Nonlinear dimension reduction for surrogate modeling using gradient information

Understand

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.