CK-IBNN surrogate model for aerodynamic performance prediction of multi-stage axial flow compressor
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摘要:
CoKriging模型在百千维变量问题上的建模代价往往难以承受，以致多级压气机叶片形状优化的百千维问题无法求解。为打破CoKriging模型的维数诅咒问题，建立了一种基于无限宽度贝叶斯神经网络（IBNN）的关联函数及基于IBNN关联函数的CoKriging模型。理论分析显示IBNN关联函数不利用欧式空间距离评价任两样本点之间的关联程度，且关联函数的超参数对任意维度问题均为3个，与变量个数无关。因此，所提方法的建模代价可显著降低。为验证所提方法的有效性和高效性，求解了5维、31维、144维和
1512 维的压气机气动建模问题，并与多保真度深度神经网络、分层Kriging等方法相比，结果显示所提方法可快速建立高精度代理模型，仅需0.1 s和7 s即可完成144维和1512 维问题的压气机气动性能预测模型的建立。-
关键词:
- 代理模型 /
- CoKriging模型 /
- 维数诅咒 /
- 多级压气机 /
- 气动建模
Abstract:The prohibitive computational cost of the model construction of CoKriging for hundreds or thousands of variables prevents the practical application of CoKriging. Therefore, the surrogate-based optimization design of multi-stage axial flow compressor with hundreds or thousands of variables cannot be conducted. To break the curse of dimensionality of the CoKriging model, a correlation function based on infinite-width Bayesian neural network (IBNN) along with CoKriging modeling method utilizing the IBNN correlation function was established. The theoretical analysis demonstrated that the IBNN correlation function was not dependent on the spatial distance to measure the similarity of any two points. Furthermore, the number of hyperparameters of the IBNN function was fixed as 3, being independent of the number of modeling variables like the existing correlation functions. Therefore, the modeling efficiency can be decreased significantly. To verify the effectiveness and efficiency of the proposed method, aerodynamic problems of axial flow compressor with 5, 31, 144 and
1512 variables were solved. Performance of the proposed method was compared with multi-fidelity deep neural network, and Hierarchical Kriging, etc. The proposed method can build the surrogate models efficiently with adequate accuracy. The surrogate models for the aero-dynamic performance prediction with 144 and1512 variables can be tuned within 0.1 and 7 seconds. -
图 2 多隐藏层无限宽度神经网络示意图
Figure 2. Illustration of multiple hidden layer infinite-width neural network
图 3 CK-IBNN模型建模及预测流程
Figure 3. Flowchart of the modeling and prediction of CK-IBNN model
图 6 10.5级压气机静叶转角调整CFD仿真用网格
Figure 6. Meshes for the CFD simulation of the 10.5-stage compressor under variable guide vane
图 11 10.5级压气机叶片变形时CFD仿真用网格
Figure 11. Meshes for the CFD simulation of the 10.5-stage compressor under blade deformation
图 13 静叶角度调整时的10.5级压气机预测值和CFD仿真值对比
Figure 13. Predicted and CFD simulated value of the aerodynamic performance of the 10.5-stage compressor under variable guide vane
图 14 Rotor 37叶片形变时气动性能预测值和CFD仿真值对比
Figure 14. Comparison of predicted and CFD simulated value of the aerodynamic performance of the Rotor 37 compressor under blade shape deformation
