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A dimension reduction method based on the "Nonlinear Level set Learning"(NLL) approach is presented for the pointwise prediction of functions whichhave been sparsely sampled. Leveraging geometric information provided by theImplicit Function Theorem, the proposed algorithm effectively reduces the inputdimension to the theoretical lower bound with minor accuracy loss, providing aone-dimensional representation of the function which can be used for regressionand sensitivity analysis. Experiments and applications are presented whichcompare this modified NLL with the original NLL and the Active Subspaces (AS)method. While accommodating sparse input data, the proposed algorithm is shownto train quickly and provide a much more accurate and informative reductionthan either AS or the original NLL on two example functions withhigh-dimensional domains, as well as two state-dependent quantities dependingon the solutions to parametric differential equations.

Nonlinear Level Set Learning for Function Approximation on Sparse Data with Applications to Parametric Differential Equations