Nonlinear dimension reduction for surrogate modeling using gradient information

Abstract We introduce a method for the nonlinear dimension reduction of high-dimensional function $u:{\mathbb{R}}^d\rightarrow{\mathbb{R}}$, $d\gg 1$. Our objective is to identify feature map $g:{\mathbb{R}}^d\rightarrow{\mathbb{R}}^m$, with prescribed intermediate $m\ll d$, so that $u$ can be well approximated by $f\circ g$ some profile $f:{\mathbb{R}}^m\rightarrow{\mathbb{R}}$. propose build aligning Jacobian $\nabla gradient u$, and we theoretically analyze properties resulting $g$. Once $g$ built, construct $f$ solving gradient-enhanced least squares problem. practical algorithm uses sample $\{{\textbf{x}}^{(i)},u({\textbf{x}}^{(i)}),\nabla u({\textbf{x}}^{(i)})\}_{i=1}^N$ builds both on adaptive downward-closed polynomial spaces, using cross validation avoid overfitting. numerically evaluate performance our across different benchmarks, explore impact $m$. show building permit more accurate approximation than linear $g$, same input data set.