The most direct method to study the geometric characters of irregularities at rail welds is to take measurements on operating lines. From previously published literature, Gao et al. obtained the most detailed data on rail weld irregularities [1]. Approximately 74 valid samples of the surface geometries at rail welds on the high-speed railway from Beijing to Shanghai in China were obtained every two months for each measurement. Upon analyzing the measured irregularity data, the irregularities at rail welds were classified into convex and concave types, which include three distinct patterns, as illustrated in Fig. 1 [1].

Fig. 1 clearly illustrates that the fundamental characteristic of the rail weld irregularities is a primary long-wavelength wave overlaid with a secondary short-wavelength wave. Based on field measurements, the classification of rail weld irregularity wave patterns included all types observed on the high-speed railway from Beijing to Shanghai. This classification was utilized in the present study.

In this study, three different models for rail weld irregularities are presented: the single cosine wave model [4], Gao’s model [1], and that developed in this study by modifying Gao’s model.

(1) Single-cosine wave model

For simplicity, a single cosine wave model is typically used to model the rail weld irregularity, which is expressed by Eq. (1): (1)z0(t)=a2[1−cos⁡(2πvt/λ)],where z₀(t) represents the vertical displacement induced by irregularities at rail welds, a and λ are the amplitude and half-wavelength of the irregularities, respectively, t is time, and v is the train speed.

(2) Gao’s model

Gao et al. proposed a more precise model to simulate the geometric features of rail weld irregularities, utilizing extensive rail surface measurements on the high-speed railway from Beijing to Shanghai. It comprises three wave patterns, that is, convex-A, convex-B, and concave (see Fig. 1), and is expressed in Eqs. (2)–(4), respectively [1]. (2)z0(t)={12a1(1−cos⁡2πvtλ1)(0≤t≤λ1−λ22v)a1+12a2[1−cos⁡2πvλ2(t−λ1−λ22v)](λ1−λ22v<t≤λ1+λ22v)12a1(1−cos⁡2πvtλ1)(λ1+λ22v<t≤λ1v), (3)z0(t)={−12a1(1−cos⁡2πvtλ1)(0≤t≤λ1−λ22v)−a1+12a2[1−cos⁡2πvλ2(t−λ1−λ22v)](λ1−λ22v<t≤λ1+λ22v)−12a1(1−cos⁡2πvtλ1)(λ1+λ22v<t≤λ1v), (4)z0(t)={12a1(1−cos⁡2πvtλ1)(0≤t≤λ1−λ22v)a1−12a2[1−cos⁡2πvλ2(t−λ1−λ22v)](λ1−λ22v<t≤λ1+λ22v)12a1(1−cos⁡2πvtλ1)(λ1+λ22v<t≤λ1v),where z₀(t), v, and t have similar physical meanings as those in Eq. (1). a₁ and a₂ are the amplitudes of the primary and secondary waves, respectively. λ₁ and λ₂ are the half-wavelengths of the primary and secondary waves, respectively. It is worth noting that the half-wavelength of the main wave was 1 m in this study because the measurements were performed on a 1-m-long base.

(3) Proposed modified Gao’s model

In this study, a novel model was developed using a parabolic function instead of the cosine function used in Gao’s model to simulate the primary wave of rail weld irregularities. The three wave patterns of rail weld irregularities are expressed by Eqs. (5)–(7), respectively. (5)z0(t)={4a1t(λ1v)(1−t(λ1v))(0≤t≤λ1−λ22v)a1+12a2[1−cos⁡2πvλ2(t−λ1−λ22v)](λ1−λ22v<t≤λ1+λ22v)4a1t(λ1v)(1−t(λ1v))(λ1+λ22v<t≤λ1v), (6)z0(t)={−4a1t(λ1v)(1−t(λ1v))(0≤t≤λ1−λ22v)−a1+12a2[1−cos⁡2πvλ2(t−λ1−λ22v)](λ1−λ22v<t≤λ1+λ22v)−4a1t(λ1v)(1−t(λ1v))(λ1+λ22v<t≤λ1v), (7)z0(t)={4a1t(λ1v)(1−t(λ1v))(0≤t≤λ1−λ22v)a1−12a2[1−cos⁡2πvλ2(t−λ1−λ22v)](λ1−λ22v<t≤λ1+λ22v)4a1t(λ1v)(1−t(λ1v))(λ1+λ22v<t≤λ1v),where all the symbols from Eqs. (5)–(7) have similar physical meanings as those from Eqs. (2)–(4).

A comparison of the geometric characteristics that were measured and simulated using convex-A, convex-B, and concave irregularities is illustrated in Fig. 2. It should be noted that the measured geometry characteristics of the irregularities can be referred to in the Literature [1].

