Algorithm 2 Bayesian optimization (BO) for optimal design

1:Input: Design space $\mathcal{D}$ and its discretized design space $\overline{\mathcal{D}}$, prior ${\mu }_0(\mathit{d})=0$, hyperparameters $l,{\sigma }_f$ of the Gaussian kernel, hyperparameter $\delta$, and maximum number of iterations $t_{\max }$. 2:for$t=1,\dots ,t_{\max }$do 3:    Find the maximizer of the acquisition function: ${\mathit{d}}_t=\arg {\max }_{d\in \mathcal{D}}{\mu }_{t-1}(\mathit{d})+\sqrt{{\beta }_{t-1}}{\sigma }_{t-1}(\mathit{d})$. 4:    Sample the objective function $U_t=\widehat{U}({\mathit{d}}_t)$ using Algorithm 1. 5:    Augment the data set ${\mathcal{S}}_{1:t}={\{{\mathit{d}}_i,U_i\}}_{i=1}^t$. 6:    Perform Bayesian update to obtain ${\mu }_t$ and ${\sigma }_t$ over $\overline{\mathcal{D}}$ using (11) and (12) respectively. 7:    Update ${\beta }_t.$ 8:end for 9:Output: Optimal design: $d^{\star }=\arg {\max }_{t=1,\dots ,t_{\max }}{U}_t$