Reduced Lagrangianand Mori-Zwanzig models: Applications to Turbulent Flows

Physics informed machine learning and data-driven reduced order modelling are rapidly evolving fields, with the potential to tackle notoriously challenging problems in engineering and the physical sciences. Many naturally occurring phenomena, such as turbulent flows, can be characterized as a high-dimensional nonlinear dynamical systems that exhibit strong coupling across a broad range of scales. In contrast to simulating the dynamics over all relevant scales, reduced-order models (ROM) seek to describe the dynamics using a low-dimensional space of variables, referred to as ”resolved variables” or observables. ROMs can be used to simulate the dynamics at substantially reduced computational costs that can be used for control and systems analysis applications. Furthermore, ROMs can provide tractable frameworks for analyzing and understanding the underlying physics. For example, ROMs can be used to extract large-scale spatio-temporal coherent structures, as well as a provide insights into the dominant physical mechanisms present in the system.Many reduced order modeling methods utilize optimization techniques or modern machine learning methods along with high fidelity data-sets in order to improve the accuracy of predictions. However, one of the main challenges of developing reduced models, such as those required in turbulence, is in their ability to generalize. For example, in turbulence applications, the Reynolds number and Mach number are key bifurcation parameters; a small smooth change made to these parameters causes a sudden topological change in its behavior. Thus, it can be challenging for ROMs to accurately capture the dependency on these bifurcation parameters in order to achieve accurate generalizations. Therefore, in order to approach accurate, interpretable and generalizable reduced models, this work blends modern machine learning with physical models and phenomenology in an attempt to achieve the best of both worlds; namely, the accuracy and performance o