Backpropagation Neural Network (BPNN) is a data-driven model that utilizes gradient descent algorithms to enhance its performance. Through the backpropagation process, BPNN adjusts the connection weights between multiple layers of neurons, effectively capturing and predicting nonlinear relationships [25]. This capability has led to its widespread application in fatigue life prediction.

The BPNN architecture processes input features, including the number of cycles N, stress, and strain to generate the predicted stiffness. The input features are not normalized in order to preserve their physical meaning; however, training efficiency is ensured by carefully tuning the learning rate. The input and output for the hidden layer are computed as follows:

where j = 1 , 2 , ⋯ , n , ω i represents the weight coefficient connecting the i t h input neuron to the j t h neuron in the hidden layer, and b i is the bias parameter associated with the j t h neuron.

where F ( ⋅ ) is the activation function. The activation function adopted in this study is the rectified linear unit (ReLU), defined as:

with its derivative:

The input and output for the output layer are given by:

where v j denotes the weight coefficient connecting the hidden layer to the output layer, and φ is the bias parameter controlling the activation of the output neuron.

The BPNN training process involves error backpropagation, where the error is propagated from the output layer to the hidden layer. The errors for the output and hidden layers are calculated as:

In this study, the BPNN model utilizes the adaptive moment estimation (Adam) optimization algorithm for the backpropagation process. Adam combines the advantages of momentum-based gradient descent and root mean square propagation (RMSProp), ensuring efficient convergence. A grid search method based on 10-fold cross-validation was employed to select the model’s hyperparameters. The optimal hyperparameters and architecture for the BPNN model were determined. The optimal number of hidden layers was set to 4, with 30 neurons in each layer. The epoch and learning rate were set to 2000 and 0.005, respectively.

In the fatigue life prediction of rubber materials, traditional empirical models and data-driven models often exhibit certain limitations. To address this issue, this study proposes a fatigue life prediction method based on Physics-Informed Neural Networks (PINN). PINN integrates physical modeling with data-driven techniques, enabling the model to learn a damage evolution process that adheres to physical laws by incorporating physical constraints. This approach allows for accurate fatigue life and stiffness degradation predictions even with limited experimental data [26].

The structure of the Physics-Informed Neural Network (PINN) based on strain energy density is shown in Figure 1. The stiffness data, load, and cycle number obtained from the experimental setup are used as inputs to the PINN model, with the predicted stiffness degradation serving as the output. The hyperparameters are determined through a grid search method based on 10-fold cross-validation. The optimal architecture was found to consist of four hidden layers, each containing 30 neurons. The number of training epochs and learning rate were set to 2000 and 0.005, respectively. The optimization process employs the Adam optimizer algorithm. The damage evolution equation and stiffness degradation model are embedded in the neural network training process, ensuring that the network’s output not only fits the experimental data but also adheres to known physical laws. Specifically, PINN uses the damage evolution equation (based on energy dissipation) as a physical constraint and fits the damage variable and stiffness through the neural network to establish the damage evolution equation, as follows:

where, D represents the damage variable, N is the number of cycles, W is

Figure 1. The architecture of the Physics-Informed Neural Network (PINN) based on strain energy density.

the strain energy density, the damage parameters α and β are treated as learnable parameters that are optimized simultaneously with the neural network weights during the training of the Physics-Informed Neural Network (PINN). This approach enables the model to adaptively adjust the damage evolution equation based on experimental data, rather than relying on predefined values obtained from prior curve fitting. Consequently, it enhances the flexibility of the physical constraints and improves the predictive accuracy of the model. Strain energy density ( W ) is a key physical quantity that describes the energy dissipation of materials. It is closely related to both the stiffness degradation and fatigue damage of the material. In this study, the strain energy density is calculated using the Yeoh model, which is suitable for modeling the large deformation behavior of materials such as rubber. The strain energy density formula of the Yeoh model is as follows:

where, I 1 represents the first invariant of the right Cauchy-Green deformation tensor, and C 1 , C 2 , and C 3 are material constants obtained through experimental data fitting. The Yeoh model accurately describes the nonlinear large deformation behavior of rubber materials and calculates the strain energy density. For the material used in the experiments, the constants are as follows: C 1 = 2.96   MPa , C 2 = 4.37   MPa , and C 3 = 3.19   MPa [27].

By calculating the stiffness variation curve and load data, the strain energy density W is computed, providing the necessary input for the damage evolution equation. During the fatigue process, the stiffness of rubber materials gradually degrades with the increase in the damage variable D . A commonly used relationship is:

where, K ( N ) represents the material stiffness after the N -th cycle, K 0 is the initial stiffness, and D ( N ) is the damage variable that describes the degree of fatigue damage in the material.

There is no unified standard for fatigue failure criteria of rubber materials. Previous studies have used static stiffness degradation as a key indicator of fatigue damage [28]. Literature [29] reports that rubber materials exhibit significant static stiffness reduction after fatigue loading. According to the TB/T 2843 standard, the tolerance limits for static property changes of rubber elastic components after fatigue are ±20% for strict tolerance and ±30% for normal tolerance. Based on the service requirements of actual vehicle suspension structures and engineering experience, this study adopts a 25% reduction in static stiffness as the fatigue failure criterion, which is expressed by the following formula:

where, Δ K represents the static stiffness degradation rate, K 0 is the initial static stiffness, and K f is the static stiffness after fatigue.

To ensure that the model training adheres to physical laws, it is necessary to specify the initial and boundary conditions. The initial condition is that the damage variable D 0 = 0 , indicating that the material is undamaged at the initial time. The initial stiffness K 0 is obtained from experimental data. The boundary condition is that when the stiffness degrades to 25%, the damage variable D reaches its maximum value of 1.

