Predicting Rail Corrugation Based on Convolutional Neural Networks Using Vehicle’s Acceleration Measurements

Citation: Haghbin, M.; Chiachío, J.; Muñoz, S.; Escalona Franco, J.L.; Guillén, A.J.; Crespo Marquez, A.; Cantero-Chinchilla, S. Predicting Rail Corrugation Based on Convolutional Neural Networks Using Vehicle’s Acceleration Measurements. Sensors 2024,24, 4627. https://doi.org/ 10.3390/s24144627 Received: 7 May 2024 Revised: 10 July 2024 Accepted: 13 July 2024 Published: 17 July 2024 Copyright: © 2024 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). sensors Article Predicting Rail Corrugation Based on Convolutional Neural Networks Using Vehicle’s Acceleration Measurements Masoud Haghbin 1,* , Juan Chiachío 1, Sergio Muñoz 2, Jose Luis Escalona Franco 2, Antonio J. Guillén 3, Adolfo Crespo Marquez 4and Sergio Cantero-Chinchilla 5 1Department of Structural Mechanics and Hydraulic Engineering, Andalusian Research Institute in Data Science and Computational Intelligence (DaSCI), University of Granada (UGR), 18001 Granada, Spain; [email protected] 2Department of Materials and Transportation Engineering, Escuela Técnica Superior de Ingeniería, University of Seville, 41092 Seville, Spain; [email protected] (S.M.); [email protected] (J.L.E.F.) 3Department of Management, Complutense University of Madrid, 28040 Madrid, Spain; [email protected] 4Department of Industrial Management, Escuela Técnica Superior de Ingeniería, University of Seville, 41092 Seville, Spain; [email protected] 5School of Electrical, Electronic and Mechanical Engineering, University of Bristol, Bristol BS8 1TR, UK; sergio.canter[email protected] *Correspondence: [email protected] Abstract: This paper presents a deep learning approach for predicting rail corrugation based on onboard rolling-stock vertical acceleration and forward velocity measurements using One-Dimensional Convolutional Neural Networks (CNN-1D). The model’s performance is examined in a 1:10 scale railway system at two different forward velocities. During both the training and test stages, the CNN-1D produced results with mean absolute percentage errors of less than 5% for both forward velocities, confirming its ability to reproduce the corrugation profile based on real-time acceleration and forward velocity measurements. Moreover, by using a Gradient-weighted Class Activation Mapping (Grad-CAM) technique, it is shown that the CNN-1D can distinguish various regions, including the transition from damaged to undamaged regions and one-sided or two-sided corrugated regions, while predicting corrugation. In summary, the results of this study reveal the potential of data-driven techniques such as CNN-1D in predicting rails’ corrugation using online data from the dynamics of the rolling-stock, which can lead to more reliable and efficient maintenance and repair of railways. Keywords: rail corrugation; deep learning; convolutional neural networks; Grad-CAM 1. Introduction Corrugation is a common defect in rail contact areas with important implications in safety, serviceability, and cost [ 1 ]. Even when the depth of the waves is typically less than 1 mm, rail corrugation can severely impact passenger comfort by causing abnormal accelerations and loud noises. In addition, it is one of the primary reasons for damage and wears both the wheels and rails, leading to higher maintenance costs and service disruption [ 2 , 3 ]. According to numerous researchers, the root cause of this defect lies in the self-excited vibrations and resonance of the wheel–rail system. However, this self-excited vibrations perspective is not universal, and many parameters such as initial regularity, bending mode, torsional vibration, and material mechanics (rail elastic modulus, rail Poisson’s ratio, stiffness of fasteners, etc.) contribute to corrugation generation [2,4–6]. Since 1895, many attempts [ 7 – 9 ] have been carried out on laboratory experiments, field tests, and analytical and numerical models to predict corrugation [ 5 ]. Before 1970, the research on this topic focused on statistical and experimental analysis as well as aspects related to rail material properties, friction, temperature variations, and other relevant Sensors 2024,24, 4627. https://doi.org/10.3390/s24144627 https://www.mdpi.com/journal/sensors Sensors 2024,24, 4627 2 of 18 factors [ 10 ]. After 1970, researchers started to use analytical and numerical models to understand the impacts of resonance, plastic deformation, and residual stress, among other factors, on railway corrugation [10–12]. For example, Igeland and Ilias (1997) [ 13 ] presented a non-linear model to predict rail corrugation on short-wavelength longitudinal irregularities. They applied various non-linear factors, such as the shift of contact point and the distribution of wear