基于多保真深度神经网络的超声速客机声爆预测

doi:10.7527/S1000-6893.2025.31936

Abstract

Abstract:

Sonic boom is a core issue restricting the development of supersonic passenger aircraft. Accurate prediction of sonic boom signatures is a prerequisite for the noise reduction design of supersonic passenger aircraft. In engineering， the sonic boom prediction method mainly combines Computational Fluid Dynamics （CFD） and acoustic propagation theory， in which the calculation accuracy of the near-field overpressure distribution is crucial， but the acquisition cost of high-precision solutions is high. To alleviate the contradiction between the cost and accuracy of sonic boom prediction， this paper uses a multi-fidelity deep neural network to construct the mapping relationship between the aerodynamic shape and the near-field overpressure distribution， and combines the acoustic propagation theory to achieve fast and accurate sonic boom prediction. Based on the analysis of the correlation characteristics between the high and low fidelity near-field overpressure data， this study explores the construction methods and prediction performance of deep neural networks under two different multi-fidelity modeling strategies. The experimental results show that by adding an adaptive search method， the MF-DNN based on the linear/nonlinear comprehensive correction strategy has both robustness and performance advantages， and by carefully designing the normalization method and regularization coefficient，the TF-DNN based on transfer learning achieves the smallest prediction error. Compared with the single-fidelity deep neural network， both models can significantly improve the prediction accuracy of the near-field overpressure distribution and sonic boom value by combining low-fidelity data under the condition of less high-fidelity data. Relevant research results provide support for the high-efficiency and low-sonic-boom design of supersonic passenger aircraft.

Key words: supersonic passenger aircraft, sonic boom, multi-fidelity, transfer learning, deep neural network

CLC Number:

V221.41

Yutong WANG, Xiao LUO, Hongyang LIU, Chao SONG, Ying ZHAO, Zhu ZHOU. Sonic boom prediction of supersonic passenger aircraft based on multi-fidelity deep neural network[J]. Acta Aeronautica et Astronautica Sinica, 2025, 46(20): 531936.

Figures/Tables 35

Fig.1

Fig.2

Fig.3

Fig.4

Fig.5

Fig.6

Fig.7

Fig.8

Fig.9

Table 1

Table 2

Single-fidelity deep neural network training results

模型	λ	Loss	Train_MSE	Test_MSE
DNN-4096	$10 - 2$	$1.3 × 10 - 1$	$3.3 × 10 - 2$	$3.0 × 10 - 2$
	$10 - 2$	$3.8 × 10 - 2$	$8.6 × 10 - 4$	$3.0 × 10 - 3$
	$10 - 4$	$3.0 × 10 - 2$	$3.7 × 10 - 3$	$6.5 × 10 - 3$
	$10 - 5$	$2.6 × 10 - 2$	$7.2 × 10 - 3$	$3.9 × 10 - 1$
DNN-8192	$10 - 2$	$5.3 × 10 - 1$	$9.2 × 10 - 3$	$1.1 × 10 - 2$
	$10 - 3$	$1.1 × 10 - 1$	$7.6 × 10 - 4$	$2.7 × 10 - 3$
	$10 - 4$	$2.5 × 10 - 2$	$4.8 × 10 - 4$	$1.2 × 10 - 1$
	$10 - 5$	$7.0 × 10 - 3$	$1.4 × 10 - 3$	$1.8 × 100$
DNN-10000	$10 - 2$	$3.8 × 100$	$1.5 × 10 - 2$	$1.0 × 10 - 1$
	$10 - 3$	$1.0 × 10 - 1$	$5.8 × 10 - 4$	$2.6 × 10 - 3$
	$10 - 4$	$4.2 × 10 - 2$	$8.1 × 10 - 3$	$4.7 × 10 - 3$
	$10 - 5$	$2.0 × 10 - 2$	$7.1 × 10 - 3$	$1.6 × 10 - 2$

