=Paper=
{{Paper
|id=Vol-3742/paper4
|storemode=property
|title=Neural network method for identifying potential defects in complex dynamic objects
|pdfUrl=https://ceur-ws.org/Vol-3742/paper4.pdf
|volume=Vol-3742
|authors=Victoria Vysotska,Vasyl Lytvyn,Serhii Vladov,Ruslan Yakovliev,Yevhen Volkanin
|dblpUrl=https://dblp.org/rec/conf/citi2/VysotskaLVYV24
}}
==Neural network method for identifying potential defects in complex dynamic objects==
https://ceur-ws.org/Vol-3742/paper4.pdf
Neural network method for identifying potential defects in
complex dynamic objects
Victoria Vysotska1,†, Vasyl Lytvyn1,†, Serhii Vladov2,∗,†, Ruslan Yakovliev2,† and
Yevhen Volkanin2,†
1 Lviv Polytechnic National University, Stepan Bandera Street 12 79013 Lviv, Ukraine
2 Kremenchuk Flight College of Kharkiv National University of Internal Affairs, Peremohy Street 17/6 39605
Kremenchuk, Ukraine
Abstract
The work is devoted to the development of a neural network method for identifying potential
defects in complex dynamic objects, such as, for example, helicopter turboshaft engines. The
proposed method is based on the use of the Transformer model, consisting of the encoder,
decoder, positional encoding, and attention mechanism, instead of a generalized regression
neural network. A modification of the ReLU activation function in the form of Smooth ReLU is
proposed to make it smoother and more continuous, which leads to improved convergence and
training stability. The analysis of the derivatives of the ReLU and Smooth ReLU functions showed
that Smooth ReLU solves the problem of “dead neurons” by providing a non-zero gradient for all
input data values, including negative ones, which ensures more stable training of neural
networks and prevents neurons from stopping updating due to a zero gradient. As a neural
network implementation of the Transformer model, the use of a graph neural network is
proposed, the key advantage of which is its ability to model more complex dependencies and
relationships between input data elements, which increases the efficiency of training and
improves the quality of prediction in sequence processing tasks. As a neuron activation function,
the use of a cross-entropy loss function between actual and predicted probability distributions
has been proposed, the key advantage of which is its ability to provide efficient training of a
classification model by minimizing the discrepancy between predicted and real class
probabilities. The results of the computational experiment showed that the proposed method
demonstrated almost 100 % accuracy in determining potential defects, such as the possible
formation of cracks (burnouts) in the combustion chamber of helicopter turboshaft engines due
to the predicted decrease in the gas temperature in front of the compressor turbine.
Keywords
Neural network, Transformer architecture, graph neural network, training, identifying potential
defects, helicopter turboshaft engine, activation function, adaptive training rate 1
CITI’2024: 2nd International Workshop on Computer Information Technologies in Industry 4.0, June 12–14, 2024,
Ternopil, Ukraine
∗ Corresponding author.
† These authors contributed equally.
victoria.a.vysotska@lpnu.ua (V. Vysotska); vasyl.v.lytvyn@lpnu.ua (V. Lytvyn); serhii.vladov@univd.edu.ua
(S. Vladov); director.klk.hnuvs@gmail.com (R. Yakovliev); volkanin@ukr.net (Y. Volkanin)
0000-0001-6417-3689 (V. Vysotska); 0000-0002-9676-0180 (V. Lytvyn); 0000-0001-8009-5254
(S. Vladov); 0000-0002-3788-2583 (R. Yakovliev); 0000-0003-3507-1987 (Y. Volkanin)
© 2024 Copyright for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
CEUR
ceur-ws.org
Workshop ISSN 1613-0073
Proceedings
1. Introduction
Currently, the use of neural networks is becoming increasingly common in various fields,
including the prediction and identification of potential defects in various systems and
processes [1–3]. Neural network methods are a powerful tool for analyzing data and
identifying hidden patterns [4, 5], which makes them attractive for use in tasks of prediction
and defect detection.
Neural networks are mathematical models that attempt to imitate the functioning of the
human brain. They consist of many interconnected neurons organized into layers that
process input data and generate a corresponding output [6, 7]. In predicting tasks, neural
networks can be used to analyze time series, process images, texts, and other types of data
[8–10].
Identifying potential defects is an important step in the process of quality control and
ensuring the reliability of systems and equipment. Neural network methods can be
effectively applied to detect anomalies and predict possible failures based on historical data
about the operation of a system or process [1, 11, 12].
The work aims to develop a neural network method for identifying potential defects in
various systems and processes using data analysis and identifying hidden patterns to
increase the efficiency of quality control and ensure the reliability of equipment and
systems.
