  In this paper we study the efficacy of combining machine-learning methods
with projection-based model reduction techniques for creating data-driven
surrogate models of computationally expensive, high-fidelity physics models.
Such surrogate models are essential for many-query applications e.g.,
engineering design optimization and parameter estimation, where it is necessary
to invoke the high-fidelity model sequentially, many times. Surrogate models
are usually constructed for individual scalar quantities. However there are
scenarios where a spatially varying field needs to be modeled as a function of
the model's input parameters. We develop a method to do so, using projections
to represent spatial variability while a machine-learned model captures the
dependence of the model's response on the inputs. The method is demonstrated on
modeling the heat flux and pressure on the surface of the HIFiRE-1 geometry in
a Mach 7.16 turbulent flow. The surrogate model is then used to perform
Bayesian estimation of freestream conditions and parameters of the SST (Shear
Stress Transport) turbulence model embedded in the high-fidelity
(Reynolds-Averaged Navier-Stokes) flow simulator, using wind-tunnel data. The
paper provides the first-ever Bayesian calibration of a turbulence model for
complex hypersonic turbulent flows. We find that the primary issues in
estimating the SST model parameters are the limited information content of the
heat flux and pressure measurements and the large model-form error encountered
in a certain part of the flow.