As observed in Fig. 2, the single cosine wave model mainly concerns the maximum value of the irregularities at rail welds on the one hand, which could not describe the geometric shapes accurately, especially that of the secondary wave. On the other hand, Gao’s model, and that proposed in this study focused on the maximum value and the geometric shapes of the irregularities, which described the main and secondary waves precisely. As illustrated in Fig. 2, among the three theoretical models, the proposed model best matches the geometric characteristics of the measured irregularities.

The fitting accuracy of the three models is described by root mean square error (RMSE) that is expressed in Eq. (8). (8)RMSE=1n∑i=1n(yi−y^i)2,where yi is the amplitude of the measured irregularity, y^i is the simulated amplitude of the irregularity and n is the number of measurement data along the measuring distance.

The RMSE of the simulated amplitudes of the irregularity corresponding to the measured irregularity amplitude are listed in Table 1. It is clearly shown that the proposed model has the smallest RMSE value, which means the proposed model has better fitting accuracy than the other two simulating models.

The results of wheel-rail vertical dynamic force induced by the three types of rail weld irregularities both modeled and measured, are shown in Figs. 3–5, respectively. They have been compared and discussed to verify the effectiveness of the proposed model.

As illustrated in Figs. 3–5, the low-frequency P₂ impact force in the time domain was simulated excellently for both the vibration waveform and maximum value using all three theoretical models. For the high-frequency P₁ impact force in the time domain, the single cosine wave model captured neither the maximum values nor the vibration waveforms for all three types of rail weld irregularities, which may have been due to a disregard of the secondary wave for the measured irregularities. Gao’s model obtained good results for convex-A irregularities but was not good for convex-B and concave irregularities, which may be owing to the large gap in the amplitude value between the primary and secondary waves (see Fig. 2). The proposed model achieved good results for all three patterns of rail weld irregularities, whether in terms of waveform or maximum value.

The error band between the amplitudes of the impact forces P₁ and P₂ induced by irregularities that were measured and modeled was analyzed, as illustrated in Fig. 6. It is worth noting that the legends for convex-A, convex-B, and concave irregularities are represented by the black, blue and red symbols, respectively.

Fig. 6 shows that the low-frequency P₂ impact forces in the time domain simulated by all three theoretical models were almost all located within the 5% error band. For the high-frequency P₁ impact force in the time domain, the simulated results of the proposed model for the three types of rail weld irregularities were almost all located within the 5% error band. The results simulated by Gao’s model were almost all located within the 10% error band, except for the convex-B irregularities. The results simulated by the single cosine wave model were all located outside the 10% error band.

Hence, from Figs. 3–6, it can be concluded that the proposed model was more appropriate for describing the vertical dynamic force than the single cosine wave model and Gao’s model, whether in the vibration waveform or amplitudes of the impact forces P₁ and P₂ in the time domain.

To further verify the effectiveness of the proposed model used in describing rail weld irregularities, the vertical dynamic forces caused by the measured and modeled irregularities in the frequency domain were analyzed and compared. The vertical dynamic forces in the frequency domain were obtained and compared using the fast Fourier transform (FFT) of the forces in the time domain shown in Figs. 3–5, which are illustrated in Figs. 7–9.

From Figs. 7–9, it was observed that the frequency of the vertical dynamic force was divided into two distinct bands by the frequency value of 100 Hz, which corresponds to the results obtained in the Literature [1]. One was in the low-frequency region, which reflected the impact force P₂ caused by the primary long-wavelength irregularity. The other was situated in the high-frequency region, representing the impact force P₁ excited by the secondary short-wavelength irregularity.

For the low-frequency impact force P₂, the vibration frequencies induced by the measured irregularities were like those of the irregularities modeled by all three theoretical models. This showed good simulation results for the low-frequency impact force P₂ in the frequency domain by all three theoretical models. The vibration frequencies associated with the low-frequency impact force P₂ were consistent across all three types of irregularities, indicating a shared characteristic among them in exciting similar vibration frequencies. The frequency range of P₂ force is concentrated between 42 to 55 Hz.

For the high-frequency impact force P₁, the vibration frequencies varied from 400 to 1000 Hz for the three types of rail weld irregularities. The vertical forces in the frequency domain caused by the proposed rail weld irregularity models corresponded closely to those induced by the measured irregularity, outperforming those induced by Gao’s model and the single cosine wave model. The vibration frequencies for the high-frequency impact force P₁ under different train running speeds of 300, 350, and 400 km/h satisfied Eq. (9), which means that the vibration frequency was proportional to the speed and inversely proportional to the irregularity wavelength. Hence, the vibration frequencies of the impact force P₁ induced by the convex-A rail weld irregularities were the highest, followed by those caused by concave irregularities. The results caused by the convex-B rail weld irregularities were the lowest, owing to the longest wavelength of the secondary wave for the convex-B rail weld irregularities adopted in this study. (9)f=vλ.