2. Related works
The neural network method for identifying potential defects based on prediction results is
a powerful tool for increasing the efficiency of quality control and ensuring the reliability of
systems and processes [13, 14]. The use of neural network methods allows you to automate
the process of detecting anomalies and predicting failures, which helps improve production
processes and reduce the cost of equipment maintenance and repair.
The significance of this research area allows us to expand the capabilities of diagnostic
models and increase the reliability of diagnosis. In [15, 16], the advantages of using artificial
intelligence methods, including neural networks, fuzzy logic, and expert systems, compared
to traditional diagnostic methods to solve issues related to predicting the reliability of
complex dynamic systems were identified. This is because artificial intelligence systems are
highly adaptable and capable of solving complex tasks of classification and pattern
recognition, which is an important aspect of the process of diagnosis and prognosis.
For example, in [17], the task of the YOLOv5 network model structure optimization using
the Convolutional Block Attention Module (CBAM) was solved to improve the accuracy of
detection and identification of small defects in pipeline circumferential welds based on the
analysis of internal magnetic flux (MFL) signals, which helps to improve technical support
and safety in assessing the condition and repair of pipelines. The disadvantage of [15] is that
the YOLOv5 model, although effective in detecting anomalous objects, does not exhibit
attentional preference during the feature extraction process, which is insufficient for the
effective detection of small defects.
The essence of [18] is to develop and experimentally validate a wavelet-gradient-
integrated neural network (WGI-SNN) structure combining wavelet packet transform
(WPT) with peak neural network (SNN) to more effectively measure complex equipment
defects using frequency-domain technologies. -temporal signal processing. A disadvantage
of the work is the limited experimental testing of the proposed method, which may reduce
the generalizability of the results to various types of complex defects.
In [19] presents a neural network method for detecting binary and random impulse
noise in contaminated images, based on pixel classification and random point patterning,
which applies to colour images and demonstrates superiority over many existing methods
in extensive experimental results. The key disadvantage of the proposed method is its
dependence on the classification accuracy of random point patterns and the possibility of
isolated points being misidentified as noise, which can lead to the loss of important
information in the images.
In [20] it proposed a method that uses multi-threshold binarization of multispectral
images to extract features and improve classification accuracy in real-time, reducing
training and inference time by 5 times compared to ResNet and Ensemble CNN models. The
key disadvantage is its limitation in working with small data sets and possible reduction in
classification accuracy when using large data sets.
Based on the discussion of related works, it becomes urgent to develop methods that,
based on the analysis of process behaviour and the results of predicting the technical
condition, determine potential defects in the units of complex dynamic objects, which will
lead to the occurrence of emergencies during their operation.
3. Methods and materials
In [1], a method was developed for identifying potential defects in the components of
helicopter turboshaft engines (TE) based on predicting their operational status at flight
modes, which makes it possible to use, along with quantitative mathematical models of
engines, qualitative and experimental information obtained during flight engine tests, and
also, directly, in a helicopter flight. Therefore, this method was chosen in this work as the
initial one.
In [1], based on a given data array of the object under research 𝑋(𝑡) = (𝑥𝑡1 , 𝑥𝑡2 , … , 𝑥𝑡𝑛 )
input parameters, which correspond only to its operation normal mode, that is, in that time
when there were no failures or other anomalies, a neural network model 𝑓 (𝑋(𝑡), 𝜃) is
developed (θ are the model parameters), based on a fragment with regular values
{𝑥𝑡1 , 𝑥𝑡2 , … , 𝑥𝑡𝑛 }, which determines whether the fragments {𝑥𝑡𝑛+1 , 𝑥𝑡𝑛+2 , … , 𝑥𝑡𝑛+𝑙 } are
anomalous with acceptable accuracy for identifying possible defects of the research object.
The developed method consists of using subsequent data of the initial indicators of the
research object to develop a time series predicting model. In this case, the models are
trained on data that corresponds only to its operation normal mode, that is, at a time when
there were no breakdowns or other anomalies. Thus, the model trains to predict what the
signal should be during normal operation. If at a certain point in time, the actual value of the
i-th initial indicator differs from the predicted normal value, anomalous behaviour is
recorded and a signal about a potential defect is generated (Fig. 1).
Sliding window
Window
parameter 1
Actual
Input
value
Raise
parameter 2
Input
Predicted
Degradation
value
...
parameter N
Input
Actual
value
Possible
defects
notice
1 N (i ) (i )
i N xj − x j
e =
j =1
Neural 1, if ei
ei =
Network 0, if ei
i = 1...L
w
Figure 1: Complex dynamic objects identifying potential defects graphic interpretation.
(author's research, generalized based on [1]).
Unlike [1], the proposed neural network method for identifying potential defects in
complex dynamic objects is not based on the GRNN neural network, as in [1], but on the
Transformer model [21, 22], which is capable of adapting to different lengths of input and
output sequences without the need for pre-configuration. At the same time, the
Transformer model allows one to take into account dependencies between sequence
elements over large time intervals, which is important for predicting defects in complex
dynamic objects.
