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A novel multi-fidelity surrogate modeling method for non-hierarchical data fusion

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Abstract

Multi-fidelity (MF) surrogate model has been widely used in simulation-based engineering design processes to reduce the computational cost, with a focus on cases involving hierarchical low-fidelity (LF) data. However, accurately identifying and sorting the fidelity of LF models is challenging when dealing with non-hierarchical cases. In this paper, we propose a novel non-hierarchical MF surrogate framework called weighted multi-bi-fidelity (WMBF) to solve this problem. The proposed WMBF has both the advantage of two non-hierarchical frameworks, the weighted sum (WS) and parallel combination (PC) techniques, leveraging an entropy-based weight to include multiple-moments statistical information. It offers not only a weight with more information but also a more individualized scaling function within the weighted-sum framework, additionally a more individualized discrepancy function compared with existing methods. Moreover, it provides the idea of exploiting Kullback–Leibler (KL) divergence (an entropy-based metric) to characterize uncertainty for calculating weight within the WS framework. To validate the performance of the WMBF, we conduct evaluations using several numerical test functions and one engineering case. The result demonstrates that the WMBF achieves both accurate and robust predictions with minimal computational cost.

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Data Availability

The MATLAB codes and the simulation data used to generate results are available upon request. All data and codes are available from the author upon reasonable request.

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Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant No. 12201656), Science and Technology Projects in Guangzhou (Grant No. SL2024A04J01579) and Key Laboratory of Information Systems Engineering (CN).

Funding

This work was supported by the National Natural Science Foundation of China (Grant No. 12201656), Science and Technology Projects in Guangzhou (Grant No. SL2024A04J01579) and Key Laboratory of Information Systems Engineering (CN).

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Authors and Affiliations

  1. School of Systems Science and Engineering, Sun Yat-Sen University, Guangzhou, 510006, Guangdong, China

    Shan Xie, Hanyan Huang, Xiaoyu Xu, Yanhui Duan & Hongbo Chen

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  1. Shan Xie
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  3. Xiaoyu Xu
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Contributions

Conceptualization and Funding acquisition: HH; Methodology and analysis: SX; Resources-simulation software design: YD; Resources-simulation software calculation: XX; Writing-original draft preparation: SX; Writing-review and editing: HH; Supervision: HC.

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Correspondence to Hanyan Huang.

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Appendices

Appendix 1: The numerical example expressions

Example 1

$$\begin{aligned} \begin{aligned} y_{{\textrm{H}}}&= \sin {x}\\ y_{{\textrm{L1}}}&= y_{H} + 0.1(x-\pi )^{2}\\ y_{{\textrm{L2}}}&= 1.2y_{H} + 0.1(x-\pi )^{2} - 0.2 \\ 0&\le x \le 2\pi \end{aligned} \end{aligned}$$
(31)

Example 2

$$\begin{aligned} \begin{aligned} y_{{\textrm{H}}}&= (6x-2)^{2}\sin {12x-4}\\ y_{{\textrm{L1}}}&= 0.5y_{H} +10(x-0.5)+5\\ y_{{\textrm{L2}}}&= 0.4y_{H} -x-1 \\ y_{{\textrm{L3}}}&= 0.3y_{H} 10x +6\\ 0&\le x \le 1 \end{aligned} \end{aligned}$$
(32)

Example 3

$$\begin{aligned} \begin{aligned} y_{{\textrm{H}}}&= 2\sin {\pi x/5}\\ y_{{\textrm{L1}}}&= x(x-5)(x-12)/30 \\ y_{{\textrm{L2}}}&= (x+2)(x-5)(x-10)/30 \\ 0&\le x \le 10 \end{aligned} \end{aligned}$$
(33)

Example 4

$$\begin{aligned} \begin{aligned} y_{{\textrm{H}}}&= 4 x_{1}^{2}-2.1x_{1}^{4}+x_{1}^{6}/3 +x_{1}x_{2} -4x_{2}^{2}+4x_{2}^{4}\\ y_{{\textrm{L1}}}&= y_{H}(0.7x_{1},0.7x_{2})+x_{1}x_{2}-65\\ y_{{\textrm{L2}}}&= y_{H}(0.8x_{1},0.6x_{2})-x_{1}^{4}+32\\ x_{1},x_{2}&\in [-2,2] \end{aligned} \end{aligned}$$
(34)

Example 5

$$\begin{aligned} \begin{aligned} y_{{\textrm{H}}}&=-\sin {x_{1}}- \exp (x_{1}/100)+x_{2}^{2}/10\\ y_{{\textrm{L1}}}&=-\sin {x_{1}}- \exp (x_{1}/100)+10.3\\ {}&\qquad +0.03(x_{1}-0.3)^{2}+(x_{2}-1)^{2}/10\\ y_{{\textrm{L2}}}&=-\sin {0.9x_{1}}- \exp (0.9x_{1}/100)+10+0.64x_{2}^{2}/10\\ x_{1},x_{2}&\in [0,1] \end{aligned} \end{aligned}$$
(35)

Example 6

$$\begin{aligned} \begin{aligned} y_{{\textrm{H}}}&= x_{1}/2\left( \sqrt{1+(x_{1}+x_{3}^{2})x_{4}/x_{1}^{20}} -1 \right) \\ {}&\qquad +(x_{1}+3x_{4})e^{(1+\sin {x_{3}})}\\ y_{{\textrm{L1}}}&= 0.79(1+sin{x_{1}}/10)y^{H} -2x_{1}+x_{2}^{2}+x_{3}^{2}+0.5\\ y_{{\textrm{L2}}}&= y_{H}+e^{x_{3}/2}-x_{1}/10\\ x_{1},x_{2},&x_{3},x_{4} \in [0.5,1] \end{aligned} \end{aligned}$$
(36)

