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Understanding, predicting and controlling laminar-turbulent boundary-layertransition is crucial for the next generation aircraft design. However, in realflight experiments, or wind tunnel tests, often only sparse sensor measurementscan be collected at fixed locations. Thus, in developing reduced models forpredicting and controlling the flow at the sensor locations, the main challengeis in accounting for how the surrounding field of unobserved variablesinteracts with the observed variables at the fixed sensor locations. This makesthe Mori-Zwanzig (MZ) formalism a natural choice, as it results in theGeneralized Langevin Equations which provides a framework for constructingnon-Markovian reduced-order models that includes the effects the unresolvedvariables have on the resolved variables. These effects are captured in the socalled memory kernel and orthogonal dynamics. In this work, we explore thedata-driven MZ formulations to two boundary layer flows obtained from DNS data;a low speed incompressible flow; and a high speed compressible flow over aflared cone at Mach 6. An array of "sensors" are placed near the surface of thesolid boundary, and the MZ operators are learned and the predictions arecompared to the Extended Dynamic Mode Decomposition (EDMD), both using delayembedded coordinates. Further comparisons are made with Long Short-Term Memory(LSTM) and a regression based projection framework using neural networks forthe MZ operators. First we compare the effects of including delay embeddedcoordinates with EDMD and Mori based MZ and provide evidence that using bothmemory and delay embedded coordinates minimizes generalization errors on therelevant time scales. Next, we provide numerical evidence that the data-drivenregression based projection MZ model performs best with respect to theprediction accuracy (minimum generalization error) on the relevant time scales.

Data-Driven Mori-Zwanzig: Reduced Order Modeling of Sparse Sensors Measurements for Boundary Layer Transition