# Concurrent surrogate model selection (COSMOS): optimizing model type, kernel function, and hyper-parameters

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## Abstract

This paper presents an automated surrogate model selection framework called the Concurrent Surrogate Model Selection or COSMOS. Unlike most existing techniques, COSMOS coherently operates at three levels, namely: 1) selecting the model type (e.g., RBF or Kriging), 2) selecting the kernel function type (e.g., cubic or multiquadric kernel in RBF), and 3) determining the optimal values of the typically user-prescribed hyper-parameters (e.g., shape parameter in RBF). The quality of the models is determined and compared using measures of median and maximum error, given by the Predictive Estimation of Model Fidelity (PEMF) method. PEMF is a robust implementation of sequential *k*-fold cross-validation. The selection process undertakes either a cascaded approach over the three levels or a more computationally-efficient one-step approach that solves a mixed-integer nonlinear programming problem. Genetic algorithms are used to perform the optimal selection. Application of COSMOS to benchmark test functions resulted in optimal model choices that agree well with those given by analyzing the model errors on a large set of additional test points. For the four analytical benchmark problems and three practical engineering applications – airfoil design, window heat transfer modeling, and building energy modeling – diverse forms of models/kernels are observed to be selected as optimal choices. These observations further establish the need for automated multi-level model selection that is also guided by dependable measures of model fidelity.

## Keywords

Automated surrogate model selection Hyper-parameter optimization Kriging Mixed-integer non-linear programming (MINLP) Predictive estimation of model fidelity (PEMF) Radial basis functions (RBF) Support vector regression (SVR)## Notes

### Acknowledgements

Support from the National Science Foundation (NSF) Awards CMMI-1642340 and CNS-1524628 is gratefully acknowledged. Any opinions, findings, conclusions, or recommendations expressed in this paper are those of the authors and do not necessarily reflect the views of the NSF.

### Author Contributions

The different core concepts underlying PEMF and COSMOS were conceived, implemented and tested (through MATLAB) by Ali Mehmani and Souma Chowdhury, with important conceptual contributions from Achille Messac with regards to the surrogate modeling paradigm. The airfoil design and building peak cooling model in this paper were developed and implemented by Ali Mehmani, with support from Christoph Meinrenken on the latter.

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