Example 7

$$\begin{aligned} \begin{aligned} y_{{\textrm{H}}}&= \left[ 100(x_{2} - x_{1}^{2})^{2} + (x_{1} -1 )^{2} +100(x_{3}-x_{2}^{2})^{2} \right. \\&\qquad \left. +(x_{2}-1)^{2} + 100(x_{4}-x_{3}^{2})^{2} +(x_{3}-1)^{2} \right. \\ {}&\qquad \left. +100(x_{5}-x_{4}^{2})^{2} +(x_{4}-1)^{2} + 100(x_{6}-x_{5}^{2})^{2} \right. \\ {}&\qquad \left. +(x_{5}-1)^{2}\right] /100000\\ y_{{\textrm{L}}1}&= \left[ 100(x_{2} - x_{1}^{2})^{2} + 4(x_{1} -1 )^{2} +100(x_{3}-x_{2}^{2})^{2} \right. \\&\qquad \left. +4(x_{2}-1)^{2} + 100(x_{4}-x_{3}^{2})^{2} +4(x_{3}-1)^{2} \right. \\ {}&\qquad \left. +100(x_{5}-x_{4}^{2})^{2} +4(x_{4}-1)^{2} + 100(x_{6}-x_{5}^{2})^{2} \right. \\ {}&\qquad \left. +4(x_{5}-1)^{2}\right] /100000\\ y_{{\textrm{L}}2}&= \left[ 80(x_{2} - x_{1}^{2})^{2} + (x_{1} -1 )^{2} +80(x_{3}-x_{2}^{2})^{2} \right. \\&\qquad \left. +(x_{2}-1)^{2} + 80(x_{4}-x_{3}^{2})^{2} +(x_{3}-1)^{2} \right. \\ {}&\qquad \left. +80(x_{5}-x_{4}^{2})^{2} +(x_{4}-1)^{2}+ 80(x_{6}-x_{5}^{2})^{2} \right. \\ {}&\qquad \left. +(x_{5}-1)^{2}\right] /100000\\ y_{{\textrm{L}}3}&= \left[ 100(x_{2} - x_{1}^{2})^{2} +100(x_{3}-x_{2}^{2})^{2} + 100(x_{4}-x_{3}^{2})^{2} \right. \\&\qquad \left. +100(x_{5}-x_{4}^{2})^{2}+ 100(x_{6}-x_{5}^{2})^{2} \right] /100000\\ x_{i}&\in [-5,10], i=1,\ldots ,6 \end{aligned} \end{aligned}$$
(37)

Example 8

$$\begin{aligned} \begin{aligned} y_{{\textrm{H}}}&= \frac{2\pi x_{3}(x_{4}-x{6})}{\ln {(x_{2}/x{1})\left[ 1+2x_{7}x_{4}/ \left( \ln {(x_{2}/x_{1})}x_{1}^{2} x_{8}\right) + x_{3}/x_{5}\right] }}\\ y_{{\textrm{L}}1}&= 0.4y_{{\textrm{H}}}+ x_{1}^{2} x_{8} + x_{1}x_{7}/x_{3} + x_{1}x_{6}/x_{2} + x_{1}^{2}x_{4}\\ y_{{\textrm{L}}2}&= 0.6y_{{\textrm{H}}} + 10 x_{1}x_{5}/x_{2} + x_{1}x_{8}/x_{4} + x_{1}^{3}x_{7}\\ x_{1}&\in [0.05,0.15];x_{2} \in [100,50000];x_{3} \in [63070,115600];\\ x_{4}&\in [990,1110];x_{5} \in [63.1,116];x_{6} \in [700,820];\\ x_{7}&\in [1120,1680];x_{8} \in [9855,12045]; \end{aligned} \end{aligned}$$
(38)

Example 9

$$\begin{aligned} \begin{aligned} y_{{\textrm{H}}}&= \sum \limits _{i=1}^{10}{x_{i}^{3}} +\left( \sum \limits _{i=1}^{10}{0.5 i x_{i}} \right) ^{2} + \left( \sum \limits _{i=1}^{10}{0.5 i x_{i}} \right) ^{4}\\ y_{{\textrm{L}}1}&= \sum \limits _{i=1}^{10}{x_{i}^{3}} +\left( \sum \limits _{i=1}^{10}{2 i x_{i}} \right) ^{2} + \left( \sum \limits _{i=1}^{10}{3 i x_{i}} \right) ^{4}\\ y_{{\textrm{L}}2}&= \sum \limits _{i=1}^{10}{x_{i}^{3}} +\left( \sum \limits _{i=1}^{10}{3 i x_{i}} \right) ^{2} + \left( \sum \limits _{i=1}^{10}{4 i x_{i}} \right) ^{4}\\ y_{{\textrm{L}}3}&= \sum \limits _{i=1}^{10}{x_{i}^{3}} +\left( \sum \limits _{i=1}^{10}{i x_{i}} \right) ^{2} + \left( \sum \limits _{i=1}^{10}{2 i x_{i}} \right) ^{4}\\ x_{i}&\in [-5,10], i=1,\ldots ,10 \end{aligned} \end{aligned}$$
(39)

Appendix 2: The sample size in Sect. 4.2.1

The fixed sample size and rate used in Sect. 4.2.1 are listed in Table 8, which are determined according to the mode of the optimal combinations of all methods.

Table 8 The fixed sample size and ratio used in Sect. 4.2.1
Full size table

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Xie, S., Huang, H., Xu, X. et al. A novel multi-fidelity surrogate modeling method for non-hierarchical data fusion. Engineering with Computers (2024). https://doi.org/10.1007/s00366-023-01937-1

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