Integrating Data-Driven Methods in Nonlinear Dynamical Systems: Control, Sparsity and Machine Learning
The goal of my thesis is to provide a theoretical demonstration of how dimension reduc- tion, control and machine learning techniques can be applied to optimize the performance of complex nonlinear systems. Specifically, integrating those methods to build mode-locked fiber lasers that are more robust and with high performances. We show that an adaptive genetic algorithm is successful in increasing pulse energies in a multi-nonlinear polarization rotation(NPR) fiber laser system and in achieving locally optimal performance. In order to maintain the local optimal performance under birefringence perturbations, we designed an extremum seeking controller(ESC). By numerical simulations of a single-NPR fiber laser, it is showed that the ESC tracks local optimal mode-locking states despite significant distur- bances to parameters. We also developed a toroidal search and a machine learning algorithm that enables us to obtain a global optimal performance when birefringence of laser cavity varies. In addition, we also demonstrated an adaptive time-stepping method for dimension reduction computation which can be used to accelerate numerical simulations for partial differential equations. Overall, these methods are all data-driven and do not rely upon un- derlying models and hence can be generalized to use in other non-optical complex systems as well in the future.
- Applied mathematics