(12f) Bonsai: A Hyper-Sample-Efficient Framework for Robust Global Optimization of Expensive Function Network Systems Under Uncertainty

Uncertainty is inevitably present in real-world design problems due to several sources including noisy and incomplete data, unknown or fluctuating model parameters, implementation errors, and environmental perturbations/disturbances (e.g., product demand, prices, and material availability). Thus, “optimal” solutions identified by solving a nominal design problem neglecting the impact of these uncertainties are potentially suboptimal or even infeasible when implemented in practice. Robust optimization is a framework that looks to overcome such limitations by searching for the robust optimal design that achieves the best worst-case value of the objective while satisfying the constraints for all possible uncertainties [1, 2, 3]. There has been a significant amount of work on developing new algorithms for robust optimization in recent years; however, a significant gap remains between theory and practice on many types of problems encountered in the real world. One of the biggest challenges is that most available robust optimization methods (especially those capable of efficiently finding the global solution) are only applicable to a narrow class of problems that exploit a large amount of problem structure such as convexity [4, 5]. As such, these algorithms are not applicable to high-fidelity simulation software (also known as “digital twins” [6]), which have quickly grown in scope due to advances in computing and increased availability of modeling software. Even though digital twins have been developed in a variety of domains including cyber-physical systems [7], robotics [8], fluid dynamics [9], building energy systems [10], and spacecraft systems [11], there has been very limited work on their use for robust design purposes due to their expensive black-box nature. Most work has instead focused on the use of high-fidelity digital twins for monitoring and/or validation of approximate design methodologies, which does not take full advantage of their powerful capabilities to extrapolate to completely new (unobserved) situations.

Previously, we developed the adversarially robust Bayesian optimization (ARBO) framework as a sample-efficient derivative-free robust optimization approach for black-box functions [12, 13]. ARBO extends traditional Bayesian optimization to explicitly consider the impact of uncertainties during the optimization process; its key innovation is the joint modeling of the dependence of the unknown functions on the design and uncertain variables, leading to a significant improvement in the choice of new query points at every iteration. ARBO has been successfully applied to a diverse set of problems involving digital twins, e.g., genome-scale reactor design [13], MPC tuning [14], and heat exchanger network flexibility tests [15]. However, as commonly observed in surrogate-based optimization routines, ARBO’s performance tends to degrade as the dimensionality of the problem increases. This performance degradation is a direct consequence of the assumption that the objective function is a black box; this generality comes at the cost performance and is rarely needed in practice since there is at least partial knowledge that can be exploited by the algorithm. Previous work has shown that significant performance gains can be achieved by leveraging a grey-box perspective within Bayesian optimization [16, 17]; however, these methods are applicable to single-level composite functions and do not consider any form of uncertainty. We look to develop a method that can straightforwardly address both of these shortcomings here.

In this work, we propose a new algorithm called Bayesian Optimization of Network Systems under uncertAInty (BONSAI) that significantly improves the sample efficiency of ARBO by exploiting prior structural knowledge available in simulation-based robust optimization problems. Instead of assuming the objective is a black-box function, BONSAI represents the objective as a directed acyclic graph (DAG) where each node can be either a black- or white-box function, enabling it to leverage intermediate information in the graph (which is currently ignored by ARBO). Note that the DAG representation is a natural way to express information flow in most engineering problems. Take, for example, a chemical manufacturing plant that converts raw materials into a set of desired products – initial, middle, and final nodes can capture the conversion, separation, and purification processes, respectively, all of which have their own local variables and share some subset of the complete set of variables in the plant. Similar graph abstractions have been developed for nonlinear optimization; however, they require an equation-based model at every node [18]. BONSAI, on the other hand, “fills in the blanks” by modeling the black-box nodes using Gaussian processes (GPs). Since the implied posterior of the objective function obtained by passing the GPs through the DAG is non-Gaussian, we cannot compute the standard acquisition functions (such as upper confidence bound) in closed form. We overcome this challenge in BONSAI by proposing a novel (scalable) Thompson sampling-based acquisition function that can be efficiently optimized using state-of-the-art gradient-based methods. Through numerical experiments on multiple synthetic and real-world problems, we demonstrate that BONSAI can dramatically accelerate optimization compared to competing methods such as ARBO (up to multiple orders of magnitude in some cases). It is worth highlighting that such improvements come simply from incorporating information that is readily available inside of the simulators but is traditionally neglected by other methods. Therefore, BONSAI provides a promising (general) path toward unlocking the full power of computationally expensive digital twins for systematic design in the presence of uncertainty.

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