Nested Bayesian Optimization for Computer Experiments

Abstract

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 paper proposes a novel nested Bayesian optimization method for complex computer experiments with multi-step 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.

Physics-Constrained Bayesian Optimization for Optimal Actuators Placement in Composite Structures Assembly

“…Their deep structure is built by composing multiple GPs; where each GP's output is modeled as the input to the next. Nested computer codes [13] were the first to apply this deep structure in Bayesian optimization. In their work, they further extended the use of Bayesian optimization to 2layer composite functions, where the inner and outer functions are assumed to be black-box and are modeled as Gaussian processes.…”

“…ω j = 1 If an actuator is placed at location j 0 Otherwise, (13) where δ represents the (i × 1) adjusted shape deviations vector, Ψ represents the (i × 1) initial shape distortions vector, U and F = [ω 1 F 1 , ω 2 F 2 , .., ω j F j ] represent the (i × j) displacement matrix, where each element corresponds to the shape correction at measurement point i given a unit actuator force at location j, and (j × 1) actuator's force vector, respectively. We define i and j as the index of the measurement point and the location of the actuator, respectively, and ω as a binary variable that is equal to one if an actuator is placed at location j and zero otherwise.…”