Table 2

Fig.10

Table 3

Table 4

Table 5

MF-DNN training results with independent normalization

模型	λ	Loss	Train_MSE	Test_MSE	α 或 β
MF-DNN-hp-ind	$10 - 2$	$2.5 × 10 - 5$	$2.5 × 10 - 5$	$5.2 × 10 - 4$	0.06
	$10 - 3$	$2.5 × 10 - 5$	$2.5 × 10 - 5$	$6.2 × 10 - 4$	0.07
	$10 - 4$	$2.5 × 10 - 5$	$2.5 × 10 - 5$	$5.9 × 10 - 4$	0.07
	$10 - 5$	$2.5 × 10 - 5$	$2.5 × 10 - 5$	$5.9 × 10 - 4$	0.07
AMF-DNN-ind	$10 - 2$	$4.2 × 10 - 5$	$2.1 × 10 - 5$	$1.1 × 10 - 3$	0.002
	$10 - 3$	$4.2 × 10 - 5$	$2.1 × 10 - 5$	$7.1 × 10 - 4$	0.002
	$10 - 4$	$4.2 × 10 - 5$	$2.1 × 10 - 5$	$1.3 × 10 - 3$	0.002
	$10 - 5$	$4.2 × 10 - 5$	$2.1 × 10 - 5$	$1.4 × 10 - 3$	0.002

Table 5

Table 6

MF-DNN training results with hybrid normalization

模型	λ	Loss	Train_MSE	Test_MSE	α 或 β
MF-DNN-hp-mix	$10 - 2$	$2.5 × 10 - 5$	$2.5 × 10 - 5$	$4.6 × 10 - 4$	0.08
	$10 - 3$	$2.5 × 10 - 5$	$2.5 × 10 - 5$	$6.4 × 10 - 4$	0.09
	$10 - 4$	$2.5 × 10 - 5$	$2.5 × 10 - 5$	$4.9 × 10 - 4$	0.09
	$10 - 5$	$2.5 × 10 - 5$	$2.5 × 10 - 5$	$4.9 × 10 - 4$	0.09
AMF-DNN-mix	$10 - 2$	$4.2 × 10 - 5$	$2.1 × 10 - 5$	$1.1 × 10 - 3$	0.002
	$10 - 3$	$4.2 × 10 - 5$	$2.1 × 10 - 5$	$9.3 × 10 - 4$	0.002
	$10 - 4$	$4.2 × 10 - 5$	$2.1 × 10 - 5$	$1.3 × 10 - 3$	0.002
	$10 - 5$	$4.2 × 10 - 5$	$2.1 × 10 - 5$	$1.5 × 10 - 3$	0.002

Table 6

Fig.11

Fig.12

Table 7

Table 8

Table 9

Low-fidelity deep neural network in pretraining phase with independent normalization

模型	λ	Loss	Train_MSE	Test_MSE
TF-NNL-ind-4096	$10 - 2$	$5.6 × 10 - 2$	$3.8 × 10 - 2$	$3.8 × 10 - 2$
	$10 - 3$	$1.8 × 10 - 2$	$1.3 × 10 - 2$	$1.4 × 10 - 2$
	$10 - 4$	$1.5 × 10 - 3$	$3.8 × 10 - 4$	$4.4 × 10 - 4$
	$10 - 5$	$4.4 × 10 - 4$	$4.6 × 10 - 6$	$1.3 × 10 - 5$
TF-NNL-ind-8192	$10 - 2$	$5.6 × 10 - 2$	$3.8 × 10 - 2$	$3.7 × 10 - 2$
	$10 - 3$	$1.8 × 10 - 2$	$1.3 × 10 - 2$	$1.2 × 10 - 2$
	$10 - 4$	$1.1 × 10 - 3$	$3.8 × 10 - 4$	$3.8 × 10 - 4$
	$10 - 5$	$6.1 × 10 - 4$	$3.5 × 10 - 6$	$1.3 × 10 - 5$
TF-NNL-ind-10000	$10 - 2$	$3.3 × 10 - 2$	$3.3 × 10 - 2$	$3.2 × 10 - 2$
	$10 - 3$	$1.6 × 10 - 2$	$4.6 × 10 - 4$	$5.7 × 10 - 4$
	$10 - 4$	$7.4 × 10 - 3$	$4.1 × 10 - 4$	$8.8 × 10 - 4$
	$10 - 5$	$1.7 × 10 - 3$	$8.7 × 10 - 5$	$2.4 × 10 - 3$