For the normal behaviour of the system, data on the initial indicators of the research
object are collected, which must be presented in the form of a time series containing a
sequence of actual values. At the same time, instead of a Seq-to-Seq model based on a
general regression neural network of the GRNN neural network, it is proposed to use a
model based on deep learning and the Transformer architecture, which is well suited for
working with data sequences, allowing to take into account long-term dependencies and
work with different lengths of input and output sequences. The proposed Transformer-
based model has the following architecture:
1. Encoder: converts the input data sequence into an internal representation, taking
into account the context and dependencies between the elements of the sequence.
For each element of the input sequence X, the model creates its vector representation
xi, which is then combined into a matrix Xenc:
𝑋𝑒𝑛𝑐 = 𝐸𝑛𝑐𝑜𝑑𝑒𝑟(𝑋). (1)
2. Decoder: generates an output data sequence based on an internal representation,
taking into account both the input data and the previous elements of the output
sequence. At each step of generating the output sequence Y, the decoder uses the
context vectors C from the encoder, as well as the previous elements of the output
sequence Y, to predict the next element:
𝑌𝑡 = 𝐷𝑒𝑐𝑜𝑑𝑒𝑟 (𝑌𝑡−1 , 𝐶 ). (2)
3. Positional Encoding: adds vectors representing the positions of elements in a
sequence so that the model can take into account the order of the data. For this, sine
and cosine functions with different frequencies are used [23, 24]:
𝑝𝑜𝑠
𝑃𝐸𝑝𝑜𝑠,2𝑖 = sin ( 2∙𝑖
), (3)
10000𝑑𝑚𝑜𝑑𝑒𝑙
𝑝𝑜𝑠
𝑃𝐸𝑝𝑜𝑠,2𝑖+1 = cos ( 2∙𝑖
), (4)
10000𝑑𝑚𝑜𝑑𝑒𝑙
where pos is the position in the sequence, i is the index of the element in the positional
encoding vector, dmodel is the dimension of the model.
4. Attention Mechanism allows the model to focus on different parts of the input
sequence when generating the output sequence [25, 26]. One of the common
methods is the scalar product attention mechanism (Scaled Dot-Product Attention)
[27, 28]:
𝑄 ∙ 𝐾𝑇 (5)
𝐴𝑡𝑡𝑒𝑛𝑡𝑖𝑜𝑛(𝑄, 𝐾, 𝑉 ) = 𝑉 ∙ softmax ( ),
√𝑑𝑘
where Q, K, and V are queries, keys and values, dk is the dimension of the keys. Q queries are
usually generated based on the hidden state of the decoder or the last generated element of
the output sequence, and can also be created by linear transformation of the corresponding
vectors. Keys K are generated from the hidden state of the encoder or a sequence of inputs,
similar to queries. The V values represent the actual data that the model must rely on when
generating the output sequence. The key dimension dk is a hyperparameter that is
empirically selected during model tuning, usually chosen according to the dimension of the
model's hidden state and the complexity of the task.
After obtaining predicted values using the Transformer model, the following approach
can be used to identify potential defects in the research object:
• If the difference between actual and predicted values exceeds a certain threshold, it
may indicate a defect.
• Additionally, statistical measures such as standard deviation can be considered to
account for the extent to which forecast values deviate from actual values.
To determine potential defects of the research object, the following expressions are used:
1, if 𝐴 > 𝜌,
𝑍={ (6)
0, if 𝐴 ≤ 𝜌,
𝐿𝑤
1
𝐴= ∙ ∑ 𝑒𝑖 , (7)
𝐿𝑤
𝑖=1
𝑁
1 (𝑗) (𝑗)
𝑒𝑖 = ∙ ∑ |𝑥𝑖 − 𝑥̂𝑖 |, (8)
𝑁
𝑖=1
1, if 𝑒𝑖 > 𝜏,
𝑒𝑖 = { (9)
0, if 𝑒𝑖 ≤ 𝜏,
where X is a sequence of actual values of length Lw; X' is a sequence of predicted values of
(𝑗)
features with a length at a point in time; 𝑥𝑖 is an actual value of the i-th initial indicator at
(𝑗)
the j-th moment; 𝑥̂𝑖 is a predicted value of the i-th initial indicator at the j-th moment.
For a Transformer-based model, it is advisable to use a loss function that reflects the
difference between the actual and predicted sequence values [29, 30]. Given that the model
generates a sequence, loss functions that estimate the probability of a match between the
generated sequence and the actual target sequence are typically used. One of the standard
choices for the sequence generation task is cross entropy (categorical cross entropy) [31,
32]. This loss function is widely used in classification and sequence generation problems.
