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A multi-fidelity surrogate modeling method based on variance-weighted sum for the fusion of multiple non-hierarchical low-fidelity data

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Abstract

Multi-fidelity (MF) surrogate models have been widely adopted in simulation-based engineering design problems to reduce the computational cost by fusing data with diverse fidelity levels. Most of the MF modeling methods only apply to the problems with hierarchical low-fidelity (LF) models. However, the LF models obtained from different simplification approaches often vary in fidelity levels throughout the design space, namely, the multiple LF models are non-hierarchical. To address this challenge, a MF surrogate modeling method based on variance-weighted sum (VWS-MFS) is developed to flexibly handle multiple non-hierarchical LF data in this work. Firstly, each set of the non-hierarchical LF data is allocated diverse weights according to uncertainties quantified by variances of constructed Kriging models, which enables all the LF data to be fused and contribute to the trend function reflecting the response trend of the true model. Secondly, for more precise scaling factor between HF and LF models and mean square error (MSE) estimation, an improved hierarchical kriging (IHK) model is introduced to construct the MF surrogate model enabling the LF model scaled by a varied scaling factor to capture the characteristics of the HF model. The performance of the proposed VWS-MFS method is compared to three MF surrogate models through several numerical examples and one engineering case. Results show that the proposed method provides more accurate MF surrogate models under the same computational cost. Additionally, the proposed method saved the computational cost by more than 59.61% with the same model accuracy compared to the Kriging model built with HF data for the engineering case.

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Funding

This research has been supported by the National Natural Science Foundation of China (NSFC) under Grant Nos. 52175231, 51775203, 51805179, and 51721092, the China Postdoctoral Science Foundation under Grant No. 2020M682396, and the Research Funds of the Maritime Defense Technologies Innovation under Grant YT19201901.

Author information

Authors and Affiliations

  1. The State Key Laboratory of Digital Manufacturing Equipment and Technology, School of Mechanical Science and Engineering, Huazhong University of Science and Technology, Wuhan, 430074, Hubei, People’s Republic of China

    Meng Cheng, Ping Jiang & Leshi Shu

  2. School of Aerospace Engineering, Huazhong University of Science and Technology, Wuhan, 430074, Hubei, People’s Republic of China

    Jiexiang Hu & Qi Zhou

Authors
  1. Meng Cheng
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  2. Ping Jiang
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  3. Jiexiang Hu
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  4. Leshi Shu
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  5. Qi Zhou
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Corresponding author

Correspondence to Qi Zhou.

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Conflict of interest

The authors declare that they have no conflict of interest.

Replication of results

The main step for applying the validation framework has been presented in Sect. 3. To help readers understand better, the toolbox could be downloaded from the website: https://pan.baidu.com/s/1SDshG37cywz1Eu42niq51g by using the code cclq.

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Responsible Editor: Byeng D Youn

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Appendix A

Appendix A

The expressions of eight examples used in Subsect. 4.2 are listed.

Example 1

$$\begin{gathered} y^{H} = 2\sin \left( {{{\pi x} \mathord{\left/ {\vphantom {{\pi x} 5}} \right. \kern-\nulldelimiterspace} 5}} \right) y_{1}^{L} = {{x\left( {x - 5} \right)\left( {x - 12} \right)} \mathord{\left/ {\vphantom {{x\left( {x - 5} \right)\left( {x - 12} \right)} {30}}} \right. \kern-\nulldelimiterspace} {30}} y_{2}^{L} = {{\left( {x + 2} \right)\left( {x - 5} \right)\left( {x - 10} \right)} \mathord{\left/ {\vphantom {{\left( {x + 2} \right)\left( {x - 5} \right)\left( {x - 10} \right)} {30}}} \right. \kern-\nulldelimiterspace} {30}} 0 \le x \le 10 \end{gathered}$$

