Ariana Mendible
Published: 2021
Total Pages: 87
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Spatiotemporal dynamical systems are ubiquitous across all areas of science and engineering. While some systems of interest can be studied through first-principles governing equations, often, the underlying dynamics are unknown. Furthermore, even for problems where equations are known, analytical solutions are rare. Computational solutions are then plagued by issues of nonlinearity, multiple relevant scales of time and space, chaos, and high-dimensionality. When parametric studies are needed, these simulations become prohibitively expensive, even under the most advance computational architectures. Often, even the most complex dynamical systems contain inherent low-dimensional coherent structure. Reduced-order models (ROMs) attempt to ease the computation burden of such simulations by uncovering this low-rank subspace, reducing the overall size of such models while retaining the critical dynamics. Many ROMs leverage the proper orthogonal decomposition (POD) to produce a linear dimensionality reduction, where a dominant set of correlated modes provide a subspace in which to project the dynamics. Dimensionality reduction techniques then enable downstream tasks such as prediction, estimation, and low-latency control. However, many dimensionality reduction techniques rely on separation of time and space variables. This assumption does not hold in systems with underlying symmetry, because the time and space variables are inherently coupled. Namely translation, or traveling waves, scaling, and rotation inhibit the effectiveness of dimensionality reduction techniques. The focus of this thesis is to develop an approach to make spatiotemporal systems with underlying symmetry amenable to traditional ROM architectures. Three objectives are addressed (1) developing a method to address translations in one spatial dimension, (2) demonstrating the effectiveness of such a method on complex dynamics in experimental data, and (3) expanding the method to approach higher-dimensional data with scaling symmetries. The Unsupervised Traveling Wave Identification with Shifting and Truncation (UnTWIST) method can be applied to pure data, yielding interpretable models for wave speeds. Several examples are explored, showing that even for systems with multiple waves, non-constant wave speeds, and nonlinear wave phenomenon, UnTWIST enables the use of traditional methods like POD for dimensionality reduction. Further, the method is demonstrated on experimental data from a rotating detonation engine, presenting complex nonlinear dynamics. UnTWIST enables the discovery of interpretable linear and nonlinear models for the traveling detonation speeds, and yields meaningful spatial modes to describe the wave shapes. Lastly, the method is augmented to accommodate higher-dimensional data and an arbitrary number of symmetries. Two-dimensional data with a scaling symmetry are investigated, and accurate models are found to improve the dimensionality reductions. Lastly, limitations to the method in the presence of interacting waves are explored, and improvements and expansions are discussed.