within the contact area. In 2003, Nielsen [ 14 ] introduced a numerical model for analyzing rail-head corrugation on straight rails based on dynamic train–track interaction. He concluded that longitudinal slip in the wheel–rail contact interface played a key factor in causing this damage, and his results showed a strong agreement when compared to field observations. Sun and Simpson (2008) [ 15 ] carried out numerical modeling and laboratory experiments, which revealed that a tighter curve can cause increased creepage between the wheel and rail, thus causing the wheel stick-slip process, which is a significant factor in the development of corrugation. Various studies have attempted to predict corrugation growth through frequency domain analysis, mainly using Fast Fourier Transform (FFT) models concentrating on the interaction between wheel and rail at various speeds [ 16 – 18 ]. Employing a field investigation and Finite Element (FE) strategies, Chen et al. (2020) [ 19 ] determined the wavelengths and frequencies of rail corrugations in three distinct field systems. Their findings indicate that corrugation is nearly certain to occur in the inner rail of a tightly curved track. However, in the case of smooth curved and tangential tracks, they found that friction-induced self-excited vibration does not occur because the creep force is not saturated. In line with this study, Wang et al. (2020) [ 4 ] developed an FE model to simulate rail corrugation in Cologne egg fastener sections in small radius curves. The model consists of three sub-models: vehicle, track and wheel–rail contact models. Their wheel–rail contact model comprises the dynamic model of the wheel-track and the non-dynamic model of friction and wear of rail material. After integrating the friction and wear model into the dynamic model, it was revealed that side wear mainly occurs on the outer rail. In contrast, corrugation tends to appear on the inner rail. Despite the success of these physics-based approaches in estimating corrugation, they are time-consuming, computationally expensive, and sensitive to simplifying assumptions and boundary conditions [ 20 ]. It is essential to investigate alternative modeling strategies further to address these drawbacks. Machine Learning (ML) models are popular nowadays due to their excellent capability to tackle complex problems by efficiently handling non-linearities in uncertain systems [ 21 ]. In early investigations, ML models, particularly those utlizing images, such as Two Dimensional Convolutional Neural Networks (CNN-2D), have been used to identify corrugation defects and fault segmentation in railways [ 22 – 25 ]. Nevertheless, there has been a clear need to measure the corrugation geometry properties using tabular data obtained through forward velocity and acceleration sensors, which are becoming increasingly accessible nowadays. Despite their importance for research, only a few studies have attempted to predict corrugation geometry properties such as depth and wavelength using acceleration and sensor data. Xie et al. (2021). [ 20 ] successfully predicted corrugation on metro lines using a novel Deep Learning (DL) model that effectively integrated a One-Dimensional Convolutional Neural Network (CNN-1D) classifier with a hybrid Kriging Surrogate Model (KSM) and Particle Swarm Optimization (PSO) algorithm. In the study, CNN-1D was applied to detect the rail corrugation’s state and classify its wavelength, followed by adopting the KSM-PSO to estimate its depth characteristics. According to their findings, the generated results were in agreement with the observed data. In line with this study, Xie et al. (2023) [ 26 ] detected the rail corrugation roughness quantitatively using a modified version of CNN-1D named Recurrent Convolutional Net (RCNet). They used axle box acceleration signals as the input parameters. The model was trained using field data and compared against other DL techniques, such as Long-Short Term Memory (LSTM) and Sensors 2024,24, 4627 3 of 18 Recurrent Neural Networks (RNNs). Their findings demonstrated that RCNet was superior to other models in accurately predicting corrugation roughness. The insights gained through ML methods have proven to be valuable in predicting corrugation on railways. However, an important research challenge remains: predicting railway corrugation properties using kinetic data from rolling stock collected during normal railway operation. In this context, this paper aims to address the following objectives: • develop a deep learning model to accurately reproduce the rail’s corrugation profile at different forward velocities using kinetic data from the vehicle; • investigate the critical railway regions that are important for corrugation prediction by DL models. In particular, the