Example 2

$$\begin{gathered} y^{H} = 4x_{1}^{2} - 2.1x_{1}^{4} + {{x_{1}^{6} } \mathord{\left/ {\vphantom {{x_{1}^{6} } 3}} \right. \kern-\nulldelimiterspace} 3} + x_{1} x_{2} - 4x_{2}^{2} + 4x_{2}^{4} y_{1}^{L} = y^{H} \left( {0.7x_{1} ,0.7x_{2} } \right) + x_{1} x_{2} - 65 y_{2}^{L} = y^{H} \left( {0.8x_{1} ,0.6x_{2} } \right) + x_{1}^{4} + 32 x_{1} ,x_{2} \in \left[ { - 2,2} \right] \end{gathered}$$

Example 3

$$\begin{gathered} y^{H} = - \sin x_{1} - e^{{{{x_{1} } \mathord{\left/ {\vphantom {{x_{1} } {100}}} \right. \kern-\nulldelimiterspace} {100}}}} + 10 + {{x_{2}^{2} } \mathord{\left/ {\vphantom {{x_{2}^{2} } {10}}} \right. \kern-\nulldelimiterspace} {10}} y_{1}^{L} = - \sin \left( {0.9x_{1} } \right) - e^{{{{9x_{1} } \mathord{\left/ {\vphantom {{9x_{1} } {1000}}} \right. \kern-\nulldelimiterspace} {1000}}}} + 9.7 + {{x_{1}^{2} } \mathord{\left/ {\vphantom {{x_{1}^{2} } {10}}} \right. \kern-\nulldelimiterspace} {10}}0 + {{x_{2}^{2} } \mathord{\left/ {\vphantom {{x_{2}^{2} } {10}}} \right. \kern-\nulldelimiterspace} {10}} \hfill \\ y_{2}^{L} = - \sin x_{1} - e^{{{{x_{1} } \mathord{\left/ {\vphantom {{x_{1} } {100}}} \right. \kern-\nulldelimiterspace} {100}}}} + 10.3 + 0.03\left( {x_{1} - 0.3} \right){{^{2} + \left( {x_{2} - 1} \right)^{2} } \mathord{\left/ {\vphantom {{^{2} + \left( {x_{2} - 1} \right)^{2} } {10}}} \right. \kern-\nulldelimiterspace} {10}} y_{3}^{L} = - \sin \left( {0.9x_{1} } \right) - e^{{{{9x_{1} } \mathord{\left/ {\vphantom {{9x_{1} } {1000}}} \right. \kern-\nulldelimiterspace} {1000}}}} + 10 + 0.64{{x_{2}^{2} } \mathord{\left/ {\vphantom {{x_{2}^{2} } {10}}} \right. \kern-\nulldelimiterspace} {10}} x_{1} ,x_{2} ,x_{3} \in \left[ {0,1} \right] \end{gathered}$$

Example 4

$$\begin{gathered} y^{H} = {{x_{1} } \mathord{\left/ {\vphantom {{x_{1} } {2\left( {\sqrt {1 + \left( {x_{1} + x_{3}^{2} } \right){{x_{4} } \mathord{\left/ {\vphantom {{x_{4} } {x_{1}^{20} }}} \right. \kern-\nulldelimiterspace} {x_{1}^{20} }}} - 1} \right)}}} \right. \kern-\nulldelimiterspace} {2\left( {\sqrt {1 + \left( {x_{1} + x_{3}^{2} } \right){{x_{4} } \mathord{\left/ {\vphantom {{x_{4} } {x_{1}^{20} }}} \right. \kern-\nulldelimiterspace} {x_{1}^{20} }}} - 1} \right)}} + \left( {x_{1} + 3x_{4} } \right)e^{{\left( {1 + \sin x_{3} } \right)}} \hfill \\ y_{1}^{L} = 0.79\left( {1 + \sin {{x_{1} } \mathord{\left/ {\vphantom {{x_{1} } {10}}} \right. \kern-\nulldelimiterspace} {10}}} \right)y^{H} - 2x_{1} + x_{2}^{2} + x_{3}^{2} + 0.5 \hfill \\ y_{2}^{L} = y^{H} + e^{{{{x_{3} } \mathord{\left/ {\vphantom {{x_{3} } 2}} \right. \kern-\nulldelimiterspace} 2}}} - {{x_{1} } \mathord{\left/ {\vphantom {{x_{1} } {100}}} \right. \kern-\nulldelimiterspace} {100}} \hfill \\ x_{1} ,x_{2} ,x_{3} ,x_{4} \in \left[ {0.5,1} \right] \hfill \\ \end{gathered}$$

