A novel and fully automated coordinate system transformation scheme for near optimal surrogate construction

Abstract

This work develops a novel coordinate system transformation scheme to improve the performance of common radial basis function surrogate models. This coordinate system transformation scheme is based on the fact that commonly used basis functions are isotropic.

Three main empirical findings are established in this study. Firstly, in general isotropic functions are inadequate to describe anisotropic data manifolds due to a mismatch between the functional form and the form of the data manifold resulting in poor generative performance. Counter-intuitively, utilising additional gradients during surrogate training often worsens the generative capability.

Secondly, component-wise scaling of isotropic model forms during cross-validation is inadequate to enhance the functional form of the data manifold form as anisotropic coupling in the data manifold remains coupled. Improving the match between the functional form and the data manifold form requires both rotation and scaling.

Thirdly, the coordinate system transformation scheme should predominantly be based on a collection of local curvature estimations and not on global curvature approximations. Gradients are critical to estimating the local curvature for identifying a near-optimal reference frame for surrogate construction, which then translates to additional benefits of gradients in gradient-enhanced surrogates.

Based on the above observations, this paper proposes an isotropic transformation for the data coordinate system that performs near-optimal transformations on lower dimensional data without requiring any cross-validation. The method is compared against commonly applied component-wise cross-validation data coordinate system scaling as well as the more modern Active Subspace Method on a carefully crafted decomposable test problem, which has a known optimal coordinate system, that varies between 2 and 16 dimensions.

The paper concludes after demonstrating that the developed transformation scheme, as well as the other common methods, will offer little benefit on non-decompose problems and offers some suggestions on future work to create a more general isotropic transformation.