proposed modeling strategy employs forward velocity and acceleration data in the x (forward) and z (vertical) directions from the vehicle, which are commonly available in real railway systems. The corrugation was predicted using CNN-1D, and the Gradient-weighted Class Activation Mapping (Grad-CAM) analysis was then used to gain a better understanding of the model’s performance. The CNN-1D model is trained using acquired data about the vehicle’s acceleration and velocity collected at a laboratory scale in the railway research group of the University of Seville. The results obtained confirm the suitability of the proposed data-driven modeling approach in predicting the complete rail’s corrugation profile using just kinetic data from the vehicle. The remainder of this paper is structured as follows. Section 2presents the research methodology, which includes a detailed description of the experimental setup and preprocessing techniques, the DL technique used, the design of the model architecture, and hyper-parameter selection. Section 3will present the main findings, discuss their implications, summarize limitations, and suggest future research directions. Finally, conclusions are provided in Section 4. The research methodology of this study is illustrated in Figure 1. Design of CNN-1D architecture & hyper-parameters selection Sections (2.3) & (2.3.1) Section (2.2) Evaluation of CNN-1D model’s results Section (3.1) Evaluation metrics for models’ performance Section (2.3.2) Grad-CAM analysis Section (3.2) Pre-process of gathered experimental data Experimental setup & gathering data Section (2.1) Figure 1. Process of the developed methodology for the prediction of railway corrugation using CNN-1D. Sensors 2024,24, 4627 4 of 18 2. Material and Methods 2.1. Description of Experimental Setup This section explains the experimental setup used to gather the datasets. Considering the difficulty of carrying out experiments on actual railroad vehicles and tracks, the railway research group of the University of Seville built a 1:10 scaled track facility, comprising a 90 m long scaled track and an instrumented scaled vehicle [ 27 ]. The 1:10 scaled track facility where the experiments were conducted is presented below. The experiments were carried out on a 5-inch wide and 90 m long scaled track built on the roof of the School of Engineering of the University of Seville. Figure 2shows the plan view of the scaled track: an aerial photograph (Figure 2a) and a schematic plan view of the track center line (Figure 2b). (a) Aerial photograph. Position x (m) 0 10 20 30 40 50 60 Position y (m) -40 -35 -30 -25 -20 -15 -10 -5 0 5 10 Corrugated region (b) Plan scheme of the track center line. Figure 2. Plan view and geometry details of the scaled track. The designed track includes straight regions, transitions, and curves. The red rectangle highlights the straight corrugated region in Figure 2b. This region has a length of 3.6 meters (m), starting at a track length s = 54.9 m and finishing at s = 58.5 m. Both rails in this region have been machined to include a realistic corrugation profile. The corrugated profile given to the surface of the rail is based on a combination of four harmonic waves of 5, 10, 20, and 30 mm wavelength with amplitudes of 30, 45, 60, and 75 microns, respectively (see Figure 3). As there is no phase difference between the four waves combination, it results in the 60 mm periodic irregularity shown in Figure 3. This profile (60 mm long) has been repeated consecutively until reaching the total length of 3.6 m. 0 10 20 30 40 50 60 s (mm) -200 -150 -100 -50 0 50 100 150 200 Corrugation profile (mm) Wave type Wavelength Amplitude 1 5 30 4 30 75 2 10 45 3 20 60 (mm) (mm) Figure 3. Profile of the designed corrugation. Figure 4a shows a detail of the corrugated region machined in the scale track. In addition to corrugation, the track has long wavelength irregularities affecting vehicle dynamics. However, since this study aims to predict rails’ corrugation, the effects of the long wavelength irregularities on the accelerometer signals are subtracted by filtering the signals with a high pass filter. The scaled railway vehicle used in this work was designed and built by the railway research group of the University of Seville. As shown in Figure 4b, the scaled vehicle is Sensors 2024,24, 4627 5 of 18 a single bogie consisting of two rigid wheelsets, a primary suspension with eight helical springs connecting both wheelsets with the bogie frame. To register the acceleration in the axle box, two uni-axial piezoelectric accelerometers have been installed in the left axle box in the front wheelset of the vehicle in longitudinal (x) and vertical (z) directions. Additionally, one high-precision encoder registers the rotation of the wheels, from which