Augmentation method of fatigue data of welded structures based on physics-informed CTGAN

Chen Lu
School of Railway Intelligent Engineering of Dalian Jiaotong University, Dalian, China
19912060618@163.com

In modern engineering practice, welded structures in aerospace, automotive industry, bridge construction and other fields typically perform under variable amplitude loading and are very susceptible to fatigue failure [1]. Fatigue damage increases cumulatively during the service of welded structures until the structure fails by fracture. The randomness and complexity in variable amplitude loading elevate the challenges associated with predicting fatigue life. Consequently, accurate prediction of fatigue life under variable amplitude loads is crucial for ensuring the project's safe operation and mitigating financial losses.
Up to now, two primary categories have been established for methods predicting fatigue life under variable amplitude loading: cumulative damage theory [2] and machine learning methods [3]. Among them, the theory of fatigue accumulation damage is categorized into two principal divisions: namely, the linear theory of damage accumulation and the nonlinear theory of damage accumulation. The most common Miner linear cumulative damage model [4] is applied extensively in engineering as it is easy to calculate. However, it overlooks the impacts of loading sequence and loading effects in its consideration. There is often a significant discrepancy between the actual damage and the predicted damage of welded structures under variable amplitude loading. Therefore, experts and scholars have proposed nonlinear fatigue damage accumulation theories rooted in damage curves, loading interactions, continuum damage mechanics, energy-based approaches, and physical property degradation to address this issue. For example, Ye et al. [5] proposed a nonlinear fatigue damage accumulation model, referred to as the Ye model, which is based on the dynamic degradation of material toughness resulting from fatigue-induced damage. This model is prevalent in engineering applications due to its straightforward form and concise physical explanation. Nonetheless, the model overlooks the interaction between loads, leaving ample room for enhancing the precision of life prediction. Several scholars have improved the model to address this issue. Lv et al. [6] integrated the influence of loading interaction into the Ye model through the introduction of a two-level loading ratio. Wang et al. [7] demonstrated the impact of load interactions on fatigue damage progression by factoring in the square of the load
ratio between successive loading stages, thereby improving prediction precision. Peng et al. [8] considered both the influence of loading sequence and the interplay between two loads when assessing residual life.
The fatigue damage models discussed above are based on specific physical mechanisms, yet they generally do not account for the uncertainties arising from various influencing factors during the fatigue analysis of welded structures [9]. Thus, methods of machine learning have been employed. For instance, Gan et al. [10] applied a data-driven model, grounded in the Kernel Extreme Learning Machine (KELM), to predict the residual lifespan of welded materials subjected to two-level loading conditions. The model autonomously learns the best correlation from the training samples, effectively describing the fatigue damage mechanism. Liu et al. [11] utilized three algorithmic frameworks-specifically, Random Forest (RF), ExtremeGradient Boosting (XGBoost), and Gradient Boosting Machines (GBM)-for forecasting the fatigue life of highstrength steels. Among these, the utilizing gradient boosting demonstrated the highest precision in estimating the fatigue lifetime of high-strength steels under extremely high cycle conditions. Matin et al. [12] used multiple machine learning algorithms to evaluate the factors influencing piston aluminum alloy specimens and the interactions that affect fatigue life values. It took into account the effect of various inputs on fatigue life. Zou et al. [13] established a method for predicting the fatigue life of welded joints, leveraging the whale optimization algorithm alongside the Support Vector Machine (SVM). It took into account the sequence and interactions of loads to estimate the fatigue life under conditions of two-level loading. However, the accuracy of fatigue life predictions made by machine learning models is often hampered by the scarcity of available fatigue samples. Obtaining a sufficient number of fatigue samples is challenging due to the large number and difficulty of various conditions required for fatigue testing, leading to inadequate accuracy in predicting fatigue life under variable loading.
Data augmentation is a method to increase the amount and improve the quality of data by transforming, augmenting or synthesizing the original data. Data augmentation methods such as Generative Adversarial Networks (GAN) [14] have emerged. GAN is a form of deep learning model, proposed by Goodfellow in 2014, with the ability to solve problems associated with limited sample sizes. Since GAN is proposed, many variants have emerged, which are widely used in fields such as computer vision, medicine, and natural language processing. In the realm of fatigue life analysis and prediction, their utilization is still in its nascent phases. For example, He et al. [15] utilized data produced by a table GAN within a machine learning framework for predicting multiaxial fatigue life. The inclusion of synthetic data enhanced the predictive capacity of the machine learning models in estimating life expectancy, the findings suggested. Sun et al. [16] employed a cyclical GAN to augment a dataset of 20 multiaxial fatigue data points, expanding it into thousands of comparable samples. This well balances the time cost of large sample sizes with prediction accuracy. He et al. [17] introduced a deep learning architecture that combines a GAN with a physical model for the purpose of predicting multiaxial fatigue life. When equipped with suitable physical constraints, the model outperforms the neural network in terms of prediction accuracy. Primarily, these models serve the purpose of predicting multiaxial fatigue life under conditions of constant amplitude loading. However, welded structures frequently encounter variable amplitude loading scenarios in practical engineering applications, resulting in fatigue data that exhibits complexity and dispersion. Therefore, it is still a problem in fatigue research applications to enhance effective training samples under the variable amplitude loading.
This work proposes a novel augmented model for fatigue data of welded structures subjected to two-step loading that integrates physical mechanisms. The cumulative damage model-Peng model is integrated into the CTGAN as a physical loss, enabling the generated fatigue data to adhere to the relevant physical mechanisms. The validity of the augmented model is confirmed through testing on machine learning models. The problem of insufficient fatigue data for residual fatigue life prediction under two-step loading is solved. The accuracy of the machine learning models for fatigue life prediction of welded structures is further improved.

