  In this work, we apply, for the first time to spatially inhomogeneous flows,
a recently developed data-driven learning algorithm of Mori-Zwanzig (MZ)
operators, which is based on a generalized Koopman's description of dynamical
systems. The MZ formalism provides a mathematically exact procedure for
constructing non-Markovian reduced-order models of resolved variables from
high-dimensional dynamical systems, where the effects due to the unresolved
dynamics are captured in the memory kernel and orthogonal dynamics. The
algorithm developed in this work applies Mori's linear projection operator and
an SVD based compression to the selection of the resolved variables
(equivalently, a low rank approximation of the two time covariance matrices).
We show that this MZ decomposition not only identifies the same spatio-temporal
structures found by DMD, but it can also be used to extract spatio-temporal
structures of the hysteresis effects present in the memory kernels. We perform
an analysis of these structures in the context of a laminar-turbulent
boundary-layer transition flow over a flared cone at Mach 6, and show the
dynamical relevance of the memory kernels. Additionally, by including these
memory terms learned in our data-driven MZ approach, we show improvement in
prediction accuracy over DMD at the same level of truncation and at a similar
computational cost. Furthermore, an analysis of the spatio-temporal structures
of the MZ operators shows identifiable structures associated with the nonlinear
generation of the so-called "hot" streaks on the surface of the flared code,
which have previously been observed in experiments and direct numerical
simulations.
