  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.
