# Plane Cascade Aerodynamic Performance Prediction Based on Metric Learning for Multi-Output Gaussian Process Regression

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## Abstract

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## 1. Introduction

## 2. Gaussian Process Regression (GPR)

#### 2.1. Single-Output Gaussian Process Regression (SOGPR)

#### 2.2. Multi-Output Gaussian Process Regression (MOGPR)

## 3. Metric Learning for MOGPR

## 4. Experiments and Analysis

#### 4.1. Neural Network Structure and Loss Function Exploration

#### 4.1.1. BPNN Experiments

#### 4.1.2. MTLNN Experiments

#### 4.2. Analysis of MOGPR Parameters

#### 4.3. Analysis of Results

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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Symbol | Description | Symbol | Description |
---|---|---|---|

D | Training dataset | ℓ | Length-scale |

${\epsilon}_{i}$ | Noises | $\nu $ | Kernel scale parameter |

$k(x,{x}^{\prime})$ | Covariance function | $\tau $ | Distance calculation parameters |

${B}_{q}$ | Positive semi-definite matrix | ${a}_{rq}^{(i,j)}$ | Amplitude parameters |

$X\in {\mathbf{R}}^{n\times d}$ | Input features | ${\phi}_{q}^{(i,j)}$ | Displacement parameters |

$Y\in {\mathbf{R}}^{n\times 1}$ | Output targets | ${\varphi}_{q}^{(i,j)}$ | Delay parameters |

${X}_{*}\in {\mathbf{R}}^{n\times d}$ | Test features | Q | Compunents |

$A\in {\mathbf{R}}^{d\times d}$ | Metrics matrix | ${R}_{q}$ | Subcompoents |

$\gamma $ | Mounting angle/(${}^{\circ}$) | t | Raster distance/(mm) |

$bax$ | String length/(mm) | $Nob$ | Number of blades |

$\beta 1$ | Inlet airflow angle/(${}^{\circ}$) | $Ma$ | Mach number |

$\alpha $ | Angle attack/(${}^{\circ}$) | $\omega $ | Loss coefficient |

$\mathsf{\Omega}$ | AVDR | ${\sigma}_{i}^{2}$ | Variance |

Num | Mounting Angle/(°) | Raster Distance/mm | Number of Blades | String Length/mm |
---|---|---|---|---|

