2021/04/29 by Anthony Gruber, Max Gunzburger, Gruber, Anthony +7 · 1 citation
Computer Science · Engineering · Physics and Astronomy · #65D15 #65D40 #Control Systems and Identification #FOS: Computer and information sciences #FOS: Mathematics #Machine Learning (cs.LG) #Machine Learning (stat.ML) #Model Reduction and Neural Networks #Neural Networks and Applications #Numerical Analysis (math.NA)
paper · pdf · doi:10.48550/arxiv.2104.14072
openalex publication_date 2021/04/29 · openalex created_date 2021/05/10 · openalex updated_date 2026/07/28
A dimension reduction method based on the "Nonlinear Level set Learning" (NLL) approach is presented for the pointwise prediction of functions which have been sparsely sampled. Leveraging geometric information provided by the Implicit Function Theorem, the proposed algorithm effectively reduces the input dimension to the theoretical lower bound with minor accuracy loss, providing a one-dimensional representation of the function which can be used for regression and sensitivity analysis. Experiments and applications are presented which compare this modified NLL with the original NLL and the Active Subspaces (AS) method. While accommodating sparse input data, the proposed algorithm is shown to train quickly and provide a much more accurate and informative reduction than either AS or the original NLL on two example functions with high-dimensional domains, as well as two state-dependent quantities depending on the solutions to parametric differential equations.