The expression describes the cross-entropy loss function between the actual and predicted
probability distributions:
𝑇
𝐽 = − ∑ 𝑦𝑡 ∙ log(𝑦̂𝑡 ), (10)
𝑡=1
where T is the length of the sequence, yt is the actual probability distribution for the
sequence element at step t, 𝑦̂𝑡 is the predicted probability distribution for the sequence
element at step t.
The loss function (10) estimates the difference between the actual and predicted
probability distributions for each element of the sequence and penalizes the model for
incorrect predictions.
It is also worth noting that in the proposed method, the root mean square error can be
used to assess the numerical prediction accuracy:
𝑇
1
𝑀𝑆𝐸 = ∙ ∑(𝑦𝑡 − 𝑦̂𝑡 )2 . (11)
𝑇
𝑡=1
Currently, Transformer-based models are successfully implemented in the form of graph
neural networks, whose ability to train on large amounts of data and effectively work with
sequences of any length make them indispensable tools in modern research and
development in the field of artificial intelligence [33–35].
Graph Neural Networks (GNN) are a powerful class of neural networks specialized for
analyzing data represented as graphs. Unlike GRNN neural networks, GNN has unique
advantages such as the ability to take into an account data structure, adaptability to various
graph structures, efficiency in modelling complex dependencies, and graph training ability.
Therefore, the proposed method uses a graph neural network (Fig. 2) in contrast to [1],
where the GRNN neural network was used.
Hidden layer Hidden layer Hidden layer Hidden layer
(encoder) (decoder) (positional encoding) (attention mechanism)
Input layer Output layer
Smooth Smooth Smooth Smooth
ReLU ReLU ReLU ReLU
... ... ... ...
Figure 2: Graph neural network general view. (author's research, based on [33–35]).
At the initial stage of GNN training based on the Transformer architecture, it is assumed
that the input data are graphs, where nodes can contain features and edges can have
weights or other information. In this case, the graphs are represented in the form of an
adjacency matrix or adjacency lists. For the model training also requires dividing the data
into training, validation and test sets.
At the next stage, the Transformer architecture is adapted to work with graph data
according to (1)–(4). This involves using attention mechanisms (5) to take into account
information about neighbouring nodes and edges in the graph, which involves creating
input and output vector representations for nodes and edges and using multilayer
perceptron to compute aggregated features.
At the next stage, the loss function is calculated according to (10) and the root mean
square error according to (11). To minimize the loss function during training, an optimizer
is then selected, for which it is advisable to use Adam [36], which has an adaptive training
rate, provides stable training through the use of gradient moments, and usually
demonstrates good performance in practice in classification tasks.
In the next stage, the model training process on the training data set is carried out
directly, followed by testing its performance on the validation set. For each training epoch,
the training set is passed through. Each element of the training set is fed to the input of the
model, then the loss function (10) is calculated for this element and the gradients of the loss
function are calculated over the model parameters as:
𝜕𝐽(𝜃)
∇𝜃 𝐽(𝜃) = . (12)
𝜕𝜃
Using the computed gradients, the model parameters are updated using an update
equation based on the Adam optimizer:
𝑚 = 𝛽1 ∙ 𝑚 + (1 − 𝛽1 ) ∙ ∇𝜃 𝐽(𝜃), (13)
2
𝑣 = 𝛽2 ∙ 𝑣 + (1 − 𝛽2 ) ∙ (∇𝜃 𝐽(𝜃)) , (14)
𝑚
𝜃 = 𝜃 − 𝛼𝑡 ∙ , (15)
√𝑣 + 𝜖
where β1 = 0.9…0.999 and β2 = 0.9…0.999 are exponential smoothing coefficients for
estimates of the first and second moments of the gradient, respectively, m and v are
estimates of the first and second moments of the gradient, ϵ is a small number for numerical
stability, αt is the adaptive training rate, proposed in this work, calculated according to the
expression:
√1 − 𝛽2𝑡
𝛼𝑡 = 𝛼 ∙ , (16)
1 − 𝛽1𝑡
where α is the specified training rate.
After each training epoch, the model is evaluated on the validation dataset using
performance metrics for the given task. For the task of identifying potential defects in
complex dynamic objects, it is proposed to use the following performance metrics [37, 38]:
𝑇𝑃 𝑇𝑃
𝑃𝑟𝑒𝑐𝑖𝑠𝑖𝑜𝑛 = , 𝑅𝑒𝑐𝑎𝑙𝑙 = , (17)
𝑇𝑃 + 𝐹𝑃 𝑇𝑃 + 𝐹𝑁
2 ∙ 𝑃𝑟𝑒𝑐𝑖𝑠𝑖𝑜𝑛 ∙ 𝑅𝑒𝑐𝑎𝑙𝑙
𝐹1 − 𝑠𝑐𝑜𝑟𝑒 = ,
𝑃𝑟𝑒𝑐𝑖𝑠𝑖𝑜𝑛 ∙ 𝑅𝑒𝑐𝑎𝑙𝑙
where Precision is accuracy, Recall is completeness, TP (True Positives) is the correctly
identified defects number, FP (False Positives) is the incorrectly identified defects number,
and FN (False Negatives) is the defects that the model number could not identify.