ANN is a computational model that mimics the structure and function of biological neural networks for processing and analyzing complex data. It mimics the workings of the human brain through a large number of simple nodes (called neurons or units) and their interconnections. ANN has powerful learning capabilities and has several advantages over other machine learning models (Support Vector Machines (SVM) and Random Forests (RF), etc.): (1) Handling complex nonlinear relationships. ANN can capture complex nonlinear relationships in data through multiple hidden layers and nonlinear transfer functions (e.g., ReLU, Sigmoid, Tanh, etc.). Compared to linear models, ANN can fit complex data more accurately. (2) Adaptive learning capability. ANN can minimize the loss function by continuously adjusting the model parameters (weights and biases). This process is adaptive and can automatically optimize the model according to the characteristics of the data and gradually improve the prediction performance. (3) Efficient parallel computing. The computational process of ANN involves a large number of matrix operations, which can be efficiently parallelized and executed on modern computing devices (e.g., GPU, TPU). This makes ANN have high computational efficiency when dealing with large-scale data.

ANN performs well in many applications, but they also have some limitations and challenges: ANN are often regarded as “black-box” models, and their internal structure and decision-making process are difficult to explain; ANN usually require a large amount of data for training, and may not perform well for tasks with a small amount of data; the training of ANN involves the tuning of many hyperparameters (e.g., learning rate, number of neurons, number of network layers, etc.), and finding the optimal combination of hyperparameters usually requires a lot of experimentation and debugging.

As shown in Fig. 10 below, the basic structure of ANN consists of three parts. (1) Input layer: receives raw data input. (2) Hidden layer: located between the input layer and the output layer and consists of a set of neurons. A neural network can have one or more hidden layers, and the number of layers and the number of neurons in each layer depends on the specific application. (3) Output layer: generates the final model output.

The input data is passed through the input layer to the hidden layer and then through the hidden layer to the output layer. After processing the input information vector X, the neuron uses the transfer function f to produce the output Y: (10)Y=f(w⋅X+b),where w and b represent the weights and biases of the neurons, respectively. The transfer function f can have various forms, including linear function, sigmoid function, and ReLU function. The training of the AI network relies on the Adam optimization algorithm. Adam is an adaptive learning rate optimization algorithm that combines the advantages of the Momentum and Root Mean Square Propagation (RMSprop) algorithms, aiming to efficiently deal with sparse and noisy gradients to speed up model convergence and improve performance.

The main steps of Adam’s optimization algorithm are to initialize the values of the parameters first-order moment estimation (momentum variable) and second-order moment estimation (squared gradient variable), followed by iterations to update the parameters, and in each iteration to compute the current gradient, gt: (11)gt=∇θft(θt−1),where ∇θ represents the gradient of the parameter θ and ft(θt−1) is the objective function being optimized. Followed by first-order moment estimation, mt, and second-order moment estimation, vt: (12)mt=β1⋅mt−1+(1−β1)⋅gt, (13)vt=β2⋅vt−1+(1−β2)⋅gt2,

The default values of the momentum parameters β1 and β2 are 0.9 and 0.999, respectively.

Since the values of mt and vt are biased towards 0 in the initial phase, a bias correction is also required: (14)m^t=mt1−β1t, (15)v^t=vt1−β2t,

Finally, the update of the parameter θt is performed:

(16)θt=θt−1−αv^t+∋⋅m^t,where the learning rate α has a default value of 0.001 and ∋ is a very small number, usually 10−8, used to prevent division by zero errors. The training process is completed when the change in the objective function (e.g., loss function) defined in the machine learning model is less than a certain threshold after many iterations by stopping the iterations.

This study involves the construction, training, and validation of an ANN model using the Python programming language. The model will include the important variables related to the generation of impact forces, i.e., the parameters associated with the weld upset modeled in this paper (a1, a2, and λ2), the different types of welds (convex-A, convex-B, and concave), and the speed at which the train is running. The outputs include impact forces P₁ and P₂. Firstly, the individual features in the database were analyzed for correlation, the correlation coefficient is defined as the covariance of two features divided by the product of their standard deviations:

(17)r(X,Y)=Cov(X,Y)Var[X]Var[Y],where Cov(X,Y) is the covariance between two features X and Y, Var[X] is the variance of X, and is the variance of Y. If the correlation coefficient is greater than 0.5, the two features are considered to be related to each other. As shown in Fig. 11 below, the correlation coefficients are shown in the form of a heat map. From the figure, it can be seen that the correlation coefficients between the five input features are very small and they are considered to be uncorrelated, but there is considered to be a weak positive correlation between the correlation parameter a1 of the weld unevenness model and the low-frequency impact force P₂, and a positive correlation between the correlation parameter a2 of the weld unevenness model and the high-frequency impact force P₁. The results show that this model can be trained using an artificial neural network model.