图 15 某3级叶片形变时气动性能预测值和CFD仿真值对比
Figure 15. Comparison of predicted and CFD simulated value of the aerodynamic performance of a 3-stage compressor under blade shape deformation
图 16 某10.5级叶片形变时气动性能预测值和CFD仿真值对比
Figure 16. Comparison of predicted and CFD simulated value of the aerodynamic performance of a 10.5-stage compressor under blade shape deformation
表 1 常见关联函数
Table 1. Representative correlation functions
函数名称 表达式 EXP $ \exp \left(\displaystyle\sum_{k=1}^{d}-\theta_{k} \left| x_{k}-x_{k}^{\prime}\right|\right) $ EXPG $ \exp \left(\displaystyle\sum_{k=1}^{d}-\theta_{k} \left| x_{k}-x_{k}^{\prime}\right|^{\theta_{d+1}}\right)\quad\quad 0<\theta_{d+1}<2 $ Gaussian $ \exp \left(\displaystyle\sum_{k=1}^{d}-\theta_{k}\left| x_{k}-x_{k}^{\prime}\right|^2\right) $ Linear $\displaystyle\prod_{k=1}^{d}\max\{0,1-\theta_k|x_k-x'_k|\} $ Cubic $ \begin{array}{l}\displaystyle\prod_{k=1}^{d} 1-3 \xi_{k}^{2}+2 \xi_{k}^{3} \\\xi_{k}=\min \left\{0, \theta_{k} \mid x_{k}-x_{k}^{\prime}\mid\right\}\end{array} $ Spline $\displaystyle\prod_{k=1}^{d}\zeta （\xi_k） ,\xi_k=\theta_k|x_k-x'_k| $ Matérn 5/2 $ \begin{array}{l}\left(1+\sqrt{5}a+\dfrac{5a^2}{3}\right)\exp\left(-\sqrt{5}a\right) \\a=\sqrt{\displaystyle\sum_{k=1}^d\theta_k\left|x_k-x'_k\right|^2}\end{array} $ 
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表 2 测试算例信息汇总
Table 2. Information of test problems
$d$ ${N_{\text{l}}}$ $N$ ${N_{\text{t}}}$ 问题名称 5 50 10 10 10StgVGV-$\pi $ 10StgVGV-$\eta $ 10StgVGV-$ \dot m $ 31 296 164 285 Rotor 37-$\eta $ Rotor 37-$\pi $ 144 1168 520 384 3StgDef-$\eta $ 3StgDef-$\pi $ 3StgDef-$ \dot m $ 1512 15120 7560 3024 10StgDef-$\pi $ 10StgDef-$\eta $ 10StgDef-$ \dot m $
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表 3 MFDNN训练参数
Table 3. Training parameters of MFDNN
层数 宽度 最大步数 问题名称 3 40 30000 10StgVGV-$\pi $ 10StgVGV-$\eta $ 10StgVGV-$ \dot m $ 10 100 40000 Rotor 37-$\eta $ Rotor 37-$\pi $ 12 600 40000 3StgDef-$\eta $ 3StgDef-$\pi $ 3StgDef-$ \dot m $ 15 800 40000 10StgDef-$\pi $ 10StgDef-$\eta $ 10StgDef-$ \dot m $
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表 4 R2指标对比
Table 4. Comparison of R2 metric
问题名称 MFDNN K-DIC HK-DIC K-IBNN CK-IBNN 10StgVGV-$\pi $ 0.9647 0.0053 0.9855 0.8684 0.9841 10StgVGV-$\eta $ 0.3624 − 0.1303 0.7684 0.7411 0.8669 10StgVGV-$ \dot m $ 0.4581 − 0.0093 0.9797 0.8594 0.9512 Rotor 37-$\eta $ 0.6179 0.8694 0.9686 0.8457 0.9554 Rotor 37-$\pi $ 0.9131 0.9834 0.9953 0.9710 0.9935 3StgDef-$\eta $ 0.9508 0.9610 0.9736 0.9054 0.9589 3StgDef-$\pi $ 0.8574 0.8715 0.9048 0.8244 0.8789 3StgDef-$ \dot m $ 0.9252 0.9386 0.9575 0.8630 0.9371 10StgDef-$\pi $ 0.8726 0.9404 0.9661 10StgDef-$\eta $ − 9.8144 0.6649 0.6808 10StgDef-$ \dot m $ 0.9118 0.9193 0.9406
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表 5 ERMSE指标对比