The loss function of the PINN model consists of two components: data loss and physical loss. The data loss ensures that the network’s predicted stiffness aligns with the experimental data. The data loss is given by the following formula:

where K measured is the stiffness experimentally measured, and K predicted is the stiffness predicted by the model.

The prior physical knowledge of fatigue failure is incorporated into the training process. During training, if the fatigue life prediction contradicts the physical knowledge, the loss function should impose the corresponding physical constraints. The physical loss is given by the following formula:

This loss term ensures that the damage variable D follows the damage evolution equation. When the stiffness degrades by 25%, the damage reaches its maximum value:

During training, the network computes the damage variable D and stiffness K through forward propagation. The network weights are then updated based on the data loss and physical loss. Through iterative training, the network is able to provide more accurate fatigue life and stiffness predictions by utilizing physical constraints, even with limited data.

To balance the influence of the data loss, physics loss, and boundary condition loss during the optimization process, weighted coefficients are applied to these loss terms. Specifically, the weight for the data loss is set to 1, while the weights for the physics loss and boundary condition loss are determined through grid search optimization as 0.4 and 0.3, respectively. This weighting strategy helps prevent any single loss term from dominating the training process, ensuring that the model fits the experimental data while strictly adhering to physical constraints, thereby improving convergence stability and predictive performance.

The major advantage of PINN lies in its ability to effectively fit the model in data-scarce situations by incorporating physical constraints, thus avoiding overfitting while enhancing the model’s generalization capability. Compared to traditional purely data-driven models, PINN better handles small sample problems and ensures physical consistency, thereby improving the accuracy of fatigue life predictions.

Table 1. Data collection.

Table 1presents the key data for the experimental specimens, including the loading amplitude range, stiffness sampling interval, and measured fatigue life. During the fatigue loading phase, the high-precision force sensor and displacement encoder of the testing machine record the force-displacement response of each loading cycle in real time. This data is used to train the PINN model and calculate the dissipated energy for each cycle.

To reflect the stiffness degradation behavior of the material during the fatigue damage process, the experiment pauses loading at several predefined cycle intervals. During these pauses, a low-speed, small-amplitude quasi-static load is applied to the specimen, and the corresponding force-displacement relationship is recorded. The regression equation fitted to the experimental data reveals a linear relationship between force and displacement, with the slope of the equation, as shown in Figure 4, representing the stiffness value.

To ensure data accuracy, the specimen temperature is stabilized and loading conditions are kept consistent before resuming fatigue loading. This method not only captures high-frequency dynamic response information but also records the static stiffness degradation characteristics that evolve with the number of cycles. Consequently, the experimental data exhibit good temporal consistency and physical interpretability, providing a reliable foundation for constructing the rubber material damage evolution model and training the Physics-Informed Neural Network (PINN).

Figure 4. Stiffness measurement.

Figure 5 shows the comparison between the stiffness predicted by the BPNN model and the PINN model versus the actual stiffness on the test set. It can be observed that the BPNN model fails to accurately capture the stiffness change trend in the later stages of stiffness degradation, resulting in significant prediction errors. In contrast, the PINN model demonstrates higher accuracy throughout the entire prediction process. Particularly during the stiffness degradation period, the predictions from the PINN model closely align with the actual values, showing a stronger fitting capability. By incorporating the physical damage evolution equation, PINN effectively captures the nonlinear fatigue degradation process of rubber materials, providing more stable and reliable predictions.

Figure 5. Comparison of stiffness predictions from BPNN and PINN on the validation set.

Figure 6 and Figure 7 display the relationship between the model’s predicted values and the actual values, helping us assess the prediction ability and error magnitude of the models. By observing the fitting performance of both the BPNN and PINN models on the training and test datasets, we can evaluate their prediction accuracy. On the training set, the BPNN model shows relatively good fitting

Figure 6. Comparison of stiffness prediction errors for BPNN and PINN on the training set.

Figure 7. Comparison of stiffness prediction errors for BPNN and PINN on the test set.

performance. However, on the test set, as the data complexity increases, the prediction error gradually increases, especially in the region of rapid stiffness degradation. This indicates that the BPNN model performs poorly when dealing with high stiffness degradation or nonlinear changes. In contrast, the PINN model exhibits better prediction performance on both the training and test sets. Notably, on the test set, the PINN model effectively captures the stiffness variation trend, maintaining high prediction accuracy even in regions of high stiffness and fatigue degradation. By comparing the two figures, it is evident that the PINN model’s predictions consistently align more closely with the actual values, especially in the regions of high stiffness and fatigue degradation, showing more stable performance.

To further evaluate the performance of BPNN and PINN, RMSE and the coefficient of determination (R²) are used as metrics for assessing prediction accuracy. Their calculation formulas are given as follows:

where n is the number of samples; y true ( i ) is the true stiffness of the i -th sample; y pred ( i ) is the stiffness predicted by the model for the i -th sample; and y ¯ true is the mean of the true values.

Table 2.Estimation results of BPNN and PINN on different datasets (%).

Root Mean Square Error (RMSE) is a commonly used evaluation metric for regression models, especially in prediction models, to measure the difference between the predicted and actual values. A smaller RMSE value indicates that the model has a smaller error and a better fit when predicting stiffness degradation during the fatigue process. The coefficient of determination (R²) represents the proportion of the variance in the output variable that can be explained by the input variables. A value close to 1 indicates a better model fit and more accurate predictions. Table 2 provides the RMSE and R² values for the BPNN and PINN models for both training and testing samples. The results show that BPNN exhibits poor predictive performance on the test samples and fails to generalize effectively. In contrast, PINN, due to the incorporation of physical damage constraints, demonstrates lower RMSE and higher R² values on both the training and testing samples, indicating that physical information enhances the model’s predictive performance and generalization ability.