# Uncertainty Quantification for Full-Flight Data Based Engine Fault Detection with Neural Networks

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. Artificial Neural Networks with Uncertainty Quantification

#### 2.1.1. Approximating the Aleatoric Uncertainty

#### 2.1.2. Approximating the Epistemic Uncertainty

#### 2.2. Fault Detection

- Identification of Regions with High Epistemic Uncertainty: Timestamps with high modeling uncertainty are identified and removed to ensure that only data points with high modeling accuracy are used for fault detection. Regions with high epistemic uncertainty are identified by defining a threshold on the confidence score ${\mathcal{L}}_{lim}$.
- Outlier Detection: The detection of outliers indicating unusual system performance is based on the Mahalanobis Distance ${d}_{M}$. The corresponding data points are considered outliers if the Mahalanobis Distance ${d}_{M}$ exceeds a predefined threshold ${d}_{M,lim}$.
- Fault Detection: The total number of outliers is computed in the last step. Since there will always be a certain number of statistical outliers, a threshold ${n}_{lim}$ on the total number of outliers is introduced. If the number of outliers detected exceeds this predefined threshold ${n}_{lim}$, the outliers are no longer considered statistical but systematic, indicating a fault.

#### 2.3. Description of the Database

## 3. Results

#### 3.1. Assessment of the Articifical Neural Network

#### 3.2. Detection Rates

#### 3.3. Sensitivity Study: Fault Initiation

## 4. Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Nomenclature and Abbreviations

## Nomenclature

Symbols | |

$Alt$ | Altitude |

$BLV$ | Bleed Setting |

${d}_{M}$ | Mahalanobis Distance |

${d}_{M,lim}$ | Threshold for the Mahalanobis Distance |

$EGT$ | Exhaust Gas Temperature |

$EAI$ | Engine Anti-Icing |

$FP$ | False Positive Detection Rate |

l | Flight Length |

$lr$ | Learning Rate |

$MN$ | Mach-Number |

$mse$ | Mean Squared Error |

${\mathcal{L}}_{lim}$ | Threshold for the Confidence Score |

${\mathcal{L}}_{setting}$ | Confidence Score |

${n}_{lim}$ | Threshold for the Number of Outliers Tolerated |

$N1$ | Shaft Speed Fan |

$N2$ | Shaft Speed Core |

$\mathcal{NLL}$ | Negative Log-Likelihood |

p | Pressure |

$PACK$ | Airflow towards the Cabin |

$PH$ | Flight Phase |

Q | Capacity |

T | Temperature |

t | Time |

${t}_{init}$ | Time Until Fault Initiation |

$TAI$ | Tail Anti-Icing |

$TP$ | True Positive Detection Rate |

$WAI$ | Wing Anti-Icing |

$Wf$ | Fuel Flow |

x | Input-Parameter |

y | Measurement |

$\Delta $ | Deviation |

$\delta $ | Correction Factor Pressure |

$\mathrm{\Sigma}$ | Correlation Matrix |

$\sigma $ | Standard Deviation |

$\mu $ | Mean Value |

$\eta $ | Efficiency |

$\mathrm{\Theta}$ | Parameter Set |

$\theta $ | Correction Factor Temperature |

Superscripts and Subscripts | |

$\overline{f}$ | Averaged Value |

c | ISA-Corrected Value |

$ISA$ | ISA Reference Value |

Acronyms | |

CC | Combustion Chamber |

HPC | High Pressure Compressor |

HPT | High Pressure Turbine |

LPC | Low Pressure Compressor |

LPT | Low Pressure Turbine |

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**Figure 2.**Architecture for approximating a univariate Gaussian probability density function with artificial neural networks.

**Figure 3.**Architecture of the neural network used for approximating the in-flight measurements of aircraft engines.

**Figure 4.**Approximation of the corrected exhaust gas temperature $EG{T}_{c}$ for an example flight with varying anti-icing setting.

**Figure 5.**Corresponding confidence score ${\mathcal{L}}_{setting}$ for the flight displayed in Figure 4.

**Figure 7.**Approximation of the corrected exhaust gas temperature $EG{T}_{c}$ for an example flight from the test dataset.

