  Data mining is routinely used to organize ensembles of short temporal
observations so as to reconstruct useful, low-dimensional realizations of an
underlying dynamical system. In this paper, we use manifold learning to
organize unstructured ensembles of observations ("trials") of a system's
response surface. We have no control over where every trial starts; and during
each trial operating conditions are varied by turning "agnostic" knobs, which
change system parameters in a systematic but unknown way. As one (or more)
knobs "turn" we record (possibly partial) observations of the system response.
We demonstrate how such partial and disorganized observation ensembles can be
integrated into coherent response surfaces whose dimension and parametrization
can be systematically recovered in a data-driven fashion. The approach can be
justified through the Whitney and Takens embedding theorems, allowing
reconstruction of manifolds/attractors through different types of observations.
We demonstrate our approach by organizing unstructured observations of response
surfaces, including the reconstruction of a cusp bifurcation surface for
Hydrogen combustion in a Continuous Stirred Tank Reactor. Finally, we
demonstrate how this observation-based reconstruction naturally leads to
informative transport maps between input parameter space and output/state
variable spaces.