Table 9

Table 10

High-fidelity deep neural network in fine-tuning phase with independent normalization

模型	λ	FL	TL	Loss	Train_MSE	Test_MSE
TF-NNH-ind-4096	$10 - 2$	3	1	$1.2 × 10 - 3$	$2.0 × 10 - 4$	$3.2 × 10 - 4$
	$10 - 2$	2	2	$4.0 × 10 - 3$	$6.6 × 10 - 4$	$1.6 × 10 - 3$
	$10 - 3$	3	1	$6.9 × 10 - 4$	$1.4 × 10 - 4$	$8.2 × 10 - 4$
	$10 - 3$	2	2	$1.0 × 10 - 3$	$2.2 × 10 - 4$	$2.7 × 10 - 3$
	$10 - 4$	3	1	$2.4 × 10 - 4$	$8.8 × 10 - 5$	$1.5 × 10 - 3$
	$10 - 4$	2	2	$2.4 × 10 - 4$	$8.2 × 10 - 5$	$1.2 × 10 - 2$
	$10 - 5$	3	1	$1.6 × 10 - 4$	$1.4 × 10 - 4$	$6.5 × 10 - 4$
	$10 - 5$	2	2	$1.4 × 10 - 4$	$1.2 × 10 - 4$	$3.4 × 10 - 3$
TF-NNH-ind-8192	$10 - 2$	3	1	$9.7 × 10 - 4$	$1.6 × 10 - 4$	$3.0 × 10 - 4$
	$10 - 2$	2	2	$3.7 × 10 - 3$	$4.4 × 10 - 4$	$8.8 × 10 - 4$
	$10 - 3$	3	1	$1.2 × 10 - 2$	$4.1 × 10 - 5$	$1.2 × 10 - 2$
	$10 - 3$	2	2	$1.3 × 10 - 3$	$2.0 × 10 - 4$	$1.2 × 10 - 3$
	$10 - 4$	3	1	$1.8 × 10 - 3$	$3.8 × 10 - 4$	$9.4 × 10 - 2$
	$10 - 4$	2	2	$6.4 × 10 - 4$	$3.9 × 10 - 4$	$3.3 × 10 - 3$
	$10 - 5$	3	1	$5.6 × 10 - 4$	$4.2 × 10 - 4$	$2.4 × 10 - 3$
	$10 - 5$	2	2	$3.3 × 10 - 4$	$3.0 × 10 - 4$	$1.9 × 10 - 3$
TF-NNH-ind-10000	$10 - 2$	3	1	$1.1 × 10 - 2$	$4.3 × 10 - 3$	$4.1 × 10 - 3$
	$10 - 2$	2	2	$4.9 × 10 - 3$	$7.6 × 10 - 4$	$2.2 × 10 - 3$
	$10 - 3$	3	1	$2.8 × 10 - 3$	$3.9 × 10 - 4$	$6.4 × 10 - 4$
	$10 - 3$	2	2	$1.9 × 10 - 3$	$3.2 × 10 - 4$	$1.2 × 10 - 3$
	$10 - 4$	3	1	$4.8 × 10 - 4$	$1.4 × 10 - 5$	$1.7 × 10 - 3$
	$10 - 4$	2	2	$5.4 × 10 - 4$	$2.8 × 10 - 4$	$1.3 × 10 - 3$
	$10 - 5$	3	1	$5.2 × 10 - 4$	$4.4 × 10 - 4$	4.2 $× 10 - 3$
	$10 - 5$	2	2	$4.0 × 10 - 4$	$3.5 × 10 - 4$	$1.3 × 10 - 3$

