Artificial Neural Networks for Predicting Asphalt Fatigue Life: Investigating Material and Loading Parameters with a Comprehensive Dataset and Addressing Model Intricacies

Artificial Neural Networks for Predicting Asphalt Fatigue Life: Investigating Material and Loading Parameters with a Comprehensive Dataset and Addressing Model Intricacies Jakub Houl´ ık[0009−0001−5956−026𝑋], Jan Valentin[0000−0001−9155−1921], Jan Kr´ ol[0000−0002−0530−2927], Piotr Pokorski[0000−0002−8985−7400], V´ aclav Neˇzerka[0000−0001−7360−0507] Abstract This study employs artificial neural networks (ANNs) to predict the fatigue life of asphalt concrete (AC), crucial for road maintenance and longevity. Leveraging a dataset from extensive laboratory tests, we optimized ANN models to address the variability in AC fatigue data. Our approach involved fine-tuning hyperparameters and adapting network architectures to best utilize a dataset of 152 samples. The models were trained with both linear and logarithmic loss functions. Results showed that modified bituminous binders significantly improve fatigue life predictions, with comprehensive input parameters being vital for accurate modeling. Although models achieved moderate overall 𝑅2scores of about 0.4, they highlighted the significant impact of binder type and content on prediction outcomes. The research demonstrates the potential of machine learning to enhance pavement engineering by providing deeper insights into AC fatigue life. Optimized models and codes are available in an open repository, encouraging further exploration and application. Jakub Houl´ ık Faculty of Civil Engineering, Czech Technical University in Prague, Th´ akurova 2077/7, 166 29 Praha 6, Czech Republic e-mail: [email protected] Jan Valentin Faculty of Civil Engineering, Czech Technical University in Prague, Th´ akurova 2077/7, 166 29 Praha 6, Czech Republic e-mail: [email protected] Jan Kr´ ol Faculty of Civil Engineering, Warsaw University of Technology, Aleja Armii Ludowej 16, 00-637 Warszawa, Poland e-mail: [email protected] Piotr Pokorski Faculty of Civil Engineering, Warsaw University of Technology, Aleja Armii Ludowej 16, 00-637 Warszawa, Poland e-mail: [email protected] V´ aclav Neˇzerka Faculty of Civil Engineering, Czech Technical University in Prague, Th´ akurova 2077/7, 166 29 Praha 6, Czech Republic e-mail: [email protected] 1 2 Jakub Houl´ ık, Jan Valentin, Jan Kr´ ol, Piotr Pokorski, V´ aclav Neˇzerka 1 Introduction Asphalt concrete (AC) serves as a fundamental material in road pavement construction, composed of aggregates like crushed aggregate, gravel, or sand, combined with a bituminous binder derived from the distillation of crude oil or enhanced with polymers and additives to form modified binders [1]. The fatigue life of AC critically influences maintenance costs and is affected by the material’s ability to resist cyclic heavy traffic loads across varying environmental conditions without significant cracking [2–4]. Degradation of AC primarily stems from the aging of the bituminous binder due to oxidation, limiting its resistance to repetitive tensile stresses [5, 6], with failure occurring when tensile strain exceeds the material’s capacity [7]. Additionally, cold conditions increase material brittleness and susceptibility to thermally-induced cracking [8], whereas high temperatures render the binder more viscous and prone to deformation [9]. Several factors such as binder content [8], air voids content [10, 11], loading strain [12, 13], and operational temperature [13, 14] influence AC’s fatigue resistance. Common laboratory assessments include four-point bending, semi-circular bending, and indirect tensile tests, each offering insights into fatigue life, stiffness, and strength under varied loading conditions [15–21]. Accurately predicting fatigue life relative to AC composition, climate, and