**Figure 8.**Comparison of the average true positive detection rate $\overline{TP}$ and number of outliers tolerated until fault detection ${n}_{lim}$.

**Figure 9.**Sensitivity of the median true positive detection rate $\overline{TP}$ with respect to the fault initiation ${t}_{init}$.

Input Parameter | |
---|---|

Parameter | Description |

$\theta ={T}_{t0}/{T}_{ISA}$ | Correction factor temperature |

$\delta ={p}_{t0}/{p}_{ISA}$ | Correction factor pressure |

$N{1}_{c}=N1/\sqrt{\theta}$ | Corrected spool speed of the fan |

$MN$ | Mach-Number |

$BLV$ | Setting bleed extraction |

$PACK$ | Airflow towards the cabin |

$EAI$ | Setting engine-anti-icing |

$TAI$ | Setting tail-anti-icing |

$WAI$ | Setting wing-anti-icing |

$PH$ | Flight phase |

Output Parameter | |

Parameter | Description |

$N{2}_{c}=N2/\sqrt{\theta}$ | Corrected spool speed of the core |

$EG{T}_{c}=EGT/\theta $ | Corrected exhaust gas temperature |

$W{f}_{c}=Wf/\left(\sqrt{\theta}\delta \right)$ | Corrected fuel flow |

**Table 2.**Definition of the OBIDICOTE test cases according to [47].

Label | Fault Description | |
---|---|---|

$\Delta \mathit{Q}$ | $\Delta \mathit{\eta}$ | |

a | $\Delta {Q}_{Fan}=-1.0\%$ | $\Delta {\eta}_{Fan}=-0.5\%$ |

$\Delta {Q}_{LPC}=-0.7\%$ | $\Delta {\eta}_{LPC}=-0.4\%$ | |

b | − | $\Delta {\eta}_{Fan}=-1.0\%$ |

c | $\Delta {Q}_{HPC}=-1.0\%$ | $\Delta {\eta}_{HPC}=-0.7\%$ |

d | − | $\Delta {\eta}_{HPC}=-1.0\%$ |

e | $\Delta {Q}_{HPC}=-1.0\%$ | − |

f | $\Delta {Q}_{HPT}=+1.0\%$ | − |

g | $\Delta {Q}_{HPT}=-1.0\%$ | $\Delta {\eta}_{HPT}=-1.0\%$ |

h | − | $\Delta {\eta}_{HPT}=-1.0\%$ |

i | − | $\Delta {\eta}_{LPT}=-1.0\%$ |

j | $\Delta {Q}_{LPT}=-1.0\%$ | $\Delta {\eta}_{LPT}=-0.4\%$ |

k | $\Delta {Q}_{LPT}=-1.0\%$ | − |

l | $\Delta {Q}_{LPT}=+1.0\%$ | $\Delta {\eta}_{LPT}=-0.6\%$ |

Climb | Cruise | Descent | |||||||
---|---|---|---|---|---|---|---|---|---|

${\mathit{mse}}_{\mathit{EGT}}$ | ${\mathit{mse}}_{\mathit{N}\mathbf{2}}$ | ${\mathit{mse}}_{\mathit{Wf}}$ | ${\mathit{mse}}_{\mathit{EGT}}$ | ${\mathit{mse}}_{\mathit{N}\mathbf{2}}$ | ${\mathit{mse}}_{\mathit{Wf}}$ | ${\mathit{mse}}_{\mathit{EGT}}$ | ${\mathit{mse}}_{\mathit{N}\mathbf{2}}$ | ${\mathit{mse}}_{\mathit{Wf}}$ | |

Training | 3.22 K | 0.13% | 0.64% | 3.20 K | 0.19% | 0.74% | 8.37 K | 0.77% | 2.36% |

Validation | 3.50 K | 0.13% | 0.69% | 3.75 K | 0.20% | 0.82% | 8.76 K | 0.78% | 2.44% |

Testing | 3.76 K | 0.16% | 0.78% | 5.06 K | 0.23% | 0.98% | 9.57 K | 0.80% | 2.55% |