Table 10

Fig.13

Table 11

Table 12

Low-fidelity deep neural network in pretraining phase with hybrid normalization

模型	λ	Loss	Train_MSE	Test_MSE
TF-NNL-mix-4096	$10 - 2$	$3.3 × 10 - 2$	$3.3 × 10 - 2$	$3.3 × 10 - 2$
	$10 - 3$	$1.3 × 10 - 2$	$8.7 × 10 - 3$	$8.9 × 10 - 3$
	$10 - 4$	$9.1 × 10 - 3$	$8.6 × 10 - 3$	$8.8 × 10 - 3$
	$10 - 5$	$3.6 × 10 - 4$	$2.5 × 10 - 4$	$2.9 × 10 - 4$
TF-NNL-mix-8192	$10 - 2$	$3.3 × 10 - 2$	$3.3 × 10 - 2$	$3.3 × 10 - 2$
	$10 - 3$	$1.3 × 10 - 2$	$8.6 × 10 - 3$	$8.9 × 10 - 3$
	$10 - 4$	$1.3 × 10 - 3$	$2.5 × 10 - 4$	$2.7 × 10 - 4$
	$10 - 5$	$1.1 × 10 - 3$	$2.6 × 10 - 4$	$2.6 × 10 - 4$
TF-NNL-mix-10000	$10 - 2$	$3.3 × 10 - 2$	$3.3 × 10 - 2$	$3.2 × 10 - 2$
	$10 - 3$	$1.8 × 10 - 2$	$3.7 × 10 - 4$	$3.8 × 10 - 4$
	$10 - 4$	$6.7 × 10 - 3$	$2.1 × 10 - 4$	$2.1 × 10 - 4$
	$10 - 5$	$1.5 × 10 - 3$	$3.1 × 10 - 5$	$1.1 × 10 - 3$

Table 12

Table 13

High-fidelity deep neural network in fine-tuning phase with hybrid normalization

模型	λ	FL	TL	Loss	Train_MSE	Test_MSE
TF-NNH-mix-4096	$10 - 2$	3	1	$7.9 × 10 - 3$	$3.5 × 10 - 3$	$3.3 × 10 - 3$
	$10 - 2$	2	2	$2.4 × 10 - 3$	$5.0 × 10 - 4$	$7.4 × 10 - 4$
	$10 - 3$	3	1	$1.1 × 10 - 3$	$2.0 × 10 - 4$	$4.8 × 10 - 3$
	$10 - 3$	2	2	$5.9 × 10 - 3$	$1.4 × 10 - 4$	$7.9 × 10 - 3$
	$10 - 4$	3	1	$2.9 × 10 - 4$	$1.5 × 10 - 4$	$6.8 × 10 - 4$
	$10 - 4$	2	2	$2.0 × 10 - 4$	$1.1 × 10 - 4$	$4.6 × 10 - 4$
	$10 - 5$	3	1	$1.3 × 10 - 4$	$9.2 × 10 - 5$	$4.4 × 10 - 4$
	$10 - 5$	2	2	$1.6 × 10 - 4$	$1.4 × 10 - 4$	$4.6 × 10 - 4$
TF-NNH-mix-8192	$10 - 2$	3	1	$1.0 × 10 - 3$	$2.4 × 10 - 4$	$3.2 × 10 - 4$
	$10 - 2$	2	2	$2.8 × 10 - 3$	$4.0 × 10 - 4$	$4.3 × 10 - 4$
	$10 - 3$	3	1	$2.2 × 10 - 4$	$5.5 × 10 - 5$	$1.1 × 10 - 4$
	$10 - 3$	2	2	$8.7 × 10 - 4$	$1.4 × 10 - 4$	$4.4 × 10 - 4$
	$10 - 4$	3	1	$5.7 × 10 - 5$	$7.3 × 10 - 6$	$5.3 × 10 - 5$
	$10 - 4$	2	2	$7.1 × 10 - 4$	$3.6 × 10 - 4$	$5.9 × 10 - 2$
	$10 - 5$	3	1	$1.2 × 10 - 5$	$8.8 × 10 - 7$	$6.6 × 10 - 5$
	$10 - 5$	2	2	$4.6 × 10 - 4$	$4.1 × 10 - 4$	$1.0 × 10 - 2$
TF-NNH-mix-10000	$10 - 2$	3	1	$3.0 × 10 - 3$	$4.2 × 10 - 4$	$9.1 × 10 - 4$
	$10 - 2$	2	2	$6.3 × 10 - 3$	$1.2 × 10 - 3$	$4.1 × 10 - 3$
	$10 - 3$	3	1	$8.7 × 10 - 4$	$1.1 × 10 - 4$	$1.2 × 10 - 3$
	$10 - 3$	2	2	$1.3 × 10 - 3$	$2.1 × 10 - 4$	$9.6 × 10 - 4$
	$10 - 4$	3	1	$2.0 × 10 - 3$	$1.3 × 10 - 3$	$2.4 × 10 - 3$
	$10 - 4$	2	2	$4.4 × 10 - 4$	$1.9 × 10 - 4$	$6.3 × 10 - 4$
	$10 - 5$	3	1	$5.3 × 10 - 4$	$4.3 × 10 - 4$	$4.1 × 10 - 2$
	$10 - 5$	2	2	$3.1 × 10 - 4$	$2.7 × 10 - 4$	$6.4 × 10 - 3$