loading rate is essential for effective mix design, though traditional testing is notably timeintensive, requiring up to 42 days for specimen preparation, with tests lasting several hours to days based on load and frequency [22]. Although these methods provide deep insights, they demand substantial time and resources. By applying machine learning (ML) models to existing datasets, predictions of AC’s fatigue life become feasible when traditional testing is not viable. Extensive reviews have been conducted on the mechanisms, characterization techniques, and mitigation strategies in AC fatigue life, pointing to the complex interplay between material components and external factors such as aging and environmental impacts [2, 3, 23, 24]. Recent developments in ML-assisted mix design have initiated models incorporating a wider array of variables to predict fatigue life [25–27]. Our research aims to utilize a comprehensive dataset to model ML relationships focusing primarily on easily measurable properties during the design phase, such as binder content (%) and air voids (%), along with the type of binder and penetration grade. These microstructural parameters were recently highlighted by Alnaqbi et al. [28] as crucial for the durability of AC pavements. To our knowledge, the detailed analysis and extensive application of neural networks in predicting laboratory specimen fatigue life have not been extensively reported in existing literature. 2 Methodology The primary aim of this study was to compare the performance of various artificial neural network (ANN) models, each utilizing different combinations of input ANNs for Asphalt Concrete Fatigue Life Predictions 3 parameters. This comparative approach allowed for the determination of which parameters are most critical for the construction of predictive models and which are less influential. Initially, a principal model were developed utilizing all available input parameters. Subsequent models were then created, each lacking one of these input parameters, to isolate and evaluate the impact of each parameter on the model’s predictive accuracy. 2.1 Data Collection Data for this study were collected from sample preparation and measurements carried out at two institutions: the Faculty of Civil Engineering at the Czech Technical University in Prague (CTU) and the Faculty of Civil Engineering at the Warsaw University of Technology (WUT). For the purposes of this study, selected data included: binder content (%), binder type—modified bituminous binders (PMB) and neat bituminous binders (NB), air voids (%), loading strain (𝜀), penetration grade (×0.1 mm), test temperature ( ◦C), initial stiffness (MPa), and number of cycles (𝑁f). The polymer-modified bitumens (PMBs) were enhanced using a styrene-butadienestyrene elastomer, improving their resilience and performance characteristics [29]. Neat bitumens (NBs), also referred to as paving grade bitumens, were prepared in various grades such as 35/50, 50/70, and 70/100, each affecting the penetration values significantly [30]. All experiments were consistently performed at a temperature of 10,◦C and a loading frequency of 10 Hz using a four-point bending setup. The initial dataset comprised 152 samples, with 102 samples from CTU and 50 from the WUT laboratory (Table1). A significant benefit of this study is the provision of these datasets alongside the codes on our GitHub repository1, making them readily available for further research. These datasets are extremely valuable, offering a robust foundation for future explorations into asphalt concrete behavior under varied conditions. Outlier detection was implemented using the Z-score technique, where: 𝑍=𝑋−𝜇 𝜎.(1) Here, 𝑍is the Z-score, representing the number of standard deviations a data point 𝑋 differs from the mean 𝜇of the dataset, with 𝜎being the standard deviation indicating the dispersion of data points around the mean. For optimal data filtration, samples with 𝑍≥3 were excluded as outliers. 