From a macro-physics standpoint, the process of fatigue damage accumulation can be interpreted as a progressive degradation and decline in the structural properties. Therefore, the alterations in material's macroscopic characteristics can be considered as damage variables to measure fatigue damage. Ye et al. [5] discovered through extensive fatigue testing that the most significant change in the material's fatigue damage history is its toughness. Consequently, a nonlinear cumulative damage model was introduced, emphasizing the dissipation of material toughness. The fatigue damage evolution equation in Ye model is shown as Eqn.(1):

where $N_f$$N_f$N_(f)N_{f}$N_f$ denotes the fatigue life under stress $\sigma ,n$$\sigma ,n$sigma,n\sigma, n$\sigma ,n$ represents the number of cycles under stress $\sigma$$\sigma$sigma\sigma$\sigma$, and $D_N$$D_N$D_(N)D_{N}$D_N$ signifies the cumulative damage variable after $n$$n$nn$n$ cycles of stress $\sigma$$\sigma$sigma\sigma$\sigma$.
Fig 1. displays the fatigue damage curves obtained under two-step loading conditions.

Based on the Ye model, after $n_1$$n_1$n_(1)n_{1}$n_1$ cycles of ${\sigma }_1$${\sigma }_1$sigma_(1)\sigma_{1}${\sigma }_1$, the fatigue cumulative damage value is:

It is equivalent to the damage incurred by $n_2{}^{\prime }$$n_2{}^{\prime }$n_(2)^(')n_{2}{ }^{\prime}$n_2{}^{\prime }$ cycles of ${\sigma }_2$${\sigma }_2$sigma_(2)\sigma_{2}${\sigma }_2$ as illustrated in Eqn.(3).

According to the principle of equivalent damage, it can be obtained that the corresponded damage at A and B is the same, i.e.

By substituting Eqn. (2) and Eqn. (3) into Eqn. (4), we obtain:

Typically, fatigue damage arises when the cumulative damage $D$$D$DD$D$ attains a critical limit, which is defined as 1 in the Ye model. Currently, the formulation for cumulative damage criterion is presented in Eqn. (6).

Thus the predicted value of the remaining life by using Ye model for two-step loading could be reached as shown in Eqn. (7):

Despite the simplicity of its form and the clear physical meaning of the Ye model, it fails to reflect the influences of load interactions under conditions of variable amplitude loading. Stress ratios that describe load interactions were used by many existing models in nonlinear cumulative damage theory. Peng et al [8] proposed the improved Ye model, accounting for both the sequence of applied loads and the mutual influence between different loads on residual life. Eqn. (8) delineates the estimated fatigue life remaining under two-step loading, according to Peng's model.

In summary, the cumulative damage models generally have a clear and explicit physical definition. However, due to the insufficient consideration of uncertainty of fatigue life influencing factors and the complexity of the formula, its application in engineering is limited. Traditional cumulative damage models' limitations are increasingly being tackled effectively through the utilization of machine learning techniques now [18]. Due to the limited number of fatigue test samples, obtaining highprecision prediction models under small sample conditions is a bottleneck problem.

A generative model known as GAN comprises two neural networks: a generator and a discriminator, engaged in a competitive learning dynamic. The optimization objective function of the GAN is presented in Eqn. (9):

where $x$$x$xx$x$ stands for the real data, $z$$z$zz$z$ signifies the potential noise, $E(\ast )$$E(\ast )$E(**)E(*)$E(\ast )$ indicates the expected value of the distribution function, $p_{\text{data}}(x)$$p_{\text{data }}(x)$p_("data ")(x)p_{\text {data }}(x)$p_{\text{data }}(x)$ embodies the distribution of authentic samples, $p_z(z)$$p_z(z)$p_(z)(z)p_{z}(z)$p_z(z)$ represents the distribution of the noise defined in the lower dimension, $D(x)$$D(x)$D(x)D(x)$D(x)$ denotes the result of the judgement of the discriminator on the real data $x$$x$xx$x$, and $G(z)$$G(z)$G(z)G(z)$G(z)$ denotes the fake data that was generated based on the random data $z$$z$zz$z$ in the generator. The discriminator $D$$D$DD$D$ needs to match the real samples as much as possible, i.e., maximize $\log D(x)$$\log D(x)$log D(x)\log D(x)$\log D(x)$, while the generator $G$$G$GG$G$ needs to maximize the loss of $D$$D$DD$D$, i.e, minimize $\log D(x)$$\log D(x)$log D(x)\log D(x)$\log D(x)$. Ultimately, a Nash equilibrium is reached between the generator and the discriminator, allowing the generator to create realistic data.
Conditional Tabular GAN (CTGAN) [19] uses a GAN-based approach to model samples from tabular data distribution. CTGAN employs a Variational Gaussian Mixture model (VGM) for each continuous variable, with the intention of determining the most suitable $k$$k$kk$k$ gaussian models to depict the data through the application of the expectation maximization algorithm. Additionally, it compels the generator to produce samples with discrete variable distributions that closely
resemble the training data and incorporates a condition vector as part of the input. The input to CTGAN comprises a condition vector, which direct the generator to create samples that belong to designated categories. The condition vector, which is encoded in one-hot format to represent all discrete columns, selects conditions by sampling from the training dataset. The generator's loss function ensures that the samples produced by the generator fulfill the specified condition. Incorporating the cross-entropy between condition vector and generated samples within loss function accomplishes this. In this work, CTGAN is employed for data augmentation to tackle the challenge posed by limited fatigue data.

