Time-varying fluid flows are ubiquitous in modern engineering and in the life sciences. Particularly challenging is the characterization of unsteady aerodynamic forces and moments as they play critical roles in, for instance, biological propulsion and bio-inspired engineering design principals. It is observed that birds, bats, insects, and fish routinely harness unsteady fluid phenomena to improve their propulsive efficiency, maximize thrust and lift, and increase maneuverability. Such observations are highly suggestive, leading to conjectures about the existence of low-dimensional structures in fluid flows. Machine learning aims to capitalize on such dominant patterns of spatio-temporal activity. When integrated with more traditional first-principals simulation, reduced-order models can be developed to accurately quantify fluid flows.
For many insects, robust flight dynamics and control is achieved despite potentially large perturbations in flight conditions (e.g., wind gusts, bodily harm). Such stable flight is accomplished despite their limited number of computational resources and spatially distributed, noisy sensors. It suggests that the low-dimensional structures in the flow-field that are sparsely sampled and exploited for robust control purposes. Our goal is to develop a mathematical framework for directly addressing this conjecture. In particular, we consider the model problem of flow around a cylinder where (i) dimensionality reduction is used to encode the low-dimensional dynamics present in the system for a wide range of fluid flows, and (ii) compressive (sparse) sampling is used to robustly decode the dynamic state of the system, thus revealing a promising, low-dimensional control architecture. The integration of such encoding (machine learning) and decoding (compressive sensing) strategies represents a novel approach for characterizing complex dynamical systems and/or fluid flows. Indeed, this approach presents a tremendous opportunity to develop the underlying mathematical theory of dynamical systems in the context of sparse data assimilation and machine learning.
Machine learning and compressive sensing provide a mathematical infrastructure that is ideal for applying and developing control laws. The machine learning provides the low-rank representation of the fluid flow, allowing for an efficient, low-dimensional control protocol. Compressive sensing can be used towards characterizing the best measurement and actuation locations capable of optimally, and with minimal actuation, manipulating the fluid flow. We continue to innovate new ideas around the integration of these methods, especially with equation-free protocols like extremum seeking controllers, towards building a better understanding of how bio-inspired flight control might occur.