If the task of identifying potential defects has several classes of defects, it is proposed to
calculate the accuracy for each class of defects to evaluate the performance of the model on
each class independently using the PerClassAccuracy metric is the percentage of correctly
identified defects for each class:
𝑇𝑃𝑖
𝑃𝑒𝑟𝐶𝑙𝑎𝑠𝑠𝐴𝑐𝑐𝑢𝑟𝑎𝑐𝑦 = , (18)
𝑇𝑃𝑖 + 𝐹𝑁𝑖
where TPi (True Positives) is the number of correctly identified defects of the i-th class, and
FNi (False Negatives) is the number of incorrectly identified defects of the i-th class.
Model training stops when a stopping criterion is reached, for example, the maximum
number of epochs, at which if the current training epoch is equal to the maximum number
of epochs, then model training is completed. Another stopping criterion is the lack of
improvement on the validation dataset for a certain number of epochs. For this criterion, an
early stopping mechanism is usually used, which allows you to stop training if the
performance of the model on the validation dataset does not improve over a certain number
of epochs P. Formally, this can be written as follows: let val_Li be the value of the loss
function on the validation dataset data set after the i-th training epoch. Then, for the i-th
training epoch, it is checked whether the model’s performance has improved on the
validation data set:
𝐼𝑚𝑝𝑟 = 𝑣𝑎𝑙𝐿𝑖−1 − 𝑣𝑎𝑙𝐿𝑖 . (19)
If an improvement occurs (Impr > 0), the epoch counter is reset to zero without
improvement (no_improvement_epochs). If there is no improvement, the counter is
incremented by 1. If the counter reaches a preset value P, then model training stops.
Once the model is trained, its performance on a test dataset is evaluated to evaluate its
ability to generalize to new data.
The work proposes to use a new neuron activation function, Smooth ReLU, based on the
ReLU function. The main idea of modifying the ReLU function is to make the activation
function smoother and more continuous to improve the convergence and robustness of
training. The expression describes the Smooth ReLU activation function:
𝑥, if 𝑥 > 0,
𝑓 (𝑥 ) = { 1 (20)
, if 𝑥 ≤ 0,
1 + 𝑒 −𝛾∙𝑥
where γ is a parameter that determines the “degree of smoothness” of the function. When
x > 0, the function behaves like a regular ReLU. For x ≤ 0, it uses a sigmoid function to
smoothly transition from 0 to negative values. This avoids sudden “steps” in the gradient
and can help speed up the neural network's training.
The proposed neural activation function Smooth ReLU retains the benefits of ReLU (no
gradient for positive values) and adds smoothness for negative values, which can be useful
for improving training in some cases.
4. Experiment
At the preliminary stage of the computational experiment, the feasibility of using the
proposed neuron activation function Smooth ReLU (20) is mathematically justified in
comparison with the traditional ReLU 𝑓(𝑥 ) = max(0, 𝑥 ), which is used in graph neural
networks. The initial data of the problem is the loss function J (10), which must be
minimized, that is, J → min. To study the activation function of neurons, it is extremely
important to analyze their derivatives. The derivative of the activation function allows you
to analyze the rate of change of the activation function of neurons depending on changes in
input data. This, in turn, helps optimize the process of updating neuron weights during
neural network training.
The derivative of the traditional ReLU neuron activation function (Fig. 3, red curve) has
the form:
1, if 𝑥 > 0,
𝑓′(𝑥 ) = { (21)
0, if 𝑥 ≤ 0.
The derivative of the proposed Smooth ReLU neuron activation function (Fig. 3, black
curve) has the form:
1, if 𝑥 > 0,
−𝛾∙𝑥
𝑓′(𝑥 ) = { 𝛾 ∙ 𝑒 (22)
, if 𝑥 ≤ 0.
(1 + 𝑒 −𝛾∙𝑥 )2
a b
Figure 3: Diagram of the derivative of traditional ReLU function (red curve) and proposed
Smooth ReLU function (black curve). (author's research).
As can be seen from (21), (22), as well as from Fig. 3, a, the problem with regular ReLU
is that the derivative is 0 for all negative values of x, which can lead to a "dead neurons"
problem in a neural network, where neurons stop updating due to a zero gradient. The
advantage of Smooth ReLU is that it always has a non-zero gradient for all values of x,
including negative values. This avoids the problem of “dead neurons” and promotes more
stable training, especially in deep neural networks. Thus, the use of Smooth ReLU (Fig. 3, b)
is mathematically justified because it provides a smooth and continuous gradient over the
entire definition domain, which can help improve convergence and model training.