Investigating fatigue performance typically necessitates a substantial number of repeated tests. However, it is currently difficult to obtain a large number of training samples due to the complexity and randomness of fatigue testing. The shortage of samples in the training dataset affects both the precision and the ability of the model to generalize. This work proposes a CTGAN generative model based on physics-informed to solve the problem of fewer training samples under two-step loading. This enables the machine learning models to effectively capture the relationship between inputs and outputs.
In this work, five fatigue life prediction models, Miner law [4], Ye model [5] and its improved model LV model [6], Wang model [7], Peng model [8], are selected to predict fatigue life under two-step loading for five welded materials [20-24]. Specific details related to the literature dataset here can be found in section Experimental results and analysis. The mean absolute percentage error (MAPE) quantifies the precision of life prediction, as defined by the equation below:

The predicted fatigue life for the i -th sample is denoted as $y_i{}^{\prime }$$y_i{}^{\prime }$y_(i)^(')y_{i}{ }^{\prime}$y_i{}^{\prime }$, while the actual fatigue life for the same sample is $y_i.\mathrm{Tab}$$y_i.\mathrm{Tab}$y_(i).Taby_{i} . \mathrm{Tab}$y_i.\mathrm{Tab}$. 1 presents the average absolute percentage error for the five materials.

According to Tab. 1, the Peng model demonstrates a notably smaller prediction error compared to the other models for materials such as butt joint, Al-2024-T42, Al-7075-T7451, and Ti-6Al-4V. The Peng model exhibits a marginally higher error on corner joint as opposed to the LV and Wang models compared to other models. In general, it seems that the Peng model has the smallest prediction error. In this work, the Peng model is selected as the physical loss component to be integrated into the generator loss function of CTGAN. The physical loss component serves as a regularizer within the network, aiding in the improvement of the training procedure. The generator is enabled by the physical loss term to produce data that adheres to physical constraints. The generator's overall loss function in this work is formulated as follows:

where $L_{\text{adversarial}}$$L_{\text{adversarial }}$L_("adversarial ")L_{\text {adversarial }}$L_{\text{adversarial }}$ is the adversarial loss function, $L_{\text{discrete}}$$L_{\text{discrete }}$L_("discrete ")L_{\text {discrete }}$L_{\text{discrete }}$ is the discrete column loss function, $L_{mmd}$$L_{mmd}$L_(mmd)L_{m m d}$L_{mmd}$ is the Max Mean Discrepancy (MMD) loss function [25] and $L_{\text{physics}}$$L_{\text{physics }}$L_("physics ")L_{\text {physics }}$L_{\text{physics }}$ is the physics loss function. The adversarial loss function and discrete column loss function in CTGAN are shown below:

where $N$$N$NN$N$ denotes the quantity of data, $x_i$$x_i$x_(i)x_{i}$x_i$ signifies the input data, $G\left(x_i\right)$$G\left(x_i\right)$G(x_(i))G\left(x_{i}\right)$G\left(x_i\right)$ represents the synthetic data generated based on the input, and $D\left(G\left(x_i\right)\right)$$D\left(G\left(x_i\right)\right)$D(G(x_(i)))D\left(G\left(x_{i}\right)\right)$D\left(G\left(x_i\right)\right)$ denotes the discriminator's probability of deeming the synthetic data as authentic. $K$$K$KK$K$ indicates the count of potential values for the discrete column, $y_{i,k}$$y_{i,k}$y_(i,k)y_{i, k}$y_{i,k}$ signifies the actual probability of the kth value in the discrete column for the i-th sample, and $y_{i,k}^{\prime }$$y_{i,k}^{\prime }$y_(i,k)^(')y_{i, k}^{\prime}$y_{i,k}^{\prime }$ denotes the probability of the same kth value generated by the generator for the i-th sample. The generator's adversarial loss function primarily strives to decrease the discriminator's likelihood of classifying the generated sample as fake. In essence, it aims to elevate the probability that discriminator will deem generated sample as authentic. In CTGAN, the loss function for discrete columns typically utilizes cross-entropy to diminish the discrepancy between the probability distribution of the generated discrete data and that of the real data. It ensures a tight correspondence between the probability distribution of the authentic data and that of the synthetic discrete data.
This work introduces two additional loss functions to the base CTGAN: the MMD loss function and the physical loss function. Below is the representation of MMD loss function:

