Dynamic mode decomposition for data-driven modeling of spatio-temporal dynamics - Araújo, Lucas Simon [UNESP] (2024)
Nonlinear phenomena are almost ubiquitous in many different engineering applications. Their underlying nonlinearities bring extra complexity to characterize, predict, and control systems. There are two standard ways to deal with this challenge: through linearization, which restricts the response; and through a nonlinear perspective, which in turn increases the analysis complexity. Still in 1931, Koopman had already proposed a third way, expanding the set of states to encompass the nonlinearities inside a Hilbert space. Towards approximate the Koopman operator, we have explored an up-and-coming technique called dynamic mode decomposition (DMD), which has been bringing attention in the last decades, especially in the fluid dynamics community. The theoretical foundations of both Koopman and DMD theories are explained, including the development of the basic algorithm. In order to challenge the performance of DMD reconstruction, four optimization benchmark functions are used to generate complex dynamic field data as input for the algorithm. Their reconstructions provide insights into the limitations of the method. In addition, a physical phenomenon is addressed in this work, the heat diffusion into a homogeneous, isotropic, and flat plate. The dynamical field is well recovered through analytically generated data by DMD with good accuracy even in the presence of noise and coarse spatial sampling. To complete the analysis, an experimental setup of an aluminum plate under a heat source is mounted and generate data for the DMD input, also getting good reconstruction results.Beyond the reconstructions, the algorithm also provided the modes, frequencies, and an approximation for the Koopman operator, useful results for identification and control. For all the cases studied, the reconstructions are compared with the original data using similarity analysis for a range of truncation levels, promoting a discussion of how this parameter influences the accuracy of the reconstruction. The results confirm the ability of DMD to extract spatio-temporal patterns and recover these original fields with high similarity even using a reduced dimensionality, which corroborates the literature by highlighting DMD as a promising tool for studying dynamic systems, especially those whose governing equations are not sufficiently known.