the distance travelled by the vehicle ( s ) and its forward velocity ( V ) is calculated. All the experimental data are acquired and synchronized by a data acquisition system mounted on the vehicle at an acquisition rate of 5000 Hz. The following paragraphs provide a brief explanation of the monitoring framework designed to track degradation in this study. (a) Detail of the corrugated segment. Accelerometers Encoder Ride direction (b) Monitored vehicle. Figure 4. Experimental setup details. The test bench to monitor track degradation involves an automated inspection vehicle equipped with various advanced sensors and measurement tools. The vehicle comprises two main parts: the body and the measuring axis, which are connected by a spherical joint to allow for rotation isolation. The body has a traction system with rigid wheels powered by a brushless electric motor and direct gear transmission, ensuring the axis stays centered on the track. The measuring axis, the main component of the vehicle, includes a linear guide and two moving carriages. It is equipped with a precision encoder, inductive sensor, Inertial Measurement Unit (IMU), inclinometer, and Linear Variable Differential Transformer (LVDT) to measure track gauge variation accurately. A NImyRIO- 1900 controller (National Instruments, Austin, TX, USA) and a mini-PC with 3G connectivity are used to control the vehicle and acquire sensor data. The cloud-based data handling and processing framework is built on Microsoft Azure, following the RAMI 4.0 hierarchical model. At Level 0, physical assets and processes are managed by sensors and devices connected via PLCs, Fieldbus, distributed systems, or servers, with data communicated Sensors 2024,24, 4627 6 of 18 to the central system using Azure IoT Edge for real-time data transmission. At Level 1, data ingestion and processing occur through Azure Data Lake Storage for long-term data retention and Azure Data Explorer for short-term analytics. Levels 2 and 3 involve model and digital twin management using Azure Machine Learning and Cosmos DB to develop predictive models and Digital Twin Definition Language (DTDL) to define digital entities. Finally, Level 4 facilitates user interaction through Power BI, providing real-time decision-making support. Figure 5illustrates the test bench configuration used in this study to monitor track degradation. System of the systems (Sensors, Signal processer, etc. ) Physical Entities (Level 0) Adapters Data Repository Metadata Repository Cloud computing Data Ingestion & Processing (Level 1) Model Repository Model Management Ontologies Model Management (Level 2) Digital Process Management Digital Services Management Twin Management (Level 3) User Interface & Control User Interaction (Level 4) Power BI Azure Machine Learning Azure Cosmos DB Stream Analytics Azure Functions Azure Data Explorer Azure IoT Central Azure Digital Twins Figure 5. Flowchart of the test bench for monitoring track degradation. 2.2. Description and Preprocessing of the Gathered Datasets To predict railway corrugation, laboratory experiments mentioned in the previous section measure three input variables: forward velocity ( V ), acceleration in the z-direction (Az), and acceleration in the x-direction (Ax) [20]. To ensure no variable dominates due to its scale, Az and Ax are normalized by V2 . It is then assumed that the corrugation depends on a group of dynamic factors as follows: Corrugation =fV,Ax V2,Az V2(1) The input and output variables are preprocessed using the moving Root Mean Square (RMS) to improve the signal quality and mitigate noise. Utilizing the RMS method to preprocess the data is useful in real-world scenarios where raw signals, such as acceleration and velocity data, are affected by interference from vibrations of other mechanical components and data sequences are not perfectly aligned in time. The RMS method aids in reducing noise and temporal misalignment by averaging the squared values of the signal over a specific time window, thereby smoothing out short-term fluctuations and emphasizing the underlying trend in the data. It is mathematically expressed as follows: xRMS[i] = v u u t 1 N i ∑ k=i−N+1 x2[k](2) Sensors 2024,24, 4627 7 of 18 where xRMS[i] refers to the RMS value at the i th point, N identifies the number of data points, and x[k] indicates the value of the signal at the k th point. These input variables were gathered at V= 0.50 m/s and V= 1.00 m/s. Table 1shows the input variable range for each velocity. Table 1. Range of input variables utilized to predict corrugation. Original Values V(m/s) Vmin Vmax Ax V2min Ax V2max Az V2min Az V2max 0.50 0.435 0.513 −4.62 7.677 −3.7249 3.297 1.00 0.941 0.988 −3.102 3.527 −2.339 2.1908 RMS Values V(m/s) Vmin Vmax Ax V2min Ax V2max Az V2min Az V2max 0.50 0.07 0.381 0.003569 0.452 0.005149 0.146 1.00 0.168 0.787 0.00649 0.749 0.1031 0.294 Figure 6illustrates a comparison between the original and the RMS of the input and output signals at V=1.00 m/s, which offers a clear vision of the data transformation. 