Climb | Cruise | Descent | |||||||
---|---|---|---|---|---|---|---|---|---|

${\mathit{\sigma}}_{\mathit{EGT}}$ | ${\mathit{\sigma}}_{\mathit{N}\mathbf{2}}$ | ${\mathit{\sigma}}_{\mathit{Wf}}$ | ${\mathit{\sigma}}_{\mathit{EGT}}$ | ${\mathit{\sigma}}_{\mathit{N}\mathbf{2}}$ | ${\mathit{\sigma}}_{\mathit{Wf}}$ | ${\mathit{\sigma}}_{\mathit{EGT}}$ | ${\mathit{\sigma}}_{\mathit{N}\mathbf{2}}$ | ${\mathit{\sigma}}_{\mathit{Wf}}$ | |

Training | 3.03 K | 0.13% | 0.66% | 2.80 K | 0.15% | 0.73% | 6.30 K | 0.67% | 2.63% |

Validation | 3.01 K | 0.13% | 0.66% | 2.80 K | 0.15% | 0.73% | 6.23 K | 0.66% | 2.60% |

Testing | 3.11 K | 0.14% | 0.68% | 2.91 K | 0.15% | 0.74% | 6.50 K | 0.70% | 2.71% |

**Table 5.**Resulting detection rates for the OBIDICOTE test cases a-l (Table 2) for aleatoric and epistemic uncertainty quantification.

TP | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

a | b | c | d | e | f | g | h | i | j | k | l | |

$FP\le 10.0\%$ | $97.7\%$ | $89.0\%$ | $84.5\%$ | $98.3\%$ | $87.7\%$ | $24.7\%$ | $98.3\%$ | $99.7\%$ | $97.0\%$ | $87.0\%$ | $91.7\%$ | $97.3\%$ |

$FP\le 7.5\%$ | $95.7\%$ | $86.7\%$ | $81.3\%$ | $96.7\%$ | $84.0\%$ | $19.7\%$ | $96.7\%$ | $97.7\%$ | $95.3\%$ | $83.7\%$ | $89.0\%$ | $95.7\%$ |

$FP\le 5.0\%$ | $94.0\%$ | $82.0\%$ | $74.3\%$ | $95.0\%$ | $79.3\%$ | $14.7\%$ | $95.0\%$ | $96.0\%$ | $94.3\%$ | $77.3\%$ | $85.7\%$ | $94.3\%$ |

$FP\le 2.5\%$ | $92.3\%$ | $75.7\%$ | $67.3\%$ | $93.7\%$ | $71.3\%$ | $9.3\%$ | $93.7\%$ | $95.3\%$ | $93.3\%$ | $71.7\%$ | $81.0\%$ | $93.7\%$ |

$FP\le 1.0\%$ | $88.3\%$ | $62.3\%$ | $52.3\%$ | $90.3\%$ | $57.0\%$ | $4.0\%$ | $89.7\%$ | $94.0\%$ | $88.0\%$ | $62.3\%$ | $74.0\%$ | $90.3\%$ |

$FP\le 0.5\%$ | $86.3\%$ | $55.3\%$ | $47.0\%$ | $87.7\%$ | $49.7\%$ | $3.3\%$ | $87.3\%$ | $92.3\%$ | $85.7\%$ | $57.0\%$ | $69.3\%$ | $88.0\%$ |

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## Share and Cite

**MDPI and ACS Style**

Weiss, M.; Staudacher, S.; Mathes, J.; Becchio, D.; Keller, C.
Uncertainty Quantification for Full-Flight Data Based Engine Fault Detection with Neural Networks. *Machines* **2022**, *10*, 846.
https://doi.org/10.3390/machines10100846

**AMA Style**

Weiss M, Staudacher S, Mathes J, Becchio D, Keller C.
Uncertainty Quantification for Full-Flight Data Based Engine Fault Detection with Neural Networks. *Machines*. 2022; 10(10):846.
https://doi.org/10.3390/machines10100846

**Chicago/Turabian Style**

Weiss, Matthias, Stephan Staudacher, Jürgen Mathes, Duilio Becchio, and Christian Keller.
2022. "Uncertainty Quantification for Full-Flight Data Based Engine Fault Detection with Neural Networks" *Machines* 10, no. 10: 846.
https://doi.org/10.3390/machines10100846