Adaptivity and uncertainty of multi-fidelity surrogate models for shape optimization

Abstract

Engineering design optimization based on expensive simulation is increasingly performed with surrogate models [1], i.e. approximate models fitted through a small dataset of simulation results. To build surrogates within the lowest possible computational budget, modern approaches use multi-fidelity data (combinations of cheap low-fidelity and expensive high-fidelity simulation results) and adaptive sampling strategies, which add simulation points one by one where they are most likely to discover the optimum [2]. Uncertainty estimation of the surrogate model is crucial for efficient adaptive sampling, since it guides the choice of new sampling points. Thus, underestimation of the uncertainty may lead to sampling in suboptimal regions, missing the true optimum. Existing surrogate models such as Gaussian process regression [3] and Stochastic Radial Basis Functions (SRBF) [4], provide uncertainty estimations, which are often used. Nevertheless, uncertainty estimation is so important for a successful surrogate model, that a more thorough investigation seems warranted. This is the main objective of this paper. This paper studies three issues with uncertainty estimation, in the context of multifidelity SRBF. First, most existing techniques rely on knowledge about the global behavior of the data, such as spatial correlations. However, the number of training points can be too small to reconstruct this global information from the data. We argue that in this situation, user-provided estimation of the function behavior is a better choice (section 3). Furthermore, the dataset may contain noise, i.e. random errors without spatial correlation (section 4). Surrogate models can filter out this noise, usually by modeling it as belonging to a normal distribution with zero mean, but this introduces two separate uncertainties: the optimum amount of noise filtering is unknown, and for a small dataset the local mean of the noisy data may not correspond to the true simulation response. Finally, in multi-fidelity models, the low-fidelity data are corrected by high-fidelity results, which could reduce the amount of uncertainty they introduce.

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