Nonlinear dimension reduction for surrogate modeling using gradient information

We introduce a method for the nonlinear dimension reduction of a high-dimensional function u : R-d -> R, d >> 1. Our objective is to identify a nonlinear feature map g : R-d -> R-m, with a prescribed intermediate dimension m << d, so that u can be well approximated by f circle g for some profile function f : R-m -> R. We propose to build the feature map by aligning the Jacobian del g with the gradient del 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 uses a sample {x((i)), u(x((i))), del 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.

Automated summary

This article introduces a method for nonlinear dimension reduction of high-dimensional functions. By building a nonlinear feature map, the function can be accurately approximated. The authors propose aligning the Jacobian matrix to construct the feature map and solve a gradient-enhanced least squares problem for the profile function. Experimental results show that a nonlinear feature map can provide a more accurate approximation of the function compared to a linear feature map.