1https://github.com/jakub-houlik/Asphalt_Fatigue_ANN 4 Jakub Houl´ ık, Jan Valentin, Jan Kr´ ol, Piotr Pokorski, V´ aclav Neˇzerka Table 1 Summary of data gathered for modelling. Both datasets (CTU and WUT) consisted of PMB and NB. The values for the Combinded dataset was filtered by applying the Z-score cut-off. Laboratory Binder content Penetration Air voids Strain Initial stiffness 𝑁𝑓Samples (%) (0.1mm) (%) (×10−6) (MPa) (×103) CTU 3.9–5.7 40–85 3.1–11.6 110–300 3,666–17,428 3.4–1,668 102 WUT 3.9–4.6 40–60 3.9–8.0 90–210 11,083–17,574 50.1–9,810 50 Combined 3.9–5.1 40–85 3.2–7.0 90–210 6,598–17,574 10.4–2,519 118 2.2 ANN The primary goal of this study was to compare the performance of various ANN models, each utilizing different combinations of input parameters to ascertain which parameters are most significant for predicting fatigue life. Initially, a principal model was created with all available input parameters, accompanied by a systematic search for optimal hyperparameters to enhance model performance and accuracy. To identify the best hyperparameters, we implemented cross-validation and split the data into three groups (folds). Each cycle of training and validation involved using one fold for validation and the others for training. This cycle was repeated, rotating the validation fold each time, which helped create multiple models with different training-validation data splits. We always allocated 20% of the dataset for testing, with the remaining 80% divided into the three folds for the cross-validation process. This approach minimized the risk of bias in model performance due to random data selection. The input data for the model are provided along with the ANN’s structure in Fig. 1, indicating the optimized architecture parameters: number of hidden layers, 𝑛h, and number of neurons in the hidden layer, 𝑛neur,h. The input parameters follow the data prepared during the experimental session (Table 1). The distribution of numeric input parameters is provided in Fig. 2. 𝑁𝑓was the only output parameter. After finding the optimum ANN size and hyperparameters, models were crafted with one input parameter omitted at a time, allowing for the evaluation of the importance of each parameter. Models were for 200,000 epochs, aiming to optimize the model to achieve the lowest validation loss. The model considered most accurate was saved at the point where it exhibited the lowest validation loss. Such an ANN model was highly effective for processing tabular numerical data; the model assigned weights to different input variables and combined them through several hidden layers to generate the output. These weights were adjusted based on validation data through multiple iterations to improve the model’s accuracy. 2.2.1 Activation Functions In neural networks, activation functions are crucial for introducing non-linear properties into the model, allowing it to learn more complex patterns. The activation ANNs for Asphalt Concrete Fatigue Life Predictions 5 Fig. 1 ANN’s architecture, input parameters, and the output parameter. 4.00 4.25 4.50 4.75 5.00 5.25 5.50 5.75 Binder content (%) 0 5 10 15 20 25 Frequency CTU WUT 4 6 8 10 12 Air voids (%) 0 5 10 15 20 25 30 35 40 Frequency CTU WUT 0.10 0.15 0.20 0.25 0.30 Loading strain 0 5 10 15 20 25 30 35 40 Frequency CTU WUT 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 Initial stiffness (GPa) ×107 0 2 4 6 8 10 12 Frequency CTU WUT Fig. 2 Distributions of numeric parameters for ANNs. 6 Jakub Houl´ ık, Jan Valentin, Jan Kr´ ol, Piotr Pokorski, V´ aclav Neˇzerka functions used in this study included ReLU, SiLU, GELU, and Mish, each with specific properties suitable for different network configurations: 1. ReLU (rectified linear unit) function, defined as 𝑓(𝑥)=max(0, 𝑥), facilitates faster training by effectively turning off the negative part of its input [31]. 