Based on [1], to conduct a computational experiment consisting of assessing the
performance of the proposed method for identifying potential defects, the research object
was TV3-117 is the helicopter TE of various helicopter modifications. To conduct the
computational experiment, we used a personal computer with an AMD Ryzen 5 5600
processor, which has 6 cores with 12 threads operating at 3.5 GHz and 32 GB of DDR-4 RAM.
At the first stage of the computational experiment, the creation, analysis and pre-processing
of a training sample is carried out, consisting of the following parameters that are recorded
on board the helicopter (Table 1): nTC is the gas generator rotor r.p.m., nFT is the free turbine
rotor speed, TG is the gas temperature in front of the compressor turbine, h is the flight
altitude, TN is the temperature, PN is the pressure, ρ is the air density [37–39]. All input
parameters are reduced to absolute values according to the theory of gas-dynamic similarity.
Table 1
The training set part (author's research, published in [37–39])
Number Gas generator rotor Free turbine rotor Gas temperature in
r.p.m. speed front of the
compressor turbine
1 0.929 0.943 0.932
2 0.933 0.982 0.964
3 0.952 0.962 0.917
4 0.988 0.987 0.908
5 0.991 0.972 0.899
… … … …
256 0.981 0.973 0.953
A detailed description of the process of preprocessing data from the training set
(Table 1) is given in [37–39]. This process includes several steps: data homogeneity
assessment, dividing them into control and test samples, and checking the
representativeness of both samples using cluster analysis. Data homogeneity is assessed
using the Fisher-Pearson test [40], the result of which confirms the homogeneity of the
samples and the hypothesis of normal distribution (at a significance level of 0.05 and 13
degrees of freedom, the resulting value χ2 = 3.588 does not exceed the critical value of 3.44).
The Fisher-Snedecor test [41] is also used to test homogeneity, confirming the homogeneity
of the training sample (at a significance level of 0.01 and 13 degrees of freedom, the
resulting value F = 1.28 does not exceed the critical value of 22.362). To assess
representativeness, data clusters identified by cluster analysis are examined (Fig. 4). After
the randomization procedure, training, and test samples are formed in a 2:1 ratio. Cluster
analysis shows that both samples contain eight classes, which indicates their
representativeness. The distances between clusters are almost the same in both samples,
which confirms their similarity and representativeness.
a b
Figure 4: Cluster analysis results: a – Original experimental dataset (I…VIII – classes); b –
Training dataset. (author's research, published in [37–39]).
In the course of graph neural network training (Fig. 2), the proposed algorithm obtained
dependences of the accuracy (Fig. 5), completeness (Fig. 6), and losses (Fig. 7) of the neural
network on the number of iterations (1000 iterations were used in the work), at which “blue
curve” means training on the training set, “orange curve” means validation on the control
set. From Fig. 5 it can be seen that the limiting value of accuracy practically reaches 1, Fig. 6
it is clear that the completeness is almost constant and equal to 1, and Fig. 7 shows that the
loss value does not exceed 0.025, which indicates the high efficiency of training the model
on the available data and its ability to accurately generalize to new data.
Figure 5: Diagram of changes in the neural network accuracy function with 1000 iterations.
(author's research).
Figure 6: Diagram of changes in the neural network recall function with 1000 iterations.
(author's research).
Figure 7: Diagram of changes in the neural network loss function with 1000 iterations.
(author's research).
Such results make the model potentially suitable for solving the task of identifying
potential defects in helicopter TE.
5. Results
Fig. 8 shows the results of identifying potential defects based on long-term prediction of the
gas temperature parameter in front of the compressor turbine TG in the components of the
TV3-117 TE. On the graph of the initial parameter TG, corresponding to the time point t =
94.65 hours, predicting is carried out. At t = 96.2 hours, there is a decrease in the TG
parameter by 4 %, and at t = 99.3 hours – by another 3%, for a total of 7 % decrease. At
times t = 96.2 hours and t = 99.3 hours Fig. 9 and 10 show the corresponding bursts,
indicating a possible malfunction of the engine combustion chamber – the potential
appearance of cracks (burnouts). In this way, normal engine operation can be guaranteed
for 8 hours. The results of these researches fully confirm previously conducted identical
research in [1].
Figure 8: The results of identification of potential defects in the combustion chamber of
TV3-117 TE. (author's research).
Figure 9: The results of identification of potential defects in the combustion chamber of
TV3-117 TE. (author's research).
Figure 10: The results of identification of potential defects in the combustion chamber of
TV3-117 TE. (author's research).
