An ensemble weighted average conservative multi-fidelity surrogate modeling method for engineering optimization

Abstract

Multi-fidelity (MF) surrogate models have been widely used in engineering optimization problems to reduce the design cost by replacing computat ional expensive simulations. Ignoring the prediction uncertainty of the MF model that is caused by a limited number of samples may result in infeasible solutions. Conservative MF surrogate model, which can effectively improve the feasibility of the constraints, has been a promising way to address this issue. In this paper, an ensemble weighted average (EWA) conservative multi-fidelity modeling method that integrates the performance of different error metrics is proposed. In the proposed method, the bootstrap method and mean-square-error method are reasonably weighted to calculate the safety margin of the MF surrogate model. The weights for the two metrics are determined through an optimization problem, which considers the performance of the two metrics in different subsets of the sample points. The effectiveness of the proposed method is illustrated through several numerical examples and a pressure vessel design problem. Results show that the proposed method constructs a more accurate conservative MF surrogate model than other methods in different problems. Furthermore, applying the constructed conservative MF surrogate model into optimization problems obtains more accurate optimal solutions while ensuring the feasibility of it.

Appendices

Appendix 1

The eight numerical examples that are used to test the performance of four error metrics in Sect. 3.1 are listed below. $y_{{\text{h}}}$ denotes the high-fidelity model that is needed to be approximated and $y_{{\text{l}}}$ denotes the low-fidelity model.

Problem 1 (P1):

$$\begin{gathered} y_{{\text{l}}} = - \sin (x) - e^{\frac{x}{100}} + 10.3 + 0.03 \times (x - 3)^{2} \hfill \\ y_{{\text{h}}} = - \sin (x) - e^{\frac{x}{100}} + 10, \, x \in [0,10]. \hfill \\ \end{gathered}$$

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Problem 2 (P2):

$$\begin{gathered} y_{{\text{l}}} = 0.5 \times \sin (12x - 4) \times (6x - 2)^{2} + 10 \times (x - 0.5) + 5 \hfill \\ y_{{\text{h}}} = \sin (12x - 4) \times (6x - 2)^{2} {, }x \in [0,1]. \hfill \\ \end{gathered}$$

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Problem 3 (P3):

$$\begin{aligned} y_{{\text{l}}} & = 4 \times (0.7x_{1} )^{2} - 2.1 \times (0.7x_{1} )^{4} + (0.7x_{1} )^{6} /3 \\ & + (0.7x_{1} \times 0.7x_{2} ) - 4 \times (0.7x_{2} )^{2} + 4 \times (0.7x_{2} )^{4} \\ y_{{\text{h}}} & = 4x_{1}^{2} - 2.1x_{1}^{4} + x_{1}^{6} /3 + x_{1} x_{2} - 4x_{2}^{2} + 4x_{2}^{4} , \, x_{1} ,x_{2} \in [ - 2,2]. \\ \end{aligned}$$

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Problem 4 (P4):

$$\begin{gathered} y_{{\text{l}}} = \left( {0.5x_{1} )^{2} + 0.8x_{2} - 11} \right)^{2} + \left( {(0.8x_{2} )^{2} + 0.5x_{1} - 7} \right)^{2} + x_{2}^{3} - (x_{1} + 1)^{2} \hfill \\ y_{{\text{h}}} = (x_{1}^{2} + x_{2} - 11)^{2} + (x_{2}^{2} + x_{1} - 7)^{2} , x_{1} ,x_{2} \in [ - 3,3]. \hfill \\ \end{gathered}$$

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Problem 5 (P5):

$$\begin{gathered} y_{{\text{l}}} = (x_{1} - 1)^{2} + 2 \times (2x_{2}^{2} - 0.75x_{1} )^{2} + 3 \times (3x_{3}^{2} - 0.75x_{2} )^{2} + 4 \times (4x_{4}^{2} - 0.75x_{3} )^{2} \hfill \\ y_{{\text{h}}} = (x_{1} - 1)^{2} + 2 \times (2x_{2}^{2} - x_{1} )^{2} + 3 \times (3x_{3}^{2} - x_{2} )^{2} + 4 \times (4x_{4}^{2} - x_{3} )^{2} \, \hfill \\ \quad x_{1} ,x_{2} ,x_{3} ,x_{4} \in [ - 10,10]. \hfill \\ \end{gathered}$$

