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Nonlinear dimension reduction for surrogate modeling using gradient information

Daniele Bigoni 1 Youssef Marzouk 1 Clémentine Prieur 2, 3 Olivier Zahm 2
2 AIRSEA - Mathematics and computing applied to oceanic and atmospheric flows
Inria Grenoble - Rhône-Alpes, UGA - Université Grenoble Alpes, LJK - Laboratoire Jean Kuntzmann, Grenoble INP - Institut polytechnique de Grenoble - Grenoble Institute of Technology
Abstract : We introduce a method for the nonlinear dimension reduction of a high-dimensional function $u:\mathbb{R}^d\rightarrow\mathbb{R}$, $d\gg1$. Our objective is to identify a nonlinear feature map $g:\mathbb{R}^d\rightarrow\mathbb{R}^m$, with a prescribed intermediate dimension $m\ll d$, so that $u$ can be well approximated by $f\circ g$ for some profile function $f:\mathbb{R}^m\rightarrow\mathbb{R}$. We propose to build the feature map by aligning the Jacobian $\nabla g$ with the gradient $\nabla u$, and we theoretically analyze the properties of the resulting $g$. Once $g$ is built, we construct $f$ by solving a gradient-enhanced least squares problem. Our practical algorithm makes use of a sample $\{x^{(i)},u(x^{(i)}),\nabla u(x^{(i)})\}_{i=1}^N$ and builds both $g$ and $f$ on adaptive downward-closed polynomial spaces, using cross validation to avoid overfitting. We numerically evaluate the performance of our algorithm across different benchmarks, and explore the impact of the intermediate dimension $m$. We show that building a nonlinear feature map $g$ can permit more accurate approximation of $u$ than a linear $g$, for the same input data set.
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https://hal.inria.fr/hal-03146362
Contributor : Olivier Zahm Connect in order to contact the contributor
Submitted on : Friday, February 19, 2021 - 8:44:01 AM
Last modification on : Thursday, January 20, 2022 - 5:31:30 PM
Long-term archiving on: : Thursday, May 20, 2021 - 9:04:27 PM

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  • HAL Id : hal-03146362, version 1

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Daniele Bigoni, Youssef Marzouk, Clémentine Prieur, Olivier Zahm. Nonlinear dimension reduction for surrogate modeling using gradient information. 2021. ⟨hal-03146362⟩

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