Advances in Data-Driven Modeling and Sensing for High-Dimensional Nonlinear Systems

Abstract

Accurate and efficient models of physical processes like fluid flows are crucial for applications ranging from forecasting the weather to controlling autonomous aircraft and suppressing combustion instabilities in liquid fueled rocket engines.These models allow us to predict what the system will do --- oftentimes in response to an input or design characteristics that we would like to choose intelligently --- as well as to detect what the real system is doing from limited and costly sensor measurements. The main challenge is that the equations governing complex systems like fluid flows that we might derive from first principles are routinely nonlinear and involve too many variables to be simulated by a computer or sensed in real time. Therefore, we aim to construct and leverage simplified models of these complex systems that capture the most important aspects of its behavior for the task at hand, while relying on a much smaller number of variables that can be simulated or sensed in real time. While highly effective and well-studied techniques exist when the system is linear, in many important cases the system is operating too far away from an equilibrium state to employ linearization or other linear approximation techniques. In this thesis, we make use of data collected from the underlying complex system or simulations performed ahead of time in order to identify patterns and construct simplified models based on them. In order to build simplified predictive models, we present a variety of techniques based on projecting the governing equations onto manifolds identified from data. Such manifolds must be nonlinear in order to find models involving the smallest number of variables. We also find that in order to build models of systems with selective sensitivity, such as shear-driven fluid flows, it is important to incorporate information from the linearized adjoint of the governing equations. We also describe an alternative viewpoint for modeling based on converting nonlinear dynamics into linear dynamics in a function space via data-driven approximation of Koopman operators. Finally, we present a constellation of data-driven techniques enabling us to find minimal sets of sensors or measurements to robustly infer what a highly nonlinear system is doing.