthermore, we propose a new hybrid ROM framework for the numerical simulation of fluid flows. This hybrid framework incorporates two closure modeling strategies: (i) A structural closure modeling component that involves the recently proposed data-driven variational multiscale ROM approach, and (ii) A functional closure modeling component that introduces an artificial viscosity term. We also utilize physical constraints for the structural ROM operators in order to add robustness to the hybrid ROM. We perform a numerical investigation of the hybrid ROM for the three-dimensional turbulent channel flow at a Reynolds number $Re = 13,750$. In addition, we focus on the mathematical foundations of ROM closures. First, we extend the verifiability concept from large eddy simulation to the ROM setting. Specifically, we call a ROM closure model verifiable if a small ROM closure model error (i.e., a small difference between the true ROM closure and the modeled ROM closure) implies a small ROM error. Second, we prove that a data-driven ROM closure (i.e., the data-driven variational multiscale ROM) is verifiable. For strategy (II), we propose new Lagrangian inner products that we use together with Eulerian and Lagrangian data to construct new Lagrangian ROMs. We show that the new Lagrangian ROMs are orders of magnitude more accurate than the standard Eulerian ROMs, i.e., ROMs that use standard Eulerian inner product and data to construct the ROM basis. Specifically, for the quasi-geostrophic equations, we show that the new Lagrangian ROMs are more accurate than the standard Eulerian ROMs in approximating not only Lagrangian fields (e.g., the finite time Lyapunov exponent (FTLE)), but also Eulerian fields (e.g., the streamfunction). We emphasize that the new Lagrangian ROMs do not employ any closure modeling to model the effect of discarded modes (which is standard procedure for low-dimensional ROMs of complex nonlinear systems). Thus, the dramatic increase in the new Lagrangian ROMs' accuracy is entirely due to the novel Lagrangian inner products used to build the Lagrangian ROM basis. / Doctor of Philosophy / Reduced order models (ROMs) are popular in physical and engineering applications: for example, ROMs are widely used in aircraft designing as it can greatly reduce computational cost for the aircraft's aeroelastic predictions while retaining good accuracy. However, for high Reynolds number turbulent flows, such as blood flows in arteries, oil transport in pipelines, and ocean currents, the standard ROMs may yield inaccurate results. In this dissertation, to improve ROM's accuracy for turbulent flows, we investigate three different types of ROMs. In this dissertation, both numerical and theoretical results show that the proposed new ROMs yield more accurate results than the standard ROM and thus can be more useful.