Nested Bayesian Optimization for Computer Experiments

Computer experiments can emulate the physical systems, help computational investigations, and yield analytic solutions. They have been widely employed with many engineering applications (e.g., aerospace, automotive, energy systems). Conventional Bayesian optimization did not incorporate the nested structures in computer experiments. This article proposes a novel nested Bayesian optimization method for complex computer experiments with multistep or hierarchical characteristics. We prove the theoretical properties of nested outputs given that the distribution of nested outputs is Gaussian or non-Gaussian. The closed forms of nested expected improvement are derived. We also propose the computational algorithms for nested Bayesian optimization. Three numerical studies show that the proposed nested Bayesian optimization method outperforms the five benchmark Bayesian optimization methods that ignore the intermediate outputs of the inner computer code. The case study shows that the nested Bayesian optimization can efficiently minimize the residual stress during composite structures assembly and avoid convergence to local optima.