An Approach to Bayesian Optimization for Design Feasibility Check on Discontinuous Black-Box Functions

Abstract

Abstract The paper presents a novel approach to apply Bayesian Optimization (BO) in predicting an unknown constraint boundary, also representing the discontinuity of an unknown function, for a feasibility check on the design space, thereby representing a classification tool to discern between a feasible and infeasible region. Bayesian optimization is an emerging field of study in the Sequential Design Methods where we learn and update our knowledge from prior evaluated designs, and proceed to the selection of new designs for future evaluation. It has been considered as a low-cost global optimization tool for design problems having expensive black-box objective functions. However, BO is mostly suited to problems with the assumption of a continuous objective function, and does not guarantee true convergence if the objective function has a discontinuity. This is because of the insufficient knowledge of the BO about the nature of the discontinuity of the unknown true function. Therefore, in this paper, we have proposed to predict the discontinuity of the objective function using a BO algorithm which can be considered as the pre-stage before optimizing the same unknown objective function. The proposed approach has been implemented in a thin tube design with the risk of creep-fatigue failure under constant loading of temperature and pressure. The stated risk depends on the location of the designs in terms of safe and unsafe regions, where the discontinuities lie at the transitions between those regions; therefore, the discontinuity has also been treated as an unknown constraint. The paper focuses on developing BO framework with maximizing the reformulated objective function on the same design space to predict the transition regions as a design methodology or classification tool between safe and unsafe designs, where we start with very limited data or no prior knowledge and then iteratively focus on sampling most designs near the transition region through better prior knowledge (training data) and thereby increase the accuracy of prediction to the true boundary while minimizing the number of expensive function evaluations. The converged model has been compared with the true solution for different design parameters and the results provided a classification error rate and function evaluations at an average of < 1% and ∼150, respectively. The results in this paper show some future research directions in extending the application of BO and considered as the proof of concept on large scale problem of complex diffusion bonded hybrid Compact Heat Exchangers.