Table 5. Comparison of ERMSE metric
问题名称 MFDNN K-DIC HK-DIC K-IBNN CK-IBNN 10StgVGV-$\pi $ 0.1784 0.9462 0.1141 0.3441 0.1197 10StgVGV-$\eta $ 0.7575 1.0086 0.4566 0.4827 0.3461 10StgVGV-$ \dot m $ 0.6983 0.9531 0.1351 0.3558 0.2097 Rotor 37-$\eta $ 0.6170 0.3607 0.1769 0.3921 0.2108 Rotor 37-$\pi $ 0.2942 0.1285 0.0681 0.1699 0.0803 3StgDef-$\eta $ 0.2216 0.1971 0.1622 0.3071 0.2024 3StgDef-$\pi $ 0.3771 0.3579 0.3082 0.4185 0.3475 3StgDef-$ \dot m $ 0.2731 0.2475 0.2060 0.3697 0.2505 10StgDef-$\pi $ 0.3569 0.2440 0.1842 10StgDef-$\eta $ 3.2880 0.5788 0.5649 10StgDef-$ \dot m $ 0.2969 0.2840 0.2437
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表 6 EMAE指标对比
Table 6. Comparison of EMAE metric
问题名称 MFDNN K-DIC HK-DIC K-IBNN CK-IBNN 10StgVGV-$\pi $ 0.4505 1.8576 0.2100 0.5880 0.2091 10StgVGV-$\eta $ 1.6343 1.4028 1.1866 0.9220 0.7485 10StgVGV-$ \dot m $ 1.8306 1.9491 0.2299 0.6632 0.3987 Rotor 37-$\eta $ 2.6992 1.9322 1.7223 1.9705 2.0445 Rotor 37-$\pi $ 0.9830 0.7303 0.4902 0.7794 0.6312 3StgDef-$\eta $ 0.6965 0.7204 0.5286 1.1033 0.7095 3StgDef-$\pi $ 2.0361 1.9271 1.9778 1.9177 2.0978 3StgDef-$ \dot m $ 1.1077 0.9200 0.8396 1.3683 0.8606 10StgDef-$\pi $ 1.5141 1.1538 1.2180 10StgDef-$\eta $ 14.3835 4.2887 3.8824 10StgDef-$ \dot m $ 1.6787 1.7340 1.9766
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表 7 建模时间对比
Table 7. Comparison of modeling time
问题名称 MFDNN K-DIC HK-DIC K-IBNN CK-IBNN 10StgVGV-$\pi $ 234.6345 0.0028 0.0087 0.0003 0.0004 10StgVGV-$\eta $ 259.8951 0.0019 0.0093 0.0004 0.0003 10StgVGV-$ \dot m $ 259.7941 0.0021 0.0097 0.0001 0.0004 Rotor 37-$\eta $ 712.4516 0.0511 0.1825 0.0051 0.0068 Rotor 37-$\pi $ 718.8171 0.0621 0.2174 0.0013 0.0073 3StgDef-$\eta $ 797.9772 2.5133 13.5913 0.0156 0.0956 3StgDef-$\pi $ 793.5154 2.2602 12.9230 0.0120 0.0932 3StgDef-$ \dot m $ 801.3981 2.1465 12.9916 0.0122 0.0953 10StgDef-$\pi $ 6723.5427 3.1202 6.5156 10StgDef-$\eta $ 6466.9472 2.9386 6.5598 10StgDef-$ \dot m $ 6801.4526 3.0376 6.6351
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表 8 不同M值的CK-IBNN模型R2值
Table 8. R2 metric value of CK-IBNN model with various M
问题名称 M=1 M=2 M=3 M=4 M=5 10StgVGV-$\pi $ 0.9823 0.9829 0.9841 0.9844 0.9844 10StgVGV-$\eta $ 0.8624 0.8561 0.8669 0.8730 0.8759 10StgVGV-$ \dot m $ 0.9354 0.9470 0.9512 0.9536 0.9554 Rotor 37-$\eta $ 0.9463 0.9531 0.9554 0.9565 0.9571 Rotor 37-$\pi $ 0.9923 0.9933 0.9935 0.9934 0.9932 3StgDef-$\eta $ 0.9560 0.9580 0.9589 0.9592 0.9593 3StgDef-$\pi $ 0.8726 0.8767 0.8789 0.8800 0.8807 3StgDef-$ \dot m $ 0.9328 0.9354 0.9371 0.9381 0.9388 10StgDef-$\pi $ 0.9691 0.9679 0.9661 0.9638 0.9614 10StgDef-$\eta $ 0.6790 0.6813 0.6808 0.6786 0.6756 10StgDef-$ \dot m $ 0.9443 0.9429 0.9406 0.9378 0.9348
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表 9 不同M值的CK-IBNN模型EMAE值
