Correction: Data-Enhanced Design Optimization Using Linear Gaussian Models
Daïgo Maruyama · 2025
Corrections for interpretation: In Eq. ( 4), the rightest side is "(| + , + )", not "(| + , + )". In Eq. ( 5), the right side is "(|′{ ( -)} + , ′)", not "(|′{ ( -)} + , ′)". In Eq. ( 6), the right side is " (), (, ) ", not " (), (, ) ". In Eq. ( 10), the right side is " (; , ), ′(, ; ) ", not " (; , ), ′(, ; ) ". The first sentence just after Eq. ( 10) should read that "Naturally (; , ) as the mean of the posterior function can be …", not "Naturally (; , ) as the mean of the posterior function can be …". In Eq. ( 12), the right side is " (; , ), ′(, ; ) ", not " (; , ), ′(, ; ) ". In Eq. ( 19), the right side is "argma ()", not "argm ()". The last sentence in the first paragraph in Section I should read that "In that context, deep Gaussian process models [16] in general have the same property", not "In that context, deep Gaussian process models [15] in general have the property". The fourth sentence in the second paragraph in Section I should read that "A simple linear function model such as () = + using least square methods is one of the most primitive approaches to deterministically approximate the target function, where and are fixed by minimizing the least square using the observed data.",not " and". The nineth sentence in the third paragraph in Section I should read that "Han et al. proposed hierarchical Kriging models using trend assistance to deal with multi-fidelity data in function approximation and optimization problems [6,7].". The second sentence just after Eq. ( 8) should read that "If it is treated as a fixed parameter, …". The sentence just before Table 2 should read that "Since the function can be any arbitray function, various types of modelling of is feasible and the choice of this function is one of the main topics of the linear Gaussian models." Table .2 should be changed into Table .1. The second last sentence just before Fig. 2 should read that "…, the linear dynamical system considers the learning and the predicition by using one sequential data sequentially, which is benefitial for the computational complexity issue ( ) as + ⋯ + , not as (( + ⋯ + ) ) in the (learning and) prediction process." The last two sentences in the third paragraph in Section III should read that "Note that the low-fidelity function of the Forrester function is a linear transformation of the high-fidelity one.On the other hand, that of the Heterogeneous function is a non-linear transformation.",not "translation" The fifth sentence in after Eq. ( 18) should read that "Note that in practice to avoid singular matrices in computing covariance matrices in the linear Gaussian models in general, when no regularization kernel is used, …", not "…, when no regularization kernel is not used, …". Corrections in references: The seventh sentence in the first paragraph in Section I should read that "Multi-fidelity surrogate models [2-15] to further reduce the number of iterations assisted by lower-fidelity analysis have been also being developed until now.",not "[2-14]". The tenth sentence in the third paragraph in Section I should read that "Zhang et al. proposed a multi-fidelity model as an extension of linear regression models [9,10].", not [10,11]. The eleventh sentence in the third paragraph in Section I should read that "These models can be regarded as kinds of linear Gaussian models [16,17] which will be introduced later.",not [17,18]. The thirteenth sentence in the third paragraph in Section I should read that "Perdikaris et al. proposed a model using a nonlinear function with respect to the parameter connecting different fidelity functions [12] with considering practical applications where multi-fidelity unknown functions are not linearly transformed to each other.",not [11]. The fifteenth sentence in the third paragraph in Section I should read that "The multi-fidelity Bayesian neural network model proposed by Menga et al. is also powerful for function approximation due to the flexibility of the neural network models [13].",not [12].