An improved quasi-sparse response surface model using the weighting method for low-dimensional simulation

Abstract

The quasi-sparse response surface (QSRS) can be employed in simulation based design and optimization of complex products since it can accurately and efficiently approach the black-box function using a few design sampling points. In the training process, a few atoms from a dictionary are chosen to fit the sampling points to construct the response surface. Due to their higher oscillation frequency, higher-order atoms are able to better fit only a few sampling points; however, the overfitting problem decreases the prediction accuracy. In this paper, the weighting method is employed for QSRS to regularize the order of atoms in the process of atom selection to reduce the problem of overfitting. The coefficients of atoms are calculated based on the ł1 minimization method. Lower-order atoms are given larger weights, while higher-order atoms are given smaller weights. Lower-order atoms with larger weights will receive smaller coefficients and be more easily chosen to construct the response surface, which can reduce overfitting. For the QSRS of low-dimensional simulation problem contains higher order atoms than high-dimensional simulation problem, the weighting method can have better results. The Gaussian kernel function is recommended as the weight function for QSRS. The Gaussian kernel weighed QSRS is compared with an inverse function weighted QSRS, un-weighted QSRS, and other 4 response surface models based on 16 benchmark functions and 1 engineering application. The results show that the weighting method can effectively improve the approximation accuracy and maintain good robustness for QSRS.