The red curve in Fig. 10 means an identified probable defect – the potential appearance
of cracks (burnouts) in the combustion chamber. The obtained data on long-term prediction
of the gas temperature parameter in front of the compressor turbine TG allows us to take
effective measures to detect and prevent potential defects in the engine combustion
chamber, which in turn helps to improve the safety and reliability of helicopter TE. Thus,
these researches have significant practical significance and can be used in the field of
maintenance and aviation equipment operation.
6. Discussion
Similar to [1], as a result of the comparative experiment, the results were obtained (Table 2)
of the developed and known methods: Simple Autoencoder, Autoencoder using the LSTM
neural network, the method developed in [1] using the GRNN neural network and the
developed method using graph neural network.
Table 2
Comparative experiment results (author's research)
SOTA-architectures Developed in [1] Developed
AutoEncoder method using method using
Simple
using the LSTM the GRNN neural the graph neural
Parameter
AutoEncoder
neural network network network
Precision
Precision
Precision
Precision
F1-scope
F1-scope
F1-scope
F1-scope
Recall
Recall
Recall
nTC 0.81 0.76 0.78 0.63 0.67 0.64 0.88 0.87 0.87 0.96 Recall
0.98 0.94
TG 0.86 0.84 0.85 0.77 0.77 0.77 0.92 0.91 0.91 0.99 0.97 0.96
nFT 0.61 0.57 0.57 0.67 0.64 0.67 0.79 0.78 0.78 0.95 1.0 0.93
Table 2 shows that the results of the proposed method using a graph neural network are
superior to almost all presented accuracy metrics. Alternative methods (Simple
AutoEncoder [42, 43], LSTM AutoEncoder [44, 45], and a similar method using the GRNN
neural network developed in [1]) also demonstrate deterioration in performance on the
same signals. Table 2 also shows that the results for all Precision, Recall, and F1-scope
metrics when using a graph neural network are better compared to Simple AutoEncoder,
LSTM AutoEncoder, and a similar method using the GRNN neural network [1]. These results
confirm the effectiveness and high accuracy of the proposed method using a graph neural
network while achieving high accuracy for all analyzed transformations. Thus, the results
of the proposed method are superior to almost all presented accuracy metrics compared to
alternative methods: Simple AutoEncoder, LSTM AutoEncoder, and a similar method using
the GRNN neural network [1].
At the final stage of the comparative experiment, the effectiveness of the proposed
method was assessed, which consisted of conducting a comparative analysis of the accuracy
of classical and neural network methods for identifying potential defects, the results of
which are given in Table 3, which displays the results of calculating the probability of errors
of I and II types when identifying defects in the compressor (nTC parameter), combustion
chamber (TG parameter) and compressor turbine (nFT parameter). In this case, errors of the
first (FPR) and second (FNR) types were calculated as [46, 47]:
𝐹𝑃 𝐹𝑁
𝐹𝑃𝑅 = , 𝐹𝑁𝑅 = , (23)
𝐹𝑃 + 𝑇𝑁 𝐹𝑁 + 𝑇𝑃
where False Positives (FP) is the number of observations that are incorrectly classified as
positive, False Negatives (FN) is the number of observations that are incorrectly classified
as negative, True Positives (TP) is the number of observations that are correctly classified
as positive, True Negatives (TN) is the number of observations that are correctly classified
as negative.
A type I error means rejecting the null hypothesis when it is true, while a type II error
means accepting the null hypothesis when it is false. In the context of identifying potential
defects, the null hypothesis is defined as: “There are no potential defects in the test item.”
This means that under the null hypothesis, there are no anomalies, errors, or defects to be
identified or corrected.
Table 3
Results of calculating errors of the I and II types (author's research)
Identification Method Probability of error in potential defect
identification, %
Compressor Combustion Compressor
defect chamber defect turbine defect
Type I Type II Type I Type II Type I Type II
error error error error error error
Classical method
1.75 1.35 2.48 1.76 2.23 1.51
(Tolerance control)
Neural Network Method:
Simple AutoEncoder 1.15 0.94 1.24 1.09 1.23 1.06
LSTM AutoEncoder 0.95 0.58 1.02 0.63 0.99 0.61
GRNN neural network [1] 0.62 0.31 0.61 0.29 0.60 0.27
Graph neural network 0.33 0.17 0.33 0.16 0.32 0.14
As can be seen from Table 3, the use of a graph neural network reduces errors of the first
and second types compared to the use of Simple AutoEncoder by 3.49...7.58 times, LSTM
AutoEncoder – by 2.88...4.36 times, and the GRNN neural network [1] – by 1.81...1.93 times.
Therefore, we can conclude that the use of a graph neural network in the method of
identifying potential defects reduces errors of the first and second types by almost 2 times
compared with the use of the GRNN neural network [1].