Example 5

$$\begin{gathered} y^{H} = \left( {x_{1} - 1} \right)^{2} + 2\left( {2x_{2}^{2} - x_{1} } \right)^{2} + 3\left( {3x_{3}^{2} - x_{2} } \right)^{2} + 4\left( {4x_{4}^{2} - x_{3} } \right) \hfill \\ y_{1}^{L} = \left( {x_{1} - 1} \right)^{2} + 2\left( {2x_{2}^{2} - 0.75x_{1} } \right)^{2} + 3\left( {3x_{3}^{2} - 0.75x_{2} } \right)^{2} + 4\left( {4x_{4}^{2} - 0.75x_{3} } \right)^{2} \hfill \\ y_{2}^{L} = \left( {x_{1} - 1} \right)^{2} + 2\left( {2x_{2}^{2} - 1.25x_{1} } \right)^{2} + 3\left( {3x_{3}^{2} - 1.25x_{2} } \right)^{2} + 4\left( {4x_{4}^{2} - 1.25x_{3} } \right)^{2} \hfill \\ y_{3}^{L} = \left( {x_{1} - 1} \right)^{2} + 2\left( {2.25x_{2}^{2} - x_{1} } \right)^{2} + 3\left( {3.25x_{3}^{2} - x_{2} } \right)^{2} + 4\left( {4.25x_{4}^{2} - x_{3} } \right)^{2} \hfill \\ x_{1} ,x_{2} ,x_{3} ,x_{4} \in \left[ { - 10,10} \right] \hfill \\ \end{gathered}$$

