**4. Surrogate modeling methods**

#### **4.1 Design of experiments**

An implicit function *<sup>g</sup>*ð Þ <sup>x</sup> is considered, where x = *<sup>x</sup>*<sup>1</sup> <sup>⋯</sup> *xs* ½ �<sup>T</sup> is an input variable vector and *s* is the number of input variables. Before a surrogate model of function *g*ð Þ x can be created, some sample points shall be generated using design of experiments (DOE). Some routinely used DOE approaches include factorial design, Latin hypercube sampling (LHS) [68], central composite design, and Taguchi orthogonal array design [69]. Assume x*<sup>i</sup>* is the input variable vector at the *i*th (*i* = 1,…*n*) sample point, the limit state function *g*ð Þ x needs to be evaluated at all the sample points to obtain the function values, i.e., *g* ¼ *g*<sup>1</sup> ⋯ *gn* � �<sup>T</sup> <sup>=</sup> ½ � *<sup>g</sup>*ð Þ x1 <sup>⋯</sup> *<sup>g</sup>*ð Þ <sup>x</sup>*<sup>n</sup>* T.