A total of 252 sets of data on the impact force P₁ and P₂ due to weld unevenness under various conditions were collected using the coupled vehicle-rail dynamics method, where the models used included both the directly measured model and the modified Gao’s model proposed in this paper. These data are randomly disrupted and divided in a ratio of 7:3, where 176 groups are used for training and the remaining 76 groups are used for testing. To reduce significant differences between the input data, the data is normalized before it enters the hidden layer. This normalization process helps to mitigate the potential impact of individual data points on network training.

The prediction results of a model are significantly affected by the internal structure and hyperparameters of the model. For ANN models, several hyperparameters must be determined, such as the selection of an appropriate activation function, the number of hidden layers, and the number of neurons within each hidden layer. The quality of the model parameters is assessed based on metrics such as RMSE and regression coefficient (R²). These metrics are benchmarks for evaluating the accuracy and goodness of fit of the model. The smaller the RMSE value or the closer the R² value is to 1, the higher the quality of the model. The equation for RESM is shown in Eq. (8) and R² is calculated as follows:

(18)R2=∑i=1n(y^i−y¯)2∑i=1n(yi−y¯)2,where yi is the expected value, y^i is the predicted value, y¯ is the mean of the expected value, and n is the number of samples.

The commonly used transfer functions in ANN include ReLU, Sigmoid, and Tanh. Among them, ReLU function is simple to compute and efficient, which can alleviate the problem of gradient vanishing and make the deep network training more efficient; the Sigmoid function can reduce the influence of extreme values and make the model more stable; Tanh has a wider dynamic range, which can allow the gradient descent algorithm to converge faster and can capture nonlinear patterns well. In this study, we adopt the ReLU function as the transfer function.

The training process of the ANN model in this paper uses an early stopping mechanism, i.e., when the loss function in the model (which is defined as the root mean spuare error RMSE in this model) no longer improves within a specified number of iterations, the training stops to avoid overfitting. In this paper, training is stopped if the loss function of the validation set does not improve within 100 iterations. As an example, the hidden layer is 3 layers, and the iterative process of the model is shown in Fig. 12.

The choice of the number of hidden layers and the allocation of neurons within each hidden layer need to be discussed in detail when exploring the optimal values of the ANN model parameters.

There are three commonly used numbers of hidden layers in AI network architectures: 2, 3, and 4. Increasing the number of hidden layers enhances the ability of the network to be able to learn more complex and abstract representations of features, thus helping to capture nuances and complex patterns in the data. However, this increase can lead to an increase in training time, and deeper networks are more likely to suffer from the problem of vanishing or exploding gradients, which may overfit the training data and lead to training difficulties. The effect of different numbers of hidden layers (2, 3, and 4) on the prediction accuracy of the neural network was comparatively evaluated in predicting the impact force due to rail weld irrregularity. The error band plots shown in Fig. 13 show that the neural network reaches its best performance when 3 hidden layers are used. Most of the data points are very close to the diagonal (y = x) and only a small fraction of the points deviates from the 5% relative error band.

In terms of the number of neurons, the number of neurons in the first layer plays a crucial role in determining the predictive performance of the ANN. We investigated the range of 20 to 100 neurons in the first layer. As shown in Fig. 14, the best performance was obtained when the first layer consisted of 34 neurons, and the sum of the root mean square errors of the two target values was 4.49. Similarly, the number of neurons in the other two layers was investigated using the same methodology. The optimal structural parameters for predicting the impact force model due to rail weld irrregularity in this study were most obtained, as shown in Table 3.

After obtaining the optimal structural parameters of the ANN model, we analyzed the testing data from the test set, which had been randomly divided earlier, and which consisted of the prediction of the impact force due to the rail weld irrregularity using two models (the measured weld irregularities model and the use of the high-style modification model proposed in this paper). The error band diagrams of the data from the test set are shown in Fig. 15. From the figure, we can see that the data predicted by the optimized machine learning model in the testing set are all within the 10% error band, and the impact forces predicted by the model for the measured weld irregularities are all within the 5% error band, and it can be seen that the optimized machine learning model based on the coupled vehicle-rail dynamics model is good for predicting the impact forces due to the rail weld irrregularity.