For four classification classes (True Positives, True Negatives, False Positives, False
Negatives) confusion matrix was developed (Table 4) [48–50]. Each cell of the confusion
matrix shows the number of times the actual class (rows) was classified as the predicted
class (columns) for each method.
Table 4
Developed confusion matrix (author's research)
Actual \ Developed Developed in Simple AutoEncoder
Predicted method using [1] method AutoEncoder using the
the graph using the LSTM neural
neural GRNN neural network
network network
True Positives 95 3 2 0
True Negatives 2 90 6 2
False Positives 1 5 85 9
False Negatives 0 2 7 91
The confusion matrix demonstrates that the GNN is the most accurate method, correctly
classifying the majority of instances in all classes with minimal misclassifications. The GRNN
network, while performing well, shows slightly more errors, particularly in distinguishing
between classes “True Negatives” and “False Positives”. The Simple Autoencoder
demonstrates moderate accuracy but has noticeable misclassifications across all classes,
particularly in misidentifying class “False Positives” instances. The Autoencoder using the
LSTM neural network exhibits the lowest accuracy, with significant misclassifications,
especially confusing instances of classes “True Positives”, “True Negatives” and “False
Positives”, while showing better performance for class “False Negatives”. Overall, the GNN
method clearly outperforms the others, followed by GRNN, Simple Autoencoder, and LSTM
Autoencoder in descending order of classification quality.
To conduct ROC analysis for four methods (Graph Neural Network, GRNN Network,
Simple Autoencoder, Autoencoder using the LSTM neural network), true positive and false
positive rates were calculated for each class and method, and then the corresponding ROC
curves were plotted. To do this, a binary classification is created for each class (this class
versus all others). For each class, True Positive Rate (TPR) and False Positive Rate (FPR)
are calculated as [49–51]:
𝑇𝑃 𝐹𝑃
𝑇𝑃𝑅 = , 𝐹𝑃𝑅 = . (24)
𝐹𝑃 + 𝑇𝑁 𝐹𝑃 + 𝑇𝑁
The area under the ROC curve (AUC) can be estimated using the trapezium equation. If
(TPRi and FPRi) are the ROC curve points coordinates, then AUC can be calculated as follows:
𝑃−1
𝑇𝑃𝑅𝑖 + 𝑇𝑃𝑅𝑖+1
𝐴𝑈𝐶 = ∑ ∙ (𝐹𝑃𝑅𝑖+1 − 𝐹𝑃𝑅𝑖 ), (25)
2
𝑖=1
where P is the points number on the ROC curve. Table 5 shows the ROC analysis results.
Table 5
ROC analysis results (author's research)
Actual \ Developed Developed in Simple AutoEncoder
Predicted method using [1] method AutoEncoder using the
the graph using the LSTM neural
neural GRNN neural network
network network
True Positives 95 90 2 0
True Negatives 3 7 8 10
False Positives 270 266 290 290
False Negatives 30 35 98 100
TPR 0.76 0.72 0.02 0
FPR 0.011 0.025 0.027 0.33
AUC 0.838 0.724 0.445 0.292
Thus, the use of GNN gives high accuracy with a low level of false positive results; the use
of GNN also gives high accuracy, but 1.16 times lower than that of GNN. Using Simple
Autoencoder gives moderate accuracy, with noticeable errors. Using LSTM Autoencoder
gives the lowest accuracy with the highest number of false positives.
7. Conclusions
The neural network method for identifying potential defects in complex dynamic objects
(using the example of helicopter turboshaft engines) has been further developed, which
differs from the existing one in that, through the use of the Transformer model rather than
the GRNN neural network, it has made it possible to identify potential defects in complex
dynamic objects with almost 100 % accuracy (using the example of the potential
appearance of cracks (burnouts) in the combustion chamber of helicopter turboshaft
engines due to a predicted decrease in gas temperature in front of the compressor turbine
values).
A neural network implementation of the Transformer model is proposed using a graph
neural network in the proposed method for identifying potential defects in complex
dynamic objects, which made it possible to reduce errors of the first and second types in
comparison with the use of Simple AutoEncoder by 3.49...7.58 times, LSTM AutoEncoder by
2.88...4.36 times, neural GRNN network – 1.81…1.93 times.
Improved ReLU activation function in the form of Smooth ReLU to make the activation
function smoother and more continuous, and, as a result, improve the convergence and
stability of training. By analyzing the derivatives of the ReLU and Smooth ReLU functions, it
is determined that the Smooth ReLU function solves the "dead neurons" task associated
with regular ReLU by providing a non-zero gradient for all input data values, including
negative ones, which provides more stable training of neural networks and prevents
neurons from stopping updating due to zero gradient.
Acknowledgements
This research was funded by the Ministry of Internal Affairs of Ukraine "Theoretical and
applied aspects of the development of the aviation sphere" under Project
No. 0123U104884.
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