%ADietrich, Felix%ADietrich, Felix [Department of Chemical and Biomolecular Engineering and Department of Applied Mathematics and Statistics, Johns Hopkins University, Baltimore, Maryland 21218, USA]%AKooshkbaghi, Mahdi [Program in Applied and Computational Mathematics, Princeton University, Princeton, New Jersey 08544, USA]%AKooshkbaghi, Mahdi%ABollt, Erik%ABollt, Erik [Department of Electrical and Computer Engineering, Clarkson Center for Complex Systems Science, Clarkson University, Potsdam, New York 13699-5815, USA]%AKevrekidis, Ioannis%AKevrekidis, Ioannis [Department of Chemical and Biomolecular Engineering and Department of Applied Mathematics and Statistics, Johns Hopkins University, Baltimore, Maryland 21218, USA]%BJournal Name: Chaos: An Interdisciplinary Journal of Nonlinear Science; Journal Volume: 30; Journal Issue: 4; Related Information: CHORUS Timestamp: 2023-06-26 21:40:20 %D2020%IAmerican Institute of Physics %JJournal Name: Chaos: An Interdisciplinary Journal of Nonlinear Science; Journal Volume: 30; Journal Issue: 4; Related Information: CHORUS Timestamp: 2023-06-26 21:40:20 %K %MOSTI ID: 10143061 %PMedium: X %TManifold learning for organizing unstructured sets of process observations %X

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 the input parameter space and output/state variable spaces.

%0Journal Article