Surrogates have been used as an approximate tool to emulate simulation responses based on a handful of response samples. However, for high fidelity simulations, often only a small number samples are affordable, and this increases the risk of extrapolation using surrogates. Frequently, most of the sampling domain is not in the interpolation domain (called coverage), usually defined as the convex hull of these samples. For example, when we build a surrogate with 20 samples in six-dimensional space, the coverage is merely 2% of the sampling domain. Multi-fidelity surrogates (MFS) may mitigate this problem, because they use large number of low fidelity simulations, so that most of the domain is covered with at least some simulations.

This paper explores the extrapolation capability of MFS frameworks through examples including algebraic functions. To examine the effects of different MFS frameworks, we consider six MFS frameworks in terms of their functional forms and frameworks for fitting the forms to data. We consider three different functional forms based on different approaches: 1) a model discrepancy function, 2) model calibration, and 3) both. Bayesian MFS frameworks based on the functional forms are considered. We include also their counterparts in simple frameworks, which have the same functional form but can be built with ready-made surrogates. We examined the effect of the high fidelity sample coverage on extrapolation while the number of high fidelity samples remains the same. The root mean square errors (RMSE) of the interpolation and extrapolation domains are calculated to see their effectiveness on the overall RMSE of whole MFS. For the examples considered, we found that the presence of a regression scalar could be important to extrapolation. Bayesian framework is useful to find a good regression scalar, which simplifies the discrepancy function.

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