Table 13

Fig.14

Table 14

Fig.A1

Fig.A2

Fig.A3

Fig.A4

Fig.A5

Fig.A7

Fig.A6

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模型	输入层	FC₁	FC₂	FC₃	FC₄	输出层
DNN-4096	33	128	512	2 048	4 096	1 000
DNN-8192	33	128	512	2 048	8 192	1 000
DNN-10000	33	128	512	2 048	10 000	1 000

样本序号	3	14	17	28	31	35	81	94
HF/PLdB	87.49	94.84	87.55	85.70	86.87	85.88	96.40	86.24
DNN/PLdB	83.95	95.53	84.46	83.14	84.80	83.17	99.23	84.88
绝对误差	3.54	0.69	3.09	2.56	2.07	2.71	2.83	1.36
相对误差/%	4.05	0.73	3.53	2.99	2.38	3.16	2.94	1.58

类名	模块名	输入维度	输出维度
NN_H1	FC₁	1 033	10 000
	输出层	10 000	1 000
NN_H2	FC₁	1 033	4 096
	FC₂	4 096	10 000
	FC₃	10 000	10 000
	输出层	10 000	1 000

样本序号	4	3	14	17	28	31	35	81	94
HF/PLdB	86.96	87.49	94.84	87.55	85.70	86.87	85.88	96.40	86.24
MF-DNN-hp/PLdB	87.05	87.52	94.70	87.56	85.83	86.41	86.20	96.38	85.83
绝对误差	0.09	0.03	-0.14	0.01	0.13	-0.46	-0.32	-0.02	-0.41
相对误差/%	0.10	0.03	0.15	0.01	0.15	0.53	0.37	0.02	0.48
AMF-DNN/PLdB	87.27	87.43	94.69	87.65	85.82	86.39	86.21	96.35	85.83
绝对误差	0.31	-0.06	-0.15	0.10	0.12	-0.48	-0.33	-0.05	-0.41
相对误差/%	0.35	0.07	0.15	0.11	0.14	0.55	0.38	0.05	0.48

模型	输入层	FL₁	FL₂	TL₁	TL₂	输出层
TF-DNN-4096	33	128	512	2 048	4 096	1 000
TF-DNN-8192	33	128	512	2 048	8 192	1 000
TF-DNN-10000	33	128	512	2 048	10 000	1 000

Sonic boom prediction of supersonic passenger aircraft based on multi-fidelity deep neural network

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References 32

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样本序号	67	3	14	17	28	31	35	81	94
HF/PLdB	101.10	87.49	94.84	87.55	85.70	86.87	85.88	96.40	86.24
TF-NNH-ind/PLdB	98.35	87.82	92.32	88.26	84.42	85.59	85.75	96.23	85.86
绝对误差	2.65	-0.33	2.52	-0.71	1.28	1.28	0.13	0.17	0.38
相对误差/%	2.62	0.38	2.66	0.81	1.49	1.47	0.15	0.18	0.44

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