Example 6

$$\begin{gathered} y^{H} = [100\left( {x_{2} - x_{1}^{2} } \right)^{2} + \left( {x_{1} - 1} \right)^{2} + 100\left( {x_{3} - x_{2}^{2} } \right)^{2} + \left( {x_{2} - 1} \right)^{2} + 100\left( {x_{4} - x_{3}^{2} } \right)^{2} \hfill \\ \, {{ + \left( {x_{3} - 1} \right)^{2} + 100\left( {x_{5} - x_{4}^{2} } \right)^{2} + \left( {x_{4} - 1} \right)^{2} + 100\left( {x_{6} - x_{5}^{2} } \right)^{2} + \left( {x_{5} - 1} \right)^{2} ]} \mathord{\left/ {\vphantom {{ + \left( {x_{3} - 1} \right)^{2} + 100\left( {x_{5} - x_{4}^{2} } \right)^{2} + \left( {x_{4} - 1} \right)^{2} + 100\left( {x_{6} - x_{5}^{2} } \right)^{2} + \left( {x_{5} - 1} \right)^{2} ]} {100000}}} \right. \kern-\nulldelimiterspace} {100000}} \hfill \\ y_{1}^{L} = [100\left( {x_{2} - x_{1}^{2} } \right)^{2} + 4\left( {x_{1} - 1} \right)^{2} + 100\left( {x_{3} - x_{2}^{2} } \right)^{2} + 4\left( {x_{2} - 1} \right)^{2} + 100\left( {x_{4} - x_{3}^{2} } \right)^{2} \hfill \\ \, {{ \, + 4\left( {x_{3} - 1} \right)^{2} + 100\left( {x_{5} - x_{4}^{2} } \right)^{2} + 4\left( {x_{4} - 1} \right)^{2} + 100\left( {x_{6} - x_{5}^{2} } \right)^{2} + 4\left( {x_{5} - 1} \right)^{2} ]} \mathord{\left/ {\vphantom {{ \, + 4\left( {x_{3} - 1} \right)^{2} + 100\left( {x_{5} - x_{4}^{2} } \right)^{2} + 4\left( {x_{4} - 1} \right)^{2} + 100\left( {x_{6} - x_{5}^{2} } \right)^{2} + 4\left( {x_{5} - 1} \right)^{2} ]} {100000}}} \right. \kern-\nulldelimiterspace} {100000}} \hfill \\ y_{2}^{L} = [80\left( {x_{2} - x_{1}^{2} } \right)^{2} + \left( {x_{1} - 1} \right)^{2} + 80\left( {x_{3} - x_{2}^{2} } \right)^{2} + \left( {x_{2} - 1} \right)^{2} + 80\left( {x_{4} - x_{3}^{2} } \right)^{2} \hfill \\ \, {{ + \left( {x_{3} - 1} \right)^{2} + 80\left( {x_{5} - x_{4}^{2} } \right)^{2} + \left( {x_{4} - 1} \right)^{2} + 80\left( {x_{6} - x_{5}^{2} } \right)^{2} + \left( {x_{5} - 1} \right)^{2} ]} \mathord{\left/ {\vphantom {{ + \left( {x_{3} - 1} \right)^{2} + 80\left( {x_{5} - x_{4}^{2} } \right)^{2} + \left( {x_{4} - 1} \right)^{2} + 80\left( {x_{6} - x_{5}^{2} } \right)^{2} + \left( {x_{5} - 1} \right)^{2} ]} {100000}}} \right. \kern-\nulldelimiterspace} {100000}} \hfill \\ y_{3}^{L} = [100\left( {x_{2} - x_{1}^{2} } \right)^{2} + 100\left( {x_{3} - x_{2}^{2} } \right)^{2} + 100\left( {x_{4} - x_{3}^{2} } \right)^{2} + 100\left( {x_{5} - x_{4}^{2} } \right)^{2} \hfill \\ \, {{ + 100\left( {x_{6} - x_{5}^{2} } \right)^{2} ]} \mathord{\left/ {\vphantom {{ + 100\left( {x_{6} - x_{5}^{2} } \right)^{2} ]} {100000}}} \right. \kern-\nulldelimiterspace} {100000}} \hfill \\ x_{1} ,x_{2} ,x_{3} ,x_{4} ,x_{5} ,x_{6} \in \left[ { - 5,10} \right] \hfill \\ \end{gathered}$$

Example 7

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

Example 8

$$\begin{gathered} y^{H} = \sum\limits_{i = 1}^{10} {x_{i}^{3} } + \left( {\sum\limits_{i = 1}^{10} {0.5ix_{i} } } \right)^{2} + \left( {\sum\limits_{i = 1}^{10} {0.5ix_{i} } } \right)^{4} \hfill \\ y_{1}^{L} = \sum\limits_{i = 1}^{10} {x_{i}^{3} } + \left( {\sum\limits_{i = 1}^{10} {2ix_{i} } } \right)^{2} + \left( {\sum\limits_{i = 1}^{10} {3ix_{i} } } \right)^{4} \hfill \\ y_{2}^{L} = \sum\limits_{i = 1}^{10} {x_{i}^{3} } + \left( {\sum\limits_{i = 1}^{10} {3ix_{i} } } \right)^{2} + \left( {\sum\limits_{i = 1}^{10} {4ix_{i} } } \right)^{4} \hfill \\ y_{3}^{L} = \sum\limits_{i = 1}^{10} {x_{i}^{3} } + \left( {\sum\limits_{i = 1}^{10} {ix_{i} } } \right)^{2} + \left( {\sum\limits_{i = 1}^{10} {2ix_{i} } } \right)^{4} \hfill \\ x_{i} \in \left[ { - 5,10} \right] \hfill \\ \end{gathered}$$

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Cheng, M., Jiang, P., Hu, J. et al. A multi-fidelity surrogate modeling method based on variance-weighted sum for the fusion of multiple non-hierarchical low-fidelity data. Struct Multidisc Optim 64, 3797–3818 (2021). https://doi.org/10.1007/s00158-021-03055-2

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