Optimization problems are ubiquitous in science and engineering applications [1]. Those also include the support to engineering tasks that are in increasing demand to meet sustainability goals such as the identification of the best design configurations to maximize the performance and minimize the environmental impact of novel engineering solutions, and the detection and identification of damages or faults to monitor the health condition of complex systems to maximize their useful life and minimize waste of resources. Over the last decades, the increase of computing power and the advances in computational modelling capabilities made available computer-based models for the accurate analysis and simulation of complex physical systems. This is the case of computational schemes for the numerical solution of governing partial differential equations as computational fluid dynamic solvers to represent viscous fluids, and finite element methods for the analysis of mechanical structures, heath transfer and electromagnetic phenomena. In principle, this computer-based representations can provide a remarkable contribution to enhance the search and identification task in simulation-based optimization. Unfortunately, the extensive adoption of these high-fidelity models during the optimization procedure is hampered by the significant computational cost and time required for their evaluation, potentially in the order of months for a single evaluation on high performance computing platforms. This issue becomes more challenging for many-query optimization problems where the demand for model evaluations grows exponentially with the number of parameters to optimize. The use of low-fidelity models constitutes a popular approach to reduce the computational resources associated with the solution of optimization problems. Low-fidelity representations introduce assumptions about the physics and/or approximate the solution of the governing equations, and relief the computational expenditure for the evaluation of the response of the system.

Science and Engineering applications are typically associated with expensive optimization problem to identify optimal design solutions and states of the system of interest. Bayesian optimization and active learning compute surrogate models through efficient adaptive sampling schemes to assist and accelerate this search task toward a given optimization goal. Both those methodologies are driven by specific infill/learning criteria which quantify the utility with respect to the set goal of evaluating the objective function for unknown combinations of optimization variables. While the two fields have seen an exponential growth in popularity in the past decades, their dualism and synergy have received relatively little attention to date. This paper discusses and formalizes the synergy between Bayesian optimization and active learning as symbiotic adaptive sampling methodologies driven by common principles. In particular, we demonstrate this unified perspective through the formalization of the analogy between the Bayesian infill criteria and active learning criteria as driving principles of both the goal-driven procedures. To support our original perspective, we propose a general classification of adaptive sampling techniques to highlight similarities and differences between the vast families of adaptive sampling, active learning, and Bayesian optimization. Accordingly, the synergy is demonstrated mapping the Bayesian infill criteria with the active learning criteria, and is formalized for searches informed by both a single information source and multiple levels of fidelity. In addition, we provide guidelines to apply those learning criteria investigating the performance of different Bayesian schemes for a variety of benchmark problems to highlight benefits and limitations over mathematical properties that characterize real-world applications.

Traditional methods for black box optimization require a considerable number of evaluations which can be time consuming, unpractical, and often unfeasible for many engineering applications that rely on accurate representations and expensive models to evaluate. Bayesian Optimization (BO) methods search for the global optimum by progressively (actively) learning a surrogate model of the objective function along the search path. Bayesian optimization can be accelerated through multifidelity approaches which leverage multiple black-box approximations of the objective functions that can be computationally cheaper to evaluate, but still provide relevant information to the search task. Further computational benefits are offered by the availability of parallel and distributed computing architectures whose optimal usage is an open opportunity within the context of active learning. This paper introduces the Resource Aware Active Learning (RAAL) strategy, a multifidelity Bayesian scheme to accelerate the optimization of black box functions. At each optimization step, the RAAL procedure computes the set of best sample locations and the associated fidelity sources that maximize the information gain to acquire during the parallel/distributed evaluation of the objective function, while accounting for the limited computational budget. The scheme is demonstrated for a variety of benchmark problems and results are discussed for both single fidelity and multifidelity settings. In particular we observe that the RAAL strategy optimally seeds multiple points at each iteration allowing for a major speed up of the optimization task.