Bayesian optimisation for expensive physical experiments and computer simulators with application in fluid dynamics

Abstract

Expensive black-box functions such as physical experiments and computer simulators are challenging to optimise as they cannot be solved analytically, and only small numbers of function evaluations are available for optimisation. This prevents use of conventional methods that rely on gradient information or larger numbers of function evaluations, requiring a specialised optimisation strategy. Bayesian optimisation is a sample-efficient strategy that represents the objective function through a surrogate model and guides the exploration of the input space with heuristics—so-called acquisition criteria—to select promising candidate points sequentially. Expensive black-box functions are a common occurrence in fluid dynamics where the underlying systems, for example the Navier-Stokes equations, can be too complex to solve explicitly and can be viewed as a black box. In addition, the expensive nature of the associated experiments and simulations makes Bayesian optimisation a prime candidate. However, the Bayesian optimisation literature is mainly geared towards statisticians and computer scientists and is potentially challenging to scrutinise and apply for non-experts. Thus, the main motivation of this thesis is to make Bayesian optimisation more accessible and answer some fundamental questions overlooked in the literature, while also developing techniques for specific challenges encountered in but not limited to fluid dynamics. This thesis studies three topics for applying Bayesian optimisation to experiments and simulators. Firstly, it investigates key choices in Bayesian optimisation empirically, such as the choice of the acquisition criterion and the number of data points used for initialisation, and applies the findings to two computer simulators with the objective of controlling air flow to maximise the skin-friction drag reduction over a flat plate—mimicking the surface of a moving vehicle such as the wing of an aeroplane. Secondly, NUBO—an open-source Python package for optimising expensive experiments and simulators aimed at practitioners of Bayesian optimisation—is presented, and its functionalities are discussed. This transparent package allows users to tailor the optimisation loop to their specific problems and supports sequential single-point, parallel multi-point and asynchronous optimisation for bounded, constrained and mixed (discrete and continuous) input parameter spaces. Lastly, problems affected by external environmental variables that cannot be controlled are investigated, and ENVBO—a novel algorithm—is introduced. ENVBO fits a global surrogate model over all controllable and environmental variables but optimises the acquisition criterion only with regard to the controllable variables while keeping the environmental variables fixed at a current measurement. Important properties of ENVBO, such as the robustness to noisy objective functions and the number of environmental variables, are studied. ENVBO is applied to a wind farm simulator to maximise energy production by (a) finding optimal positions for four wind turbines within a complex terrain with changing wind directions and (b) setting optimal derating factors of a row of five wind turbines subject to changing wind speeds.