where $P$$P$PP$P$ denotes the probability distribution of the data produced by the generator and $Q$$Q$QQ$Q$ represents the probability distribution of the real data. $N$$N$NN$N$ and $M$$M$MM$M$ are the number of samples obtained by sampling from $P$$P$PP$P$ and $Q$$Q$QQ$Q$, respectively. $y_i$$y_i$y_(i)y_{i}$y_i$ and $y_j$$y_j$y_(j)y_{j}$y_j$ are the samples obtained by sampling from $P$$P$PP$P$ and $Q$$Q$QQ$Q$, respectively. $\varphi (\ast )$$\varphi (\ast )$phi(**)\phi(*)$\varphi (\ast )$ serves as a feature mapping function, transforming the samples into a higher-dimensional feature space. MMD aims to minimize the distance separating two probability distributions, ensuring that generator's sample distribution closely approximates genuine sample distribution. Often, it is integrated into GAN as a component of the generator's loss function, with the goal of decreasing the discrepancy between generated and true values.
The physical loss function is derived from the Peng fatigue life prediction model and is applied to this loss component as detailed below:

where ${\sigma }_1$${\sigma }_1$sigma_(1)\sigma_{1}${\sigma }_1$ denotes the first stress level, ${\sigma }_2$${\sigma }_2$sigma_(2)\sigma_{2}${\sigma }_2$ denotes the second stress level, $N_{f1}$$N_{f1}$N_(f1)N_{f 1}$N_{f1}$ is the fatigue life under ${\sigma }_1,N_{f2}$${\sigma }_1,N_{f2}$sigma_(1),N_(f2)\sigma_{1}, N_{f 2}${\sigma }_1,N_{f2}$ is the fatigue life under ${\sigma }_2$${\sigma }_2$sigma_(2)\sigma_{2}${\sigma }_2$, and $n_1$$n_1$n_(1)n_{1}$n_1$ is the number of cycles under the first level of stress. Incorporating physical loss function allows the generated fatigue data to adhere to physical laws, resulting in generated residual fatigue life values that are closer to the real data $n_2$$n_2$n_(2)n_{2}$n_2$. It also reflects the fatigue behavior features and the prior training process is more stable.
In CTGAN, the discriminator's loss function is usually based on the adversarial training principle. The cross-entropy loss function is commonly employed, and it is expressed as follows:

where, $x_i^{\prime }$$x_i^{\prime }$x_(i)^(')x_{i}^{\prime}$x_i^{\prime }$ is the input generated data, label ${}_i$${}_i$_(i){ }_{i}${}_i$ is the real label, the real data takes the value of 1 , the generated fake data takes the value of 0 , and $D\left(x_i^{\prime }\right)$$D\left(x_i^{\prime }\right)$D(x_(i)^('))D\left(x_{i}^{\prime}\right)$D\left(x_i^{\prime }\right)$ is the probability of judging that the data is real data. This loss function boosts the discriminator's capacity to differentiate accurately between genuine and synthesized samples. At the same time, it minimizes the probability of incorrectly classifying a generated sample as real.
In contrast to original GAN, the input for CTGAN generator in this work consists of real experimental data, instead of random variables. The physics loss $L_{\text{physics}}$$L_{\text{physics }}$L_("physics ")L_{\text {physics }}$L_{\text{physics }}$ is combined with the MMD loss $L_{\text{mmd}}$$L_{\text{mmd }}$L_("mmd ")L_{\text {mmd }}$L_{\text{mmd }}$ to make the generated data by the generator closer to the real values, adhering to the physical principles governing fatigue data under variable amplitude loading during training phase. Fatigue data exhibits high discreteness. CTGAN's initial discrete column loss $L_{\text{discrete}}$$L_{\text{discrete }}$L_("discrete ")L_{\text {discrete }}$L_{\text{discrete }}$ guarantees that the synthesized fatigue data corresponds to the real data distribution under specified conditions, enhancing the generator network's efficiency and elevating the quality of the generated fatigue data.

A fatigue data augmentation model for welded structures fused with physical mechanism is developed to target the problem of less fatigue data in machine learning models under two-step loading. The proposed model's overall architecture is illustrated in Fig. 2. This model generates data that reflects fatigue behavior under variable amplitude loading, making the generated fatigue data consistent with the physical results. It can be seen that the overall experimental process is divided into four main parts. First we get the fatigue data. The data attributes included are specifically shown therein. Then the data is input into the CTGAN model. And it has been validated effectively on four machine learning models KELM, SVM, RF, and Back Propagation (BP), respectively. Finally its effect on fatigue prediction accuracy is evaluated by two indicators. The specific steps of the framework are outlined as follows:
Step1: Gather literature and laboratory test data on welded structures subjected to variable amplitude loading to create a fatigue dataset for these structures. The attributes of the fatigue dataset are: ${\sigma }_1$${\sigma }_1$sigma_(1)\sigma_{1}${\sigma }_1$ and ${\sigma }_2$${\sigma }_2$sigma_(2)\sigma_{2}${\sigma }_2$ are the first and second stress levels, respectively, $N_{f1}$$N_{f1}$N_(f1)N_{f 1}$N_{f1}$ and $N_{f2}$$N_{f2}$N_(f2)N_{f 2}$N_{f2}$ are the fatigue life at the first and second stress levels, respectively, $n_1$$n_1$n_(1)n_{1}$n_1$ is the number of cycles under ${\sigma }_1$${\sigma }_1$sigma_(1)\sigma_{1}${\sigma }_1$, and $n_{2p}$$n_{2p}$n_(2p)n_{2 p}$n_{2p}$ is the number of cycles under ${\sigma }_2$${\sigma }_2$sigma_(2)\sigma_{2}${\sigma }_2$, namely, the residual fatigue life.