Table 9. EMAE metric value of CK-IBNN model with various M
问题名称 M=1 M=2 M=3 M=4 M=5 10StgVGV-$\pi $ 0.2690 0.1986 0.2091 0.2358 0.2570 10StgVGV-$\eta $ 0.7283 0.7683 0.7485 0.7192 0.6884 10StgVGV-$ \dot m $ 0.4798 0.4392 0.3987 0.3630 0.3658 Rotor 37-$\eta $ 1.9332 2.0129 2.0445 2.0585 2.0645 Rotor 37-$\pi $ 0.5648 0.6011 0.6312 0.6562 0.6777 3StgDef-$\eta $ 0.7090 0.7089 0.7095 0.7103 0.7115 3StgDef-$\pi $ 2.1331 2.1117 2.0978 2.0883 2.0817 3StgDef-$ \dot m $ 0.8754 0.8690 0.8606 0.8522 0.8445 10StgDef-$\pi $ 1.3230 1.2661 1.2180 1.1755 1.1375 10StgDef-$\eta $ 3.8006 3.8451 3.8824 3.9146 3.9429 10StgDef-$ \dot m $ 1.9297 1.9557 1.9766 1.9941 2.0093
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表 10 不同${\boldsymbol{\sigma}} _{\bf{w}}^{\bf{2}}$值的CK-IBNN模型精度指标值
Table 10. Metric value of CK-IBNN model with various ${\boldsymbol{\sigma}} _{\bf{w}}^{\bf{2}}$
问题名称 R2 MAE $\sigma _{\text{w}}^2$=5 $\sigma _{\text{w}}^2$=10 $\sigma _{\text{w}}^2$=15 $\sigma _{\text{w}}^2$=5 $\sigma _{\text{w}}^2$=10 $\sigma _{\text{w}}^2$=15 10StgVGV-$\pi $ 0.9851 0.9841 0.9834 0.1935 0.2091 0.2146 10StgVGV-$\eta $ 0.8717 0.8669 0.8635 0.7312 0.7485 0.7600 10StgVGV-$ \dot m $ 0.9538 0.9512 0.9494 0.3966 0.3987 0.4022 Rotor 37-$\eta $ 0.9548 0.9554 0.9555 2.0493 2.0445 2.0425 Rotor 37-$\pi $ 0.9936 0.9935 0.9934 0.6268 0.6312 0.6325 3StgDef-$\eta $ 0.9588 0.9589 0.9589 0.7181 0.7095 0.7057 3StgDef-$\pi $ 0.8777 0.8789 0.8792 2.1007 2.0978 2.0969 3StgDef-$ \dot m $ 0.9361 0.9371 0.9374 0.8663 0.8606 0.8586 10StgDef-$\pi $ 0.9671 0.9661 0.9656 1.2385 1.2180 1.2104 10StgDef-$\eta $ 0.6813 0.6808 0.6804 3.8591 3.8824 3.8913 10StgDef-$ \dot m $ 0.9419 0.9406 0.9400 1.9707 1.9766 1.9784
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表 11 不同${\boldsymbol{\sigma}} _{\bf{b}}^{\bf{2}}$值的CK-IBNN模型精度指标值
Table 11. Metric value of CK-IBNN model with various ${\boldsymbol{\sigma}} _{\bf{b}}^{\bf{2}}$
问题名称 R2 MAE $\sigma _{\text{b}}^2$=1.2 $\sigma _{\text{b}}^2$=1.6 $\sigma _{\text{b}}^2$=2.0 $\sigma _{\text{b}}^2$=1.2 $\sigma _{\text{b}}^2$=1.6 $\sigma _{\text{b}}^2$=2.0 10StgVGV-$\pi $ 0.9837 0.9841 0.9843 0.2127 0.2091 0.2058 10StgVGV-$\eta $ 0.8650 0.8669 0.8681 0.7550 0.7485 0.7437 10StgVGV-$ \dot m $ 0.9501 0.9512 0.9519 0.4003 0.3987 0.3978 Rotor 37-$\eta $ 0.9555 0.9554 0.9552 2.0429 2.0445 2.0459 Rotor 37-$\pi $ 0.9934 0.9935 0.9935 0.6321 0.6312 0.6303 3StgDef-$\eta $ 0.9589 0.9589 0.9589 0.7069 0.7095 0.7116 3StgDef-$\pi $ 0.8791 0.8789 0.8786 2.0972 2.0978 2.0984 3StgDef-$ \dot m $ 0.9373 0.9371 0.9368 0.8592 0.8606 0.8619 10StgDef-$\pi $ 0.9657 0.9661 0.9663 1.2128 1.2180 1.2227 10StgDef-$\eta $ 0.6805 0.6808 0.6809 3.8885 3.8824 3.8769 10StgDef-$ \dot m $ 0.9402 0.9406 0.9409 1.9778 1.9766 1.9753
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