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Problem 6 (P6):

$$\begin{aligned} f(x_{1} , \ldots ,x_{6} ) & = - \sum\limits_{i = 1}^{4} {c_{i} \exp \left[ { - \sum\limits_{j = 1}^{6} {ai_{j} (x_{j} - p_{ij} )^{2} } } \right]} , \, x_{j} \in [0,1] \\ [ci] & = [1 \, 1.2 \, 3 \, 3.2]^{{\text{T}}} , \, [aij] = \left[ \begin{gathered} \, 10 \, 3 \, 17 \, 3.05 \, 1.7 \, 8 \hfill \\ 0.05 \, 10 \, 17 \, 0.1 \, 8 \, 4 \hfill \\ \, 3 \, 3.5 \, 1.7 \, 10 \, 17 \, 8 \hfill \\ \, 17 \, 8 \, 0.05 \, 10 \, 0.1 \, 14 \hfill \\ \end{gathered} \right] \\ [p_{ij} ] & = \left[ \begin{gathered} 0.1312 \, 0.1696 \, 0.5569 \, 0.0124 \, 0.8283 \, 0.5886 \hfill \\ 0.2329 \, 0.4139 \, 0.8307 \, 0.3736 \, 0.1004 \, 0.9991 \hfill \\ 0.2348 \, 0.1451 \, 0.3522 \, 0.2883 \, 0.3047 \, 0.6650 \hfill \\ 0.4047 \, 0.8828 \, 0.8732 \, 0.5743 \, 0.1091 \, 0.0381 \hfill \\ \end{gathered} \right] \\ [lc_{i} ] & = [1.1 \, 0.8 \, 2.5 \, 3]^{{\text{T}}} , \, [l_{j} ] = [0.75 \, 1 \, 0.8 \, 1.3 \, 0.7 \, 1.1]^{T} \\ y_{l} & = - \sum\limits_{i = 1}^{4} {lc_{i} \exp [ - \sum\limits_{j = 1}^{6} {a_{ij} (l_{j} x_{j} - p_{ij} )^{2} } ]} \\ y_{h} & = f(x_{1} , \ldots ,x_{6} ), \, x_{j} \in [0,1]. \\ \end{aligned}$$

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Problem 7 (P7):

$$\begin{gathered} f_{{{\text{borehole}}}} = \frac{{2\pi x_{3} (x_{4} - x_{6} )}}{{\ln (x_{2} /x_{1} )\left[ {1 + 2x_{7} x_{4} /\left( {\ln (x_{2} /x_{1} )x_{1}^{2} x_{8} } \right) + x_{3} /x_{5} } \right]}} \hfill \\ y_{{\text{l}}} = 0.4f_{{{\text{borehole}}}} (x) + 0.07x_{1}^{2} x_{8} + x_{1} x_{7} /x_{3} + x_{1} x_{6} /x_{2} + x_{1}^{2} x_{4} \hfill \\ y_{{\text{h}}} = f_{{{\text{borehole}}}} (x) \hfill \\ x_{1} \in [0.05,0.15], \, x_{2} \in [100,50000], \hfill \\ x_{3} \in [63070,115600], \, x_{4} \in [990,1110], \hfill \\ x_{5} \in [63.1,116], \, x_{6} \in [700,820], \hfill \\ x_{7} \in [1120,1680], \, x_{8} \in [9855,12045]. \hfill \\ \end{gathered}$$

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Problem 8 (P8):

$$\begin{gathered} y_{l} = \sum\limits_{i = 1}^{10} {x_{i}^{3} } + \left( {\sum\limits_{i = 1}^{10} {2ix_{i} } } \right)^{2} + \left( {\sum\limits_{i = 1}^{10} {3ix_{i} } } \right)^{4} \hfill \\ y_{h} = \sum\limits_{i = 1}^{10} {x_{i}^{2} } + \left( {\sum\limits_{i = 1}^{10} {0.5ix_{i} } } \right)^{2} + \left( {\sum\limits_{i = 1}^{10} {0.5ix_{i} } } \right)^{4} , \, - 5 \le x_{i} \le 10. \hfill \\ \end{gathered}$$

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Appendix 2

2.1 Influence of the number of LF samples

The test setting for investigating the influence of the number of LF samples is given in Table 9. The numbers of HF samples in P1–P6 and P7–P8 are fixed to 6d and 9d, respectively. While the number of LF samples ranges from one to five times that of HF samples. The comparison results are listed in Table 10 and Fig. 16. In the table, the p values that are larger than 0.05 are marked in bold, while p values that are smaller than 0.05 are marked in italic.