Step2: The CTGAN model was used to data augmentation based on the original welded structure fatigue dataset that has been built. The fatigue attributes ${\sigma }_1,{\sigma }_2,n_1,N_{f1},N_{f2},n_{2p}$${\sigma }_1,{\sigma }_2,n_1,N_{f1},N_{f2},n_{2p}$sigma_(1),sigma_(2),n_(1),N_(f1),N_(f2),n_(2p)\sigma_{1}, \sigma_{2}, n_{1}, N_{f 1}, N_{f 2}, n_{2 p}${\sigma }_1,{\sigma }_2,n_1,N_{f1},N_{f2},n_{2p}$ are input into the CTGAN model. The trained model finally obtains fatigue data with outputs ${\sigma }_{1m},{\sigma }_{2m},n_{1m},n_{2m}$${\sigma }_{1m},{\sigma }_{2m},n_{1m},n_{2m}$sigma_(1m),sigma_(2m),n_(1m),n_(2m)\sigma_{1 m}, \sigma_{2 m}, n_{1 m}, n_{2 m}${\sigma }_{1m},{\sigma }_{2m},n_{1m},n_{2m}$. Where ${\sigma }_{1m}$${\sigma }_{1m}$sigma_(1m)\sigma_{1 m}${\sigma }_{1m}$ and ${\sigma }_{2m}$${\sigma }_{2m}$sigma_(2m)\sigma_{2 m}${\sigma }_{2m}$ are the generated first and second stress levels, respectively, $n_{1m}$$n_{1m}$n_(1m)n_{1 m}$n_{1m}$ is the generated number of cycles under ${\sigma }_{1m}$${\sigma }_{1m}$sigma_(1m)\sigma_{1 m}${\sigma }_{1m}$, and $n_{2m}$$n_{2m}$n_(2m)n_{2 m}$n_{2m}$ is the generated number of cycles under ${\sigma }_{2m}$${\sigma }_{2m}$sigma_(2m)\sigma_{2 m}${\sigma }_{2m}$. Filtering in fatigue data from ${\sigma }_1,{\sigma }_2,n_{1m},N_{f1},N_{f2},n_{2m}$${\sigma }_1,{\sigma }_2,n_{1m},N_{f1},N_{f2},n_{2m}$sigma_(1),sigma_(2),n_(1m),N_(f1),N_(f2),n_(2m)\sigma_{1}, \sigma_{2}, n_{1 m}, N_{f 1}, N_{f 2}, n_{2 m}${\sigma }_1,{\sigma }_2,n_{1m},N_{f1},N_{f2},n_{2m}$ and the data were normalized.
Step3: The machine learning models were trained using the normalized data values ${\sigma }_1,{\sigma }_2,n_{1m},N_{f1},N_{f2},n_{2m}$${\sigma }_1,{\sigma }_2,n_{1m},N_{f1},N_{f2},n_{2m}$sigma_(1),sigma_(2),n_(1m),N_(f1),N_(f2),n_(2m)\sigma_{1}, \sigma_{2}, n_{1 m}, N_{f 1}, N_{f 2}, n_{2 m}${\sigma }_1,{\sigma }_2,n_{1m},N_{f1},N_{f2},n_{2m}$ as inputs, with the normalized fatigue life $n_{2m}$$n_{2m}$n_(2m)n_{2 m}$n_{2m}$ serving as the output. Subsequent training sessions were then conducted for these models.
Step4: Experiments were conducted on fatigue data from the test set using machine learning models obtained after data augmentation and original data training, respectively. The work examined how the generated samples influenced the predictive accuracy of the machine learning models. Furthermore, the proposed model's performance was validated by
comparing it with other conventional physical models, namely the Miner law, the Ye model, and its improved model, the Peng model.
Within the four outlined steps, Step 2 encompasses the following two specific subprocesses:
Step2.1: Fatigue data at the same stress level as the original dataset were chosen from the output of the CTGAN generation model that is when ${\sigma }_{1m}$${\sigma }_{1m}$sigma_(1m)\sigma_{1 m}${\sigma }_{1m}$ equals ${\sigma }_1$${\sigma }_1$sigma_(1)\sigma_{1}${\sigma }_1$ and ${\sigma }_{2m}$${\sigma }_{2m}$sigma_(2m)\sigma_{2 m}${\sigma }_{2m}$ equals ${\sigma }_2$${\sigma }_2$sigma_(2)\sigma_{2}${\sigma }_2$. And insert the corresponding $N_{f1}$$N_{f1}$N_(f1)N_{f 1}$N_{f1}$ and $N_{f2}$$N_{f2}$N_(f2)N_{f 2}$N_{f2}$ in the four columns of the output data which are the fatigue life under the first stage load and the second stage load. Finally, the complete data from ${\sigma }_1,{\sigma }_2,n_{1m},N_{f1},N_{f2},n_{2m}$${\sigma }_1,{\sigma }_2,n_{1m},N_{f1},N_{f2},n_{2m}$sigma_(1),sigma_(2),n_(1m),N_(f1),N_(f2),n_(2m)\sigma_{1}, \sigma_{2}, n_{1 m}, N_{f 1}, N_{f 2}, n_{2 m}${\sigma }_1,{\sigma }_2,n_{1m},N_{f1},N_{f2},n_{2m}$ is obtained.
Step2.2: The welded structures' fatigue dataset was split into training and test sets. A part of the fatigue test data was selected as the test set and the remaining data was used as the training set. Following this, the validated augmented data was combined with original training data to create a new training sample set. The data was at the same time normalized and the normalization was calculated by the formula:

where $x$$x$xx$x$ denotes the pre-normalization value of the sample data, $x_{\text{max}}$$x_{\text{max }}$x_("max ")x_{\text {max }}$x_{\text{max }}$ and $x_{\text{min}}$$x_{\text{min }}$x_("min ")x_{\text {min }}$x_{\text{min }}$ represent the maximum and minimum values of the fatigue test samples for a given data attribute, respectively, and $x_{\text{new}}$$x_{\text{new }}$x_("new ")x_{\text {new }}$x_{\text{new }}$ is the value of the sample data after normalization.

In this work, two welded joints and three welded materials are selected for experiments on the proposed framework, aluminum alloy butt joint [20,21] and corner joint [20,21], Al-2024-T42 [22], Al-7070-T7451 [23], and Ti-6Al-4V [24] respectively. These welded joints and materials are widely utilized in aerospace, marine, automotive, and various other industrial sectors. All experimental materials were tested for fatigue in a stress-controlled mode at room temperature. Additional information regarding the other test data is available in the accompanying references. The distribution of data from five types of experimental data is represented in terms of loading cycle ratios in Fig. 3.

In this work, all experiments were made on a personal computer with Windows 10 (64bit) operating system, hardware platform parameters of Intel(R) Core (TM) i5-7300HQ CPU, 2.50 GHz main frequency and 16 G RAM. The CTGAN model employs the Adam optimization algorithm for training the neural network. The discrete feature embedding dimension is set to 128 , while the hidden layer dimensions for both the generator and discriminator are ( 256,256 ). The learning rates for the generator and discriminator are both set to $2\mathrm{e}-4$$2\mathrm{e}-4$2e-42 \mathrm{e}-4$2\mathrm{e}-4$, with a learning rate decay of $1\mathrm{e}-6$$1\mathrm{e}-6$1e-61 \mathrm{e}-6$1\mathrm{e}-6$ for both. The batch size during training is 600 , and the number of epochs is 2000.300 fatigue data for each material is generated. The relevant parameters of the machine learning models are presented in Tab. 2:

In this work, a part of aluminum butt and corner joints, Al-2024-T42, Al-7075-T7451 and Ti-6Al-4V are selected as the test set. A part of the data of the test set is shown in Tab. 3. The rest of the data is the original training set. The generated fatigue dataset is then integrated with the original training set to produce the augmented training set.

The evaluation indicators in this work are absolute percentage error Error and mean absolute percentage error $ME$$ME$MEM E$ME$ to assess the performance of the model. Its calculation formula is:

where $\frac{n_2}{N_{f2}}$$\frac{n_2}{N_{f2}}$(n_(2))/(N_(f2))\frac{n_{2}}{N_{f 2}}$\frac{n_2}{N_{f2}}$ is the experimental cycle ratio under stress ${\sigma }_2,\frac{n_{2m}}{N_{f2}}$${\sigma }_2,\frac{n_{2m}}{N_{f2}}$sigma_(2),(n_(2m))/(N_(f2))\sigma_{2}, \frac{n_{2 m}}{N_{f 2}}${\sigma }_2,\frac{n_{2m}}{N_{f2}}$ is the predicted cycle ratio under stress ${\sigma }_2$${\sigma }_2$sigma_(2)\sigma_{2}${\sigma }_2$, and $m$$m$mm$m$ is the sample size of the dataset. The smaller the values of Error and ME mean that the model performs better.

The experimental data utilized for the tests comprised the fatigue test results of welded joints from the aluminum alloy bodies of high-speed trains in papers [20,21]. The welded joints used in the test contain two types of welding: butt joint and corner joint. The samples' base material is ENAW6005, an aluminum alloy employed in the construction of CRH2 highspeed train bodies. Based on fatigue testing outcomes, the butt joint fatigue life of $549,300,880,500$$549,300,880,500$549,300,880,500549,300,880,500$549,300,880,500$, and $1,540,100$$1,540,100$1,540,1001,540,100$1,540,100$ cycles at stress levels of 104,89 , and 74 MPa , respectively. Similarly, the corner joint's fatigue life were $619,800,952,300$$619,800,952,300$619,800,952,300619,800,952,300$619,800,952,300$, and $1,546,100$$1,546,100$1,546,1001,546,100$1,546,100$ cycles at stress levels of 93,83 , and 73 MPa , respectively. The error between experimental and predicted values for butt joint and corner joint are shown in Fig 4 and 5.