55 55.5 56 56.5 57 57.5 58 Distance travelled (m) 0.94 0.95 0.96 0.97 0.98 V (m/s) Original signal 55 55.5 56 56.5 57 57.5 58 Distance travelled (m) 0.17 0.175 0.18 0.185 0.19 RMS-V (m/s) RMS signal (a) Top: original velocity signal; bottom: RMS velocity signal. 55 55.5 56 56.5 57 57.5 58 Distance travelled (m) -4 -2 0 2 4 Ax/V2 (1/m) Original signal 55 55.5 56 56.5 57 57.5 58 Distance travelled (m) 0 0.2 0.4 0.6 0.8 RMS-Ax/ V2 (1/m) RMS signal (b) Top: original Ax V2signal; bottom: RMS processed Ax V2signal. Figure 6. Cont. Sensors 2024,24, 4627 8 of 18 55 55.5 56 56.5 57 57.5 58 Distance travelled (m) -2 -1 0 1 2 Az/V2 (1/m) Original signal 55 55.5 56 56.5 57 57.5 58 Distance travelled (m) 0 0.1 0.2 0.3 RMS-Az/V2 (1/m) RMS signal (c) Top: original Az V2signal; bottom: RMS processed Az V2signal. 55 55.5 56 56.5 57 57.5 58 Distance travelled (m) -2 -1 0 1 2 Corrugation (m) 10-4 Original signal 55 55.5 56 56.5 57 57.5 58 Distance travelled (m) 7 7.5 8 8.5 9 RMS-Corrugation (m) 10-5 RMS signal (d) Top: original corrugation signal; bottom: RMS processed corrugation signal. Figure 6. Comparative analysis of original signals and their corresponding RMS transformation for V=1.00 m/s. 2.3. Deep Learning Predictive Models for Corrugation Prediction Various deep learning architectures, such as Recurrent Neural Networks (RNNs) and Long Short-Term Memory (LSTM), were initially adopted and optimally configured to address the problem. However, results showed poor generalization due to the complexity of the problem and due to the fact that the past values of the target variable (corrugation) cannot be used as input here, since real-time measurements of corrugation are not available in real scenarios. This issue made it impractical to use an autoregressive component approach in time series using RRNs and LSTM, which relies on past values of the dependent variable (target) as input. Instead, one-dimensional convolutional neural networks (CNN-1D) showed superior predictive capability for this problem and were adopted in this research. Convolutional Neural Network 1D (CNN-1D) is a powerful deep learning model designed to deal with the difficulties of time series regression [ 28 ]. This capability is particularly important because time series data often contains complex temporal patterns, noises, and outliers that can challenge traditional machine learning methods to capture the patterns accurately [ 29 ]. The CNN-1D model is a modified version of CNN-2D that can adjust convolutional layers typically used in videos and images to 1D sequences [ 30 , 31 ]. Sensors 2024,24, 4627 9 of 18 This capability allows the CNN model to handle temporal data, making it a suitable choice for predicting one-dimensional sequences at each time step [ 32 ]. Further details of the CNN-1D model can be found in [30]. This study adopts the triple-layer CNN-1D architecture, which comprises three consecutive convolutional layers and two dense layers. Each convolutional layer has appropriate kernel sizes and is equipped with Rectified Linear Unit (ReLU) activation functions to enhance the learning process. After the convolutional layers, the outputs of these layers are flattened and then processed through two fully connected layers, which use linear activation functions. Finally, the output of the last layer is the prediction of corrugation. The architecture of the developed CNN-1D is illustrated in Figure 7. This figure shows the process of passing input data through multiple convolutional layers to extract patterns. The flattening layer then converts the output from the convolutional layers into a one-dimensional vector, preparing it for input into the fully connected layers. In a fully connected layer, each neuron in one layer is connected to every neuron in the next layer. The number of neurons or nodes in each layer is denoted by the term “unit”. This is the final step in the processing to generate the output. Each layer has different hyper-parameters, which are explained in detail in Section 2.3.1. Layer-1: Kernel Number=256 Kernel Size=4 Activation=ReLU Strides=1 Padding=Valid Layer-2: Kernel Number=128 Kernel Size=4 Activation=ReLU Strides=1 Padding=Valid Layer-3: Kernel Number=64 Kernel Size=4 Activation=ReLU Strides=1 Padding=Valid Layer-4: Number of Units=10 Activation=Linear Layer-5: Number of Units=1 Activation=Linear V=0.5 m/s : 3 × 47,952 V=1.0 m/s : 3 ×22,816 Input layers Convolutional layers Flatten layer Fully connected layers Output layer Figure 7. Architecture of the proposed CNN-1D model. 