Table 9 The test setting for investigating the influence of the number of LF samples

Fig. 16

Performance comparison of the four error metrics under different number of HF samples: a P1 problem; b P2 problem; c P3 problem; d P4 problem; e P5 problem; f P6 problem; g P7 problem; h P8 problem

Table 10 T test results under different number of LF samples

Several results can be obtained from the comparison results: (1) The estimated errors by the MSE method are a good representation of the variation of the true errors in all of the eight test examples, and thus, the MSE method performs the best among the four error metrics. (2) The results of the bootstrap method and the PEMF method can only reflect the variation of the true error in some of the test problems. The results of the LOO method are always higher than the true error. (3) From the t test results, it can be seen that the uncertainty quantification results of the bootstrap method and the LOO method are significantly different from the true errors. While the MSE method and the PEMF method perform slightly better. In a word, the MSE method can better reflect the variation of the true error of the MF surrogate model under different numbers of LF samples. However, no error metric can accurately estimate the magnitude of the true error.

2.2 Influence of the noise

During the sample generation process in engineering design, there are inevitably some noises existing in the response of sample points. To study the influence of the noise on the performance of these error metrics, different levels of noises are added into the HF responses:

$$\hat{y}_{{{\text{nmf}}}} (x) = \hat{y}_{{{\text{mf}}}} (x) + l^{\prime}\delta ,$$

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where $\hat{y}_{{{\text{mf}}}} ( \cdot )$ is the original response from the MF surrogate model, $\hat{y}_{{{\text{mf}}\_{\text{n}}}} ( \cdot )$ is the responses with noises, $l^{\prime}$ is the noisy level ranging from 0 to 15%, and $\delta$ is randomly selected from the normal distribution $N(0,1)$. Four different noise levels are set for comparison, as shown in Table 11. The numbers of HF samples and LF samples are fixed to 9d ($N_{{\text{h}}} = 9d$) and 25d ($N_{{\text{l}}} = 25d$) respectively. The test results are shown in Fig. 17 and Table 12. In the table, the p values that are larger than 0.05 are marked in bold, while p values that are smaller than 0.05 are marked in italic.

Table 11 The test setting for investigating the influence of noise

Fig. 17

Performance comparison of the four error metrics under different noise levels

Table 12 T test results under different noise levels

According to the uncertainty quantification results in P1 and P4 problems, it can be seen that the results of the LOO method cannot accurately reflect the variation of the true error in low-dimensional problems. In Fig. 17d, the true error of the MF surrogate model becomes larger with the increasing noise level, but the results of the LOO method gradually decrease. On the other hand, the bootstrap method and the MSE method can reflect the change of the true error accurately. The results in Table 12 also confirm this conclusion: the estimated uncertainty by the bootstrap method is not significantly different from the true error in P1, P3, and P5 problems, especially the pvalue in P5 problem is 0.9729, which is far larger than the confidence level 0.05. The MSE method and the PEMF method perform well in two problems, while the LOO method only performs well in one problem.

2.3 Influence of the correlation between the HF and LF model

The LF model is generally used to reflect the trend of the HF responses; therefore, the correlation between the HF model and the LF model has a vital influence on the accuracy of the MF model. P3 and P6 test problems are used to test the performance of the four error metrics under different correlation coefficients between the HF model and the LF model. Seven different LF models are selected for each HF model, among which the correlation coefficient between the HF and LF model ranges from 0.2 to 0.6. The numbers of HF and LF samples are set to 6d and 25d, respectively. The comparison results are listed in Fig. 18.

Fig. 18

Performance comparison of the four error metrics under different correlation coefficients between the HF and LF model: a comparison results in P3 problem; b comparison results in P6 problem

It can be seen that the errors estimated by the bootstrap method are the closest to the change of the true error in the P2 example, while the results from the LOO method differ most from the variation of the true error. When the correlation coefficient between the HF and LF model changes from 0.3 to 0.4, the true error tends to increase. Among all of the four error metrics, only the results from the bootstrap method and the MSE method can reflect this change. In the P6 problem, it can be seen that the estimated errors by the bootstrap method and MSE method are very close, which can reflect the change of the true error well.