As observed in the predicted results of the welded joints presented in Figs. 4 and 5. For the butt joint, most of the four machine learning models utilizing data augmentation predicted points within the $10\mathrm{\%}$$10\mathrm{\%}$10%10 \%$10\mathrm{\%}$ error band, with three points nearly
aligning with the $0\mathrm{\%}$$0\mathrm{\%}$0%0 \%$0\mathrm{\%}$ error band. In contrast, the majority of predictions made by the Miner and Ye models falls outside the $20\mathrm{\%}$$20\mathrm{\%}$20%20 \%$20\mathrm{\%}$ error band. For corner joint, four machine learning models predictions with data augmentation are comparatively closer to the $0\mathrm{\%}$$0\mathrm{\%}$0%0 \%$0\mathrm{\%}$ error band. The machine learning models exhibited more consistent predictive performance compared to traditional models, displaying minimal variance in their results.

The Al-2024-T42 material [22] offers advantages such as high strength and excellent temperature tolerance. It is used to manufacturing a variety of components that hold high loads and is mainly used in aerospace applications. Fully reversed fatigue loads of different amplitudes were applied to the polished thin plate samples, setting the experimental frequency at 25 Hz and the stress ratio $\mathrm{R}=-1$$\mathrm{R}=-1$R=-1\mathrm{R}=-1$\mathrm{R}=-1$. The results of fatigue testing indicate that, under an applied stress of 150 MPa , Al-2024T42 has a fatigue life of 430,000 cycles, whereas at 200 MPa , its fatigue life is 150,000 cycles. The error between the experimental and predicted values of Al-2024-T42 is shown in Fig 6.

As shown in the prediction results in Fig. 6, most of the predicted values from the four machine learning models utilizing data augmentation are concentrated within the $20\mathrm{\%}$$20\mathrm{\%}$20%20 \%$20\mathrm{\%}$ error band. Conversely, the majority of predictions from the Miner and Ye models lie beyond the $20\mathrm{\%}$$20\mathrm{\%}$20%20 \%$20\mathrm{\%}$ error band. And there are five points in the augmented machine learning models that are closer to the fatigue experiment results and better stability relative to the Peng model.

Al-7050-T7451 is a high strength aluminum alloy material. It has good corrosion resistance and process-ability and is widely used in aerospace and automotive industries. The Al-7050-T7451 material from the paper [23] was tested for single stress amplitude change in flexural fatigue at room temperature. The results of fatigue testing reveal that, when subjected to applied stresses of $176\mathrm{MPa},133\mathrm{MPa}$$176\mathrm{MPa},133\mathrm{MPa}$176MPa,133MPa176 \mathrm{MPa}, 133 \mathrm{MPa}$176\mathrm{MPa},133\mathrm{MPa}$, and 85 MPa , the fatigue life corresponds to 27,300 cycles, 61,400 cycles, and 225,800 cycles, respectively. The error between the experimental and predicted values of Al-7050-T7451 is shown in Fig 7.
The prediction results in Fig. 7 show that most of the predicted values from the four machine learning models using data augmentation are concentrated within the $10\mathrm{\%}$$10\mathrm{\%}$10%10 \%$10\mathrm{\%}$ error band. Conversely, the majority of the predicted values from both the Miner and Ye models lie outside and are distant from the $20\mathrm{\%}$$20\mathrm{\%}$20%20 \%$20\mathrm{\%}$ error band. Furthermore, it is evident that the KELM and SVM models demonstrate superior prediction performance compared to the RF and BP models among the four machine learning approaches.

Ti-6Al-4V is the pioneering titanium alloy material that has been successfully developed and implemented. Its exceptional heat and corrosion resistance qualities make it predominantly utilized in aviation engines, rockets, and various other industries. The aero-engine compressor blade material $\mathrm{Ti}-6\mathrm{Al}-4\mathrm{V}$$\mathrm{Ti}-6\mathrm{Al}-4\mathrm{V}$Ti-6Al-4V\mathrm{Ti}-6 \mathrm{Al}-4 \mathrm{~V}$\mathrm{Ti}-6\mathrm{Al}-4\mathrm{V}$ titanium alloy from the literature [24] was tested in a room temperature environment. Its variable amplitude loading fatigue test consists of two types of loading: high-to-low and low-to-high. The load levels for the high-low loading were $595-517\mathrm{MPa}$$595-517\mathrm{MPa}$595-517MPa595-517 \mathrm{MPa}$595-517\mathrm{MPa}$ and $647-517\mathrm{MPa}$$647-517\mathrm{MPa}$647-517MPa647-517 \mathrm{MPa}$647-517\mathrm{MPa}$, while for the low-high loading, they were $517-595\mathrm{MPa}$$517-595\mathrm{MPa}$517-595MPa517-595 \mathrm{MPa}$517-595\mathrm{MPa}$ and $517-647\mathrm{MPa}$$517-647\mathrm{MPa}$517-647MPa517-647 \mathrm{MPa}$517-647\mathrm{MPa}$, respectively. Where the fatigue life at 647,517 , and 595 MPa stresses were 37200 , 143633, and 64467 cycles respectively. Fig. 8 displays the error between the experimental and predicted values for $\mathrm{Ti}-6\mathrm{Al}-$$\mathrm{Ti}-6\mathrm{Al}-$Ti-6Al-\mathrm{Ti}-6 \mathrm{Al}-$\mathrm{Ti}-6\mathrm{Al}-$ 4 V .