2. SiLU (sigmoid linear unit) function, defined as 𝑓(𝑥)=𝑥 1+𝑒−𝑥, introduces a nonmonotonic function that helps the network learn more complex patterns [32]. 3. GELU (Gaussian error linear unit) function, using the formula 𝑓(𝑥)=𝑥Φ(𝑥) where Φ(𝑥)is the cumulative distribution function of the standard Gaussian, offers a probabilistic approach to gating inputs [33]. 4. Mish function is defined as 𝑓(𝑥)=𝑥tanh(ln(1+𝑒𝑥)), providing a smooth and non-monotonic activation that has shown success in deeper or more complex architectures [34]. 2.2.2 Optimizers An optimizer is a crucial algorithm used to minimize a loss function, which is a measure of how well a model’s predictions match the actual outcomes. The optimizer adjusts the weights of the network with the aim of reducing the loss, thereby improving the model’s accuracy during training. The choice of optimizer is critical for training neural networks efficiently and effectively, as it directly influences the speed and quality of learning [35]. In this study, the following optimizers were evaluated: 1. Adam: An optimizer that computes adaptive learning rates for each parameter from estimates of first and second moments of the gradients. 2. AdamW: A variant of Adam with decoupled weight decay regularization, helping to improve training stability. 3. RMSprop: Utilizes the moving average of squared gradients to normalize the gradient itself, a good choice for recurrent neural networks. 4. Stochastic gradient descent (SGD): Utilizes mini-batched gradient descent with a defined learning rate and momentum, known for its simplicity and effectiveness in various scenarios. 2.2.3 Loss Functions Selecting the appropriate loss function is crucial when training neural networks because it defines the measure of error between the predicted values and the actual values that the model aims to minimize. This error quantification directly influences how the model learns during training, guiding the adjustment of weights to optimize performance. A well-chosen loss function ensures that the model accurately captures the underlying patterns of the dataset, particularly in tasks where the scale of data varies significantly. In this study, two types of loss functions were utilized to tailor the model’s performance to the diverse scales of fatigue life data, typically represented on a logarithmic scale: ANNs for Asphalt Concrete Fatigue Life Predictions 7 1. Commonly used mean square error (MSE): This loss function calculates the average squared difference between the predicted and actual values, suitable for regression problems. It is defined as: 𝐿MSE (𝑁f,ˆ 𝑁f)=1 𝑛 𝑛 ∑︁ 𝑖=1 (𝑁f,𝑖 −ˆ 𝑁f,𝑖)2,(2) where 𝑁f,𝑖 is the measured number of cycles, ˆ 𝑁f,𝑖 is the predicted number of cycles, and 𝑛is the number of samples. 2. Mean square logarithmic error (MSLE): This loss function is particularly useful when the target variable, like fatigue life, spans several orders of magnitude. It computes the mean of the squared logarithmic differences between the predicted and actual values, helping to prevent large errors in the predictions of large true values. It is defined as: 𝐿MSLE (𝑁f,ˆ 𝑁f)=1 𝑛 𝑛 ∑︁ 𝑖=1 (log (𝑁f,𝑖 +1) − log (ˆ 𝑁f,𝑖 +1))2.