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Manifold learning techniques seek to discover structure-preserving mappingsof high-dimensional data into low-dimensional spaces.  While the new sets of coordinates specified by these mappings can closelyparameterize the data, they are generally complicated nonlinear functions ofthe original variables. This makes them difficult to interpret physically.  Furthermore, in data-driven model reduction applications the governingequations may have structure that is destroyed by nonlinear mapping intocoordinates on an inertial manifold, creating a computational bottleneck forsimulations.  Instead, we propose to identify a small collection of the original variableswhich are capable of uniquely determining all others either locally viaimmersion or globally via embedding of the underlying manifold.  When the data lies on a low-dimensional subspace the existing discreteempirical interpolation method (DEIM) accomplishes this with recent variantsemploying greedy algorithms based on pivoted QR (PQR) factorizations.  However, low-dimensional manifolds coming from a variety of applications,particularly from advection-dominated PDEs, do not lie in or near anylow-dimensional subspace.  Our proposed approach extends DEIM to data lying near nonlinear manifolds byapplying a similar pivoted QR procedure simultaneously on collections ofpatches making up locally linear approximations of the manifold, resulting in anovel simultaneously pivoted QR (SimPQR) algorithm.  The immersion provided by SimPQR can be extended to an embedding by applyingSimPQR a second time to a modified collection of vectors.  The SimPQR method for computing these `nonlinear DEIM' (NLDEIM) coordinatesis successfully applied to real-world data lying near an inertial manifold in acylinder wake flow as well as data coming from a viscous Burgers equation withdifferent initial conditions.

A Discrete Empirical Interpolation Method for Interpretable Immersion and Embedding of Nonlinear Manifolds