As shown in the prediction results in Fig. 8, the majority of the predicted values from the four machine learning models utilizing data augmentation fall within the $20\mathrm{\%}$$20\mathrm{\%}$20%20 \%$20\mathrm{\%}$ error band. Four of the data are closer to being within the $10\mathrm{\%}$$10\mathrm{\%}$10%10 \%$10\mathrm{\%}$ error band. Alternatively, the majority of the forecasts generated by the Miner and Ye models fall outside the $20\mathrm{\%}$$20\mathrm{\%}$20%20 \%$20\mathrm{\%}$ error band and are notably remote from it. It is apparent that the model augmented with data demonstrates a reduced error rate. The proposed augmented model is pertinent for predicting the fatigue life of both welded aluminum alloy structures and welded structures made from various other alloy materials.

For the four machine learning models with data augmentation, the $ME$$ME$MEM E$ME$ for almost all materials are near $10\mathrm{\%}$$10\mathrm{\%}$10%10 \%$10\mathrm{\%}$, with the exception of Al-7070-T7451, which is slightly higher. It is worth noting that the ME values for butt and corner joints are particularly low after data augmentation, with minimum errors of $6.82\mathrm{\%}$$6.82\mathrm{\%}$6.82%6.82 \%$6.82\mathrm{\%}$ and $4.65\mathrm{\%}$$4.65\mathrm{\%}$4.65%4.65 \%$4.65\mathrm{\%}$ on the KELM and RF、BP models, respectively. When compared to the conventional Miner, Ye, and Peng models, as well as the original machine learning models, the machine learning models that incorporate data augmentation exhibit a notable increase in prediction accuracy. It has been improved to some extent for the low fatigue data leading to poor accuracy of machine learning predictive models.

In order to avoid the uncertainty associated with the fatigue test data extracted above, five-fold cross-validation is added to this section. Thereby the stability and generalizability of the data augmentation model is verified on the machine learning models. ME is used here as an evaluation indicator to validate the generalization ability of the augmented model on four machine learning models. The results are shown in Tab. 6 and Tab. 7.

As can be seen from Tab. 6 and Tab. 7, the ME on the five-fold cross-validated machine learning models with augmented data overall less than $10\mathrm{\%}$$10\mathrm{\%}$10%10 \%$10\mathrm{\%}$, and the errors are greatly reduced. And the KELM and SVM models have better prediction results relative to the RF and BP models. The data-augmented machine learning models are still more advantaged compared to the traditional Miner, Ye and Peng models. Comparing with Tab. 5, the ME values in Tab. 7 are further reduced in general. The five-fold cross-validation reduces the randomness of model performance evaluation and provides more reliable performance estimates. The results show the stability and generalization ability of data-augmented machine learning models.

Limited fatigue data frequently impacts the precision and universal applicability of machine learning models for life prediction. To tackle this challenge, this work introduces a physics-based Generative Adversarial Networks (GAN) model aimed at generating fatigue data under two-step loading. The generated data served as input for machine learning algorithms, enabling predictions of the fatigue life of two types of welded joints and three welded materials under variable amplitude loading conditions.
The model combines a traditional model with machine learning models to better characterize fatigue behavior relative to a simple GAN model. The life prediction Peng model is integrated within the loss function of Conditional Tabular GAN (CTGAN) to guarantee that the generated data conforms to the physical relationships between stress and life. The final valid data that aligns with the characteristics of the original dataset is obtained through a careful selection process. The Peng model can consider the load sequence and the interaction between loads, which makes generated data meet the characteristics of fatigue data under two-step loading. Meanwhile, it effectively solves the limitation that machine learning models rely on large samples. The experimental findings indicate that the generated data notably augment the models' predictive accuracy.
The experiments are validated on two welded joints and three welded materials. The prediction indicators absolute percentage error Error and mean absolute percentage error $ME$$ME$MEM E$ME$ decreased obviously for each material. The $ME$$ME$MEM E$ME$ values of both welded joints decreased to less than $10\mathrm{\%}$$10\mathrm{\%}$10%10 \%$10\mathrm{\%}$, and the $ME$$ME$MEM E$ME$ values of titanium alloy materials also decreased by almost $10\mathrm{\%}$$10\mathrm{\%}$10%10 \%$10\mathrm{\%}$ on average. The results show that it is not only suitable for aluminum alloy materials, but also apparently effective for titanium alloy materials. In comparison to the traditional Miner model, Ye model, and Peng model, the augmented machine learning model exhibits improved accuracy and stability. And the accuracy of model performance evaluation was improved using five-fold cross-validation. This model markedly improves the precision of fatigue life prediction and is highly appropriate for augmenting fatigue data under two-step loading conditions.
The data produced by CTGAN effectively addresses the challenge of limited fatigue samples in machine learning applications under variable amplitude loading, all while maintaining clear physical significance. Using generated data as input for machine learning to predict fatigue life holds significant potential in engineering, as it improves accuracy and addresses the challenge of data scarcity. Future studies ought to concentrate on comprehensive evaluations and assessments of the reliability of fatigue life predictions for welded materials subjected to multistage loading, to bolster the robustness and applicability of models.

This research was supported by the National Science Foundation of China under Grant (52005071) and Liaoning Provincial Educational Department Project under Grant (2023JH2/101300236).

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