(3) The ANN model configuration and training were executed as follows: The Sequential model from the Keras library (version 2.12.0rc1) was utilized, and before training, the data were normalized using the MinMaxScaler function from the sklearn.preprocessing module to scale the input variables between 0 and 1. The configured hyperparameters included the number of hidden layers (𝑛h), number of neurons within the hidden layers (𝑛neur,h), activation function, and the optimization algorithm, as detailed in Table 2. Table 2 Hyperparameters tested during the model development and values selected for the final model. Hyperparameter Grid Selected Loss function 𝐿MSE,𝐿MSLE both Optimizer Adam, AdamW, RMSprop, SGD SGD (for 𝐿MSE) and RMSprop (for 𝐿MSLE) Activation function ReLU, SiLU, GELU, Mish SiLU (for 𝐿MSE) and ReLU (for 𝐿MSLE) 𝑛h1, 2, 3 2 𝑛neur,h10, 20, 50, 100, 200 10 (for 𝐿MSE) and 20 (for 𝐿MSLE) 3 Results and discussion 3.1 Hyperparameters The use of SiLU and ReLU activation functions was instrumental in mitigating the vanishing gradient problem, facilitating effective error backpropagation through the 8 Jakub Houl´ ık, Jan Valentin, Jan Kr´ ol, Piotr Pokorski, V´ aclav Neˇzerka network [36]. Our detailed examination aimed to determine the optimal settings for the number of hidden layers (𝑛h) and the number of neurons per hidden layer (𝑛neur,h), tested against both 𝐿MSE and 𝐿MSLE loss functions. For the model employing 𝐿MSE, the configuration with 𝑛h=2 and 𝑛neur,h=10 proved most effective, as depicted in Fig. 3. This structure achieved the highest average 𝑅2score across the three folds, where some combinations of hyperparameters resulted in 𝑅2scores below zero, indicating a predictive ability worse than the mean of the dataset. 123 nh −1.00 −0.75 −0.50 −0.25 0.00 0.25 0.50 0.75 1.00 R2 25 50 75 100 125 150 175 200 nneur,h −1.00 −0.75 −0.50 −0.25 0.00 0.25 0.50 0.75 1.00 R2 nneur,h= 10, nh= 1 nneur,h= 20, nh= 1 nneur,h= 50, nh= 1 nneur,h= 100, nh= 1 nneur,h= 200, nh= 1 nneur,h= 10, nh= 2 nneur,h= 20, nh= 2 nneur,h= 50, nh= 2 nneur,h= 100, nh= 2 nneur,h= 200, nh= 2 nneur,h= 10, nh= 3 nneur,h= 20, nh= 3 nneur,h= 50, nh= 3 nneur,h= 100, nh= 3 nneur,h= 200, nh= 3 Fig. 3 Variation of the model performance, measured by the 𝑅2score, in relation to 𝑛hand 𝑛neur,h when employing 𝐿MSE as the loss function. Conversely, the logarithmic model using 𝐿MSLE with the same number of hidden layers and 20 neurons per layer showed optimal performance, as demonstrated in Fig. 4. Certain configurations failed to converge and were excluded from the results, while some resulted in negative 𝑅2scores, similarly indicating less predictive power than the average dataset value. When assessing different optimizers, the SGD optimizer paired with the SiLU activation function showcased superior performance, particularly with 𝐿MSE. In contrast, the RMSprop optimizer was more effective with the ReLU activation function for 𝐿MSLE, as depicted in Fig. 5. RMSprop’s method of adjusting the learning rate using a moving average of squared gradients provides stability in noisy data environments [35]. ANNs for Asphalt Concrete Fatigue Life Predictions 9 123 nh −1.00 −0.75 −0.50 −0.25 0.00 0.25 0.50 0.75 1.00 R2 25 50 75 100 125 150 175 200 nneur,h −1.00 −0.75 −0.50 −0.25 0.00 0.25 0.50 0.75 1.00 R2 nneur,h= 10, nh= 1 nneur,h= 20, nh= 1 nneur,h= 50, nh= 1 nneur,h= 100, nh= 1 nneur,h= 200, nh= 1 nneur,h= 10, nh= 2 nneur,h= 20, nh= 2 nneur,h= 50, nh= 2 nneur,h= 100, nh= 2 nneur,h= 200, nh= 2 nneur,h= 10, nh= 3 nneur,h= 20, nh= 3 nneur,h= 50, nh= 3 nneur,h= 100, nh= 3 nneur,h= 200, nh= 3 Fig. 4 Variation of the model performance, measured by the 𝑅2score, in relation to 𝑛hand 𝑛neur,h when employing 𝐿MSLE as the loss function. ReLU SiLU GELU Mish −0.1 0.0 0.1 0.2 0.3 R2 LMSE(y, ˆy) ReLU SiLU GELU Mish 0.00 0.05 0.10 0.15 0.20 0.25 R2 LMSLE(y, ˆy) Adam AdamW RMSprop SGD Fig. 5 Model performance in relation to activation functions and optimizers used, showcasing different predictive abilities and optimal hyperparameter combinations for both loss functions. 3.2 Model accuracy Table 3 presents the 𝑅2scores for models trained on the entire dataset post Z-score filtering as listed in Table 1. These models were optimized for architecture and hyperparameters, using all inputs depicted in Fig. 1 as well as scenarios where individual parameters were omitted. Despite substantial efforts to optimize the architecture and 16 Jakub Houl