Nonlinear dimension reduction for surrogate modeling using gradient information

TL;DR

This paper presents a novel nonlinear dimension reduction method for surrogate modeling that leverages gradient information to construct feature maps, enabling more accurate high-dimensional function approximation than linear methods.

Contribution

The paper introduces a gradient-aligned nonlinear feature map construction method for dimension reduction, with theoretical analysis and an adaptive polynomial-based algorithm.

Findings

Nonlinear feature maps improve approximation accuracy over linear methods.

The method effectively utilizes gradient information for dimension reduction.

Numerical experiments demonstrate the approach's effectiveness across benchmarks.

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.