1 | 63.2 | 43.35 | 9 | 70.32 |

2 | 58.29 | 48 | 8 | 79.18 |

3 | 40.86 | 50 | 10 | 64 |

4 | 76.67 | 48.05 | 6 | 51.08 |

5 | 76.9 | 35.44 | 11 | 78.11 |

Loss Function | Number of Samples | Network Structure | RMSE |
---|---|---|---|

MSE | 200 | (8,4) | 0.37188 |

(16,8,4) | 0.31296 | ||

(32,16,8,4) | 0.32353 | ||

(64,32,16,8,4) | 0.34634 | ||

(32,16,8) | 0.30392 | ||

(64,32,16) | 0.33974 | ||

250 | (8,4) | 0.33172 | |

(16,8,4) | 0.29141 | ||

(32,16,8,4) | 0.36742 | ||

(64,32,16,8,4) | 0.38399 | ||

(32,16,8) | 0.27811 | ||

(64,32,16) | 0.30556 | ||

300 | (8,4) | 0.35633 | |

(16,8,4) | 0.26045 | ||

(32,16,8,4) | 0.28259 | ||

(64,32,16,8,4) | 0.34756 | ||

(32,16,8) | 0.25457 | ||

(64,32,16) | 0.31949 | ||

MAE | 200 | (8,4) | 0.10589 |

(16,8,4) | 0.09234 | ||

(32,16,8,4) | 0.09827 | ||

(64,32,16,8,4) | 0.14207 | ||

(32,16,8) | 0.08824 | ||

(64,32,16) | 0.12416 | ||

250 | (8,4) | 0.09043 | |

(16,8,4) | 0.08657 | ||

(32,16,8,4) | 0.11009 | ||

(64,32,16,8,4) | 0.12838 | ||

(32,16,8) | 0.08124 | ||

(64,32,16) | 0.12254 | ||

300 | (8,4) | 0.08937 | |

(16,8,4) | 0.08163 | ||

(32,16,8,4) | 0.11042 | ||

(64,32,16,8,4) | 0.12196 | ||

(32,16,8) | 0.07949 | ||

(64,32,16) | 0.10877 |

Loss Function | Number of Samples | RMSE |
---|---|---|

MSE | 200 | 0.63451 |

250 | 0.51326 | |

300 | 0.40387 | |

MAE | 200 | 0.16859 |

250 | 0.14789 | |

300 | 0.12659 |

Number of Samples | Model | Components | ||||
---|---|---|---|---|---|---|

$\mathit{Q}\mathbf{=}\mathbf{1}$ | $\mathit{Q}\mathbf{=}\mathbf{2}$ | $\mathit{Q}\mathbf{=}\mathbf{3}$ | $\mathit{Q}\mathbf{=}\mathbf{4}$ | $\mathit{Q}\mathbf{=}\mathbf{5}$ | ||

200 | MOSM | 0.09031 | 0.09326 | 0.08927 | 0.09372 | 0.09688 |

MOHSM | 0.13041 | 0.28731 | 0.16916 | 0.12024 | 0.09843 | |

ML_MOSM | 0.08766 | 0.08711 | 0.09436 | 0.08838 | 0.09224 | |

ML_MOHSM | 0.09469 | 0.11852 | 0.08281 | 0.10147 | 0.08331 | |

250 | MOSM | 0.09026 | 0.08724 | 0.09125 | 0.09081 | 0.09422 |

MOHSM | 0.09732 | 0.08811 | 0.12228 | 0.08904 | 0.08612 | |

ML_MOSM | 0.08492 | 0.08081 | 0.08425 | 0.08932 | 0.08716 | |

ML_MOHSM | 0.08721 | 0.09859 | 0.07804 | 0.08115 | 0.08103 | |

300 | MOSM | 0.08519 | 0.08698 | 0.08959 | 0.09309 | 0.09124 |

MOHSM | 0.09401 | 0.08361 | 0.08665 | 0.07821 | 0.09062 | |

ML_MOSM | 0.08356 | 0.07579 | 0.08394 | 0.09046 | 0.09662 | |

ML_MOHSM | 0.08232 | 0.08362 | 0.07653 | 0.07729 | 0.07649 |

Samples | SOGPR | SVR | BPNN | MTLNN | MOSM | MOHSM | ML_MOSM | ML_MOHSM |
---|---|---|---|---|---|---|---|---|

200 | 0.8676 | 5.2402 | 0.08824 | 0.16859 | 0.08927 | 0.09843 | 0.08711 | 0.08281 |

250 | 0.8289 | 4.6859 | 0.08124 | 0.14789 | 0.08724 | 0.08612 | 0.08081 | 0.07804 |

300 | 0.6487 | 4.2881 | 0.07949 | 0.12659 | 0.08519 | 0.07821 | 0.07579 | 0.07649 |

Samples | SOGPR | SVR | BPNN | MTLNN | MOSM | MOHSM | ML_MOSM | ML_MOHSM |
---|---|---|---|---|---|---|---|---|

200 | 0.6241 | 0.8801 | 0.08976 | 0.12591 | 0.087561 | 0.06565 | 0.07405 | 0.04776 |

250 | 0.6101 | 0.7095 | 0.08876 | 0.10143 | 0.07246 | 0.05216 | 0.07126 | 0.04671 |

300 | 0.5736 | 0.6028 | 0.08347 | 0.09629 | 0.06635 | 0.05024 | 0.06284 | 0.04452 |

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## Share and Cite

**MDPI and ACS Style**

Liu, L.; Yang, C.; Xiang, H.; Lin, J.
Plane Cascade Aerodynamic Performance Prediction Based on Metric Learning for Multi-Output Gaussian Process Regression. *Symmetry* **2023**, *15*, 1692.
https://doi.org/10.3390/sym15091692

**AMA Style**

Liu L, Yang C, Xiang H, Lin J.
Plane Cascade Aerodynamic Performance Prediction Based on Metric Learning for Multi-Output Gaussian Process Regression. *Symmetry*. 2023; 15(9):1692.
https://doi.org/10.3390/sym15091692

**Chicago/Turabian Style**

Liu, Lin, Chunming Yang, Honghui Xiang, and Jiazhe Lin.
2023. "Plane Cascade Aerodynamic Performance Prediction Based on Metric Learning for Multi-Output Gaussian Process Regression" *Symmetry* 15, no. 9: 1692.
https://doi.org/10.3390/sym15091692