2.3.1. The Training Process and Hyper-Parameter Selection This section discusses the selection of hyper-parameters to clarify how we improve the model’s performance and ensure its reliability. The kernel unit is the core of the CNN- 1D model used for time series analysis; therefore, the accurate selection of the number of kernels and their sizes plays a crucial role in the model’s efficiency. Capturing the features and patterns relied on the number of kernels. Increasing the number of kernels when the data is insufficient can lead to overfitting issues [ 33 ]. Determining the optimal number of kernels is crucial based on task complexity, network depth, and computational resources [34]. Kernel size refers to the window length that convolves over the data. The size of this window can be different for each convolutional layer. The larger kernel size leads to capturing the broader pattern, while the smaller ones are better for particular details [ 35 ]. In this study, various numbers of kernels and kernel sizes were identified for the proposed model. The model employed 256, 128, and 64 kernels to predict corrugation in three consecutive CNN layers. The size of the kernels was set to 4 for the model across different forward velocities. The size of the convolution kernel is fixed at 4 to balance the trade-off between capturing sufficient features and computational efficiency. In practical applications, the choice of convolution kernel size depends on the input data’s specific characteristics and the desired detail level. It is often beneficial to experiment with different kernel sizes and even design convolutional layers with varying kernel sizes to capture diverse features at multiple scales. To tackle the dynamic challenges, such as shifting between observed and predicted results, a trial and error strategy has been applied to arrive at the aforementioned hyper-parameters. Our trial-and-error approach was based on the initial ranges of hyper- Sensors 2024,24, 4627 16 of 18 The tuning of hyper-parameters is also a challenging stage. In this study, selecting the appropriate kernel size and learning rate proved crucial in solving data-shifting and reducing generalization errors. The failure of the models to capture the damage patterns at high forward velocities due to fewer data points emphasizes the need for more laboratory experiments. In future works, physics-based models and data-driven techniques shall be fused to enhance the accuracy and efficiency of predictions for both low and high velocities. This development helps us exploit both approaches’ potential to improve model predictability. Furthermore, the corrugation predictions can be integrated into a higher-level railway maintenance model to analyze the impact of different levels of damage on the overall railway operation and maintenance. 4. Conclusions The modelling of vehicle-track dynamics is still a fundamental challenge in the railway industry. In this sense, incorporating the new advances in artificial intelligence and data analysis opens a new door that complements the results of physics-based models and leads to a better understanding of degradation processes and their causes. This work demonstrates how new artificial intelligence techniques can be useful in solving challenging engineering problems, such as railway corrugation prediction, which has been partially unsolved using physics-based methods. The present study evaluates the capability of CNN- 1D to reproduce corrugation damage in railway systems based on vehicle kinetic data taken at two different forward velocities. The study uses the moving Root Mean Square of forward velocity and acceleration in the longitudinal and vertical directions, normalized by the square of velocity, as input variables to predict corrugation. The findings reveal that the CNN-1D model can reproduce the corrugation profiles with relatively high accuracy. Grad-CAM analysis was also applied to determine regions with a higher impact on predicted results. The Grad-CAM analysis revealed CNN-1D’s capabilities to distinguish between different railway regions, such as undamaged as well as one-sided and two-sided damaged regions. Regardless of informing the model about various transition regions on the railway, the CNN-1D model successfully detected and assigned different activation intensity values to data in transition regions to predict corrugation, which shed light on its robust internal decision-making process to map the relationship between inputs and outputs. The results of this study contribute to enhancing proactive maintenance in railways by accurately predicting rail corrugation and detecting damage during normal railway operation. Specifically, the proposed predictive corrugation modeling approach enables monitoring of the state of degradation of extensive railway segments and even the entire network since it is based on kinematic data collected during operation. This predictive modeling can also be integrated within a whole-system condition-based maintenance modeling scheme, employing Petri nets or similar methods to support anticipated and data-driven maintenance decision making. Author Contributions: M.H.: Conceptualization, Methodology, Writing—original draft, and Visualization. J.C.: Supervision, Conceptualization, Methodology, Validation, and Writing—review & editing. S.M.: Laboratory experiments, Formal analysis, and Writing—original draft. J.L.E.F.: Laboratory experiments, Formal analysis, and Writing—review. A.J.G.: Formal analysis and Writing—review & editing. A.C.M.: Formal analysis and Writing—review, S.C.-C.: Methodology, Validation, and Writing—review. All authors have read and agreed to the published version of the manuscript. Funding: This work has been developed within the framework of the project Geminhi (Digital model for Intelligent Maintenance based on Hybrid prognostics models) (Grant US-1381456, founded by Junta de Andalucía, Andalucía FEDER 2014–2020) and the AMADIT Project (PID2022-137748OB-C32), funded by MCIN/AEI/10.13039/501100011033/FEDER, EU. Institutional Review Board Statement: Not applicable. Sensors 2024,24, 4627 17 of 18 Informed Consent Statement: Not applicable. Data Availability Statement: Dataset available on request from the authors. Acknowledgments: The authors would like to acknowledge the valuable contribution given by Daniel Molina Cabrera for his comments and useful discussions. Conflicts of Interest: The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. References 1. Grassie, S. Rail corrugation: Characteristics, causes, and treatments. Proc. Inst. Mech. Eng. Part F J. Rail Rapid Transit 2009, 223, 581–596. [CrossRef] 2. Zhao, Y.; Zhao, C.; Wang, L.; Wang, P. A rail corrugation evaluation method using fractal characterization based on structure function method. Wear 2022,506, 204454. [CrossRef] 3. Liu, X.; Wang, P.; Wan, F. Formation mechanism of rail corrugations in heavy-haul rail line. J. China Railw. Soc. 2000,22, 98. 4. Wang, Z.; Lei, Z.; Zhao, Y.; Xu, Y. Rail corrugation characteristics of Cologne egg fastener section in small radius curve. Shock. Vib. 2020,2020, 1827053. [CrossRef] 5. Wang, P.; Liu, X.; Wan, F. Torsional vibration of wheelsets and curved rail corrugations. J. Southwest Jiaotong Univ. 1996,31, 58. 6. Cui, X.; Yao, J.W.; Hu, X.Y.; Sun, L.; Chang, C. Rotation effect of flexible wheelset on wheel-rail force in Euler coordinate system. China Railw. Sci. 2019,40, 120–128. 7. Liu, Q.; Zhang, B.; Zhou, Z. An experimental study of rail corrugation. Wear 2003,255, 1121–1126. [CrossRef] 8. Li, W.; Wang, H.; Wen, Z.; Du, X.; Wu, L.; Li, X.; Jin, X. An investigation into the mechanism of metro rail corrugation using experimental and theoretical methods. Proc. Inst. Mech. Eng. Part F J. Rail Rapid Transit 2016,230, 1025–1039. [CrossRef] 9. Diana, G.; Cheli, F.; Bruni, S.; Collina, A. Experimental and numerical investigation on subway short pitch corrugation. Veh. Syst. Dyn. 1998,29, 234–245. [CrossRef] 10. Sato, Y.; Matsumoto, A.; Knothe, K. Review on rail corrugation studies. Wear 2002,253, 130–139. [CrossRef] 11. Grassie, S.; Gregory, R.; Harrison, D.; Johnson, K. The dynamic response of railway track to high frequency vertical excitation. J. Mech. Eng. Sci. 1982,24, 77–90. [CrossRef] 12. Bhaskar, A.; Johnson, K.; Wood, G.; Woodhouse, J. Wheel-rail dynamics with closely conformal contact Part 1: Dynamic modelling and stability analysis. Proc. Inst. Mech. Eng. Part F J. Rail Rapid Transit 1997,211, 11–26. [CrossRef] 13. Igeland, A.; Ilias, H. Rail head corrugation growth predictions based on non-linear high frequency vehicle/track interaction. Wear 1997,213, 90–97. [CrossRef] 14. Nielsen, J. Numerical prediction of rail roughness growth on tangent railway tracks. J. Sound Vib. 2003,267, 537–548. [CrossRef] 15. Sun, Y.; Simson, S. Wagon–track modelling and parametric study on rail corrugation initiation due to wheel stick-slip process on curved track. Wear 2008,265, 1193–1201. [CrossRef] 16. Batten, R.; Bellette, P.; Meehan, P.; Horwood, R.; Daniel, W. Field and theoretical investigation of the mechanism of corrugation wavelength fixation under speed variation. Wear 2011,271, 278–286. [CrossRef] 17. Song, N.; Meehan, P. A closed form analytical solution for a simplified wear-type rail corrugation model. In Proceedings of the ACOUSTICS, Gold Coast, Australia, 3–5 April 2004; pp. 227–232. 18. Bellette, P.; Meehan, P.; Daniel, W. Effects of variable pass speed on wear-type corrugation growth. J. Sound Vib. 2008,314, 616–634. [CrossRef] 19. Chen, G.; Zhang, S.; Wu, B.; Zhao, X.; Wen, Z.; Ouyang, H.; Zhu, M. Field measurement and model prediction of rail corrugation. Proc. Inst. Mech. Eng. Part F J. Rail Rapid Transit 2020,234, 381–392. [CrossRef] 20. Xie, Q.; Tao, G.; He, B.; Wen, Z. Rail corrugation detection using one-dimensional convolution neural network and data-driven method. Measurement 2022,200, 111624. [CrossRef] 21. Kalogirou, S.A. Artificial intelligence for the modeling and control of combustion processes: A review. Prog. Energy Combust. Sci. 2003,29, 515–566. [CrossRef] 22. Gibert, X.; Patel, V.M.; Chellappa, R. Deep multitask learning for railway track inspection. IEEE Trans. Intell. Transp. Syst. 2016, 18, 153–164. [CrossRef] 23. Wei, X.; Yang, Z.; Liu, Y.; Wei, D.; Jia, L.; Li, Y. Railway track fastener defect detection based on image processing and deep learning techniques: A comparative study. Eng. Appl. Artif. Intell. 2019,80, 66–81. [CrossRef] 24. Yuan, H.; Chen, H.; Liu, S.; Lin, J.; Luo, X. A deep convolutional neural network for detection of rail surface defect. In Proceedings of the 2019 IEEE Vehicle Power and Propulsion Conference (VPPC), Hanoi, Vietnam, 14–17 October 2019; IEEE: Piscataway, NJ, USA, 2019; pp. 1–4. [CrossRef] 25. Qi, S.; Yang, J.; Zhong, Z. A review on industrial surface defect detection based on deep learning technology. In Proceedings of the 2020 3rd International Conference on Machine Learning and Machine Intelligence, Hangzhou, China, 18–20 September 2020; pp. 24–30. [CrossRef] 26. Xie, Q.; Tao, G.; Lo, S.M.; Yang, X.; Wen, Z. A data-driven convolutional regression scheme for on-board and quantitative detection of rail corrugation roughness. Wear 2023,524, 204770. [CrossRef] Sensors 2024,24, 4627 18 of 18 27. Muñoz, S.; Ros, J.; Urda, P.; Escalona, J.L. Estimation of lateral track irregularity using a Kalman filter. Experimental validation. J. Sound Vib. 2021,504, 116122. [CrossRef] 28. Wang, K.; Li, K.; Zhou, L.; Hu, Y.; Cheng, Z.; Liu, J.; Chen, C. Multiple convolutional neural networks for multivariate time series prediction. Neurocomputing 2019,360, 107–119. [CrossRef] 29. Gao, J.; Song, X.; Wen, Q.; Wang, P.; Sun, L.; Xu, H. Robusttad: Robust time series anomaly detection via decomposition and convolutional neural networks. arXiv 2020, arXiv:2002.09545. [CrossRef] 30. Kiranyaz, S.; Avci, O.; Abdeljaber, O.; Ince, T.; Gabbouj, M.; Inman, D.J. 1D convolutional neural networks and applications: A survey. Mech. Syst. Signal Process. 2021,151, 107398. [CrossRef] 31. Kiranyaz, S.; Ince, T.; Gabbouj, M. Real-time patient-specific ECG classification by 1-D convolutional neural networks. IEEE Trans. Biomed. Eng. 2015,63, 664–675. [CrossRef] [PubMed] 32. Avci, O.; Abdeljaber, O.; Kiranyaz, S.; Inman, D. Convolutional neural networks for real-time and wireless damage detection. In Proceedings of the Dynamics of Civil Structures, Volume 2: Proceedings of the 37th IMAC, A Conference and Exposition on Structural Dynamics, Orlando, FL, USA, 28–31 January 2019; Springer: Cham, Switzerland, 2020; pp. 129–136. [CrossRef] 33. Khessiba, S.; Blaiech, A.G.; Ben Khalifa, K.; Ben Abdallah, A.; Bedoui, M.H. Innovative deep learning models for EEG-based vigilance detection. Neural Comput. Appl. 2021,33, 6921–6937. [CrossRef] 34. Hssayni, E.H.; Joudar, N.E.; Ettaouil, M. KRR-CNN: Kernels redundancy reduction in convolutional neural networks. Neural Comput. Appl. 2022,34, 2443–2454. [CrossRef] 35. Tran, V.T.; Tsai, W.H.; Furletov, Y.; Gorodnichev, M. End-to-end train horn detection for railway transit safety. Sensors 2022, 22, 4453. [CrossRef] [PubMed] 36. Kingma, D.P.; Ba, J. Adam: A method for stochastic optimization. arXiv 2014, arXiv:1412.6980. [CrossRef] 37. Ibrahim, Z.; Fahmy, Y. Enhanced learning for recurrent neural network-based polar decoder. In Proceedings of the 2022 13th International Conference on Electrical Engineering (ICEENG), Cairo, Egypt, 29–31 March 2022; IEEE: Piscataway, NJ, USA, 2022; pp. 105–109. [CrossRef] 38. Hao, H.; Fuzhou, F.; Junzhen, Z.; Xun, Z.; Pengcheng, J.; Feng, J.; Jun, X.; Yazhi, L.; Guanghui, S. Research on Fault Diagnosis Method Based on Improved CNN. Shock. Vib. 2022,2022, 9312905. [CrossRef] 39. Kim, J.K.; Jung, S.; Park, J.; Han, S.W. Arrhythmia detection model using modified DenseNet for comprehensible Grad-CAM visualization. Biomed. Signal Process. Control 2022,73, 103408. [CrossRef] 40. Selvaraju, R.R.; Cogswell, M.; Das, A.; Vedantam, R.; Parikh, D.; Batra, D. Grad-cam: Visual explanations from deep networks via gradient-based localization. In Proceedings of the IEEE International Conference on Computer Vision, Venice, Italy, 22–29 October 2017; pp. 618–626. 41. Prechelt, L. Early stopping-but when? In Neural Networks: Tricks of the Trade; Springer: Berlin/Heidelberg, Germany , 2002; pp. 55–69. [CrossRef] Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.