Nonlinear Level Set Learning for Function Approximation on Sparse Data with Applications to Parametric Differential Equations
Year: 2021
Author: Yuankai Teng, Zhu Wang, Anthony Gruber, Max Gunzburger, Lili Ju, Yuankai Teng, Zhu Wang
Numerical Mathematics: Theory, Methods and Applications, Vol. 14 (2021), Iss. 4 : pp. 839–861
Abstract
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.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/nmtma.OA-2021-0062
Numerical Mathematics: Theory, Methods and Applications, Vol. 14 (2021), Iss. 4 : pp. 839–861
Published online: 2021-01
AMS Subject Headings:
Copyright: COPYRIGHT: © Global Science Press
Pages: 23
Keywords: Nonlinear level set learning function approximation sparse data nonlinear dimensionality reduction.
Author Details
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Level Set Learning with Pseudoreversible Neural Networks for Nonlinear Dimension Reduction in Function Approximation
Teng, Yuankai
Wang, Zhu
Ju, Lili
Gruber, Anthony
Zhang, Guannan
SIAM Journal on Scientific Computing, Vol. 45 (2023), Iss. 3 P.A1148
https://doi.org/10.1137/21M1459198 [Citations: 2]