Nonlinear dimension reduction for surrogate modeling using gradient information

Daniele Bigoni (1) , Youssef Marzouk (1) , Clémentine Prieur (2, 3) , Olivier Zahm (2)

1 MIT - Massachusetts Institute of Technology
2 AIRSEA - Mathematics and computing applied to oceanic and atmospheric flows
3 GdR MASCOT-NUM - Méthodes d'Analyse Stochastique des Codes et Traitements Numériques

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.

Mots clés: high-dimensional approximation; nonlinear dimension reduction; feature map; Poincaré inequality; adaptive polynomial approximation

Domaines: Mathématiques [math]; Informatique [cs]

Daniele Bigoni, Youssef Marzouk, Clémentine Prieur, Olivier Zahm. Nonlinear dimension reduction for surrogate modeling using gradient information. Information and Inference, 2022. ⟨hal-03146362⟩