Extended Dynamic Mode Decomposition (EDMD) is a powerful data-driven numerical method based on the theoretical concepts of Koopman operator theory. It can be used to approximate and predict the behavior of a dynamical system using sample data without requiring any knowledge of the underlying dynamics. This is particularly useful for predicting the dynamics of nonlinear systems by summarizing them into a linear opera- tor, which acts on a space of observable functions to create a reduced-order model of the original system. Kernel EDMD is an extension of EDMD that uses a kernel function to efficiently utilize a high-dimensional dictionary of observables. The theory of Kernel EDMD is well established; however, the accuracy of its practical implementation is highly dependent on the choice of functions in the dictionary. This thesis details a method for finding an optimal choice of dictionary functions through the use of kernel parameter learning with a gradient method called kernel EDMD-DL. This method is applied to numerically generated data from two different dynamical systems, a hopf bifurcation and a hydrogen combustion reaction in a constant volume transient CSTR. The data is successfully reconstructed for both systems, and the accuracy is reported along with an overview of the techniques used for optimal hyperparameter tuning and data cleanup, which are critical components to the success of the method. A strong emphasis is made on using the kernel EDMDL-DL method for making out-of-sample predictions. The knowledge gained through this work can be used to enhance an understanding of the capabilities of kernel EDMDL-ML and how it can be applied to complex nonlinear systems existing in engineering and science applications.     «

Extended Dynamic Mode Decomposition (EDMD) is a powerful data-driven numerical method based on the theoretical concepts of Koopman operator theory. It can be used to approximate and predict the behavior of a dynamical system using sample data without requiring any knowledge of the underlying dynamics. This is particularly useful for predicting the dynamics of nonlinear systems by summarizing them into a linear opera- tor, which acts on a space of observable functions to create a reduced-order mo...     »