Nonlinear dimension reduction for surrogate modeling using gradient information
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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∘ g for some profile function f:ℝ^m→ℝ. We propose to build the feature map by aligning the Jacobian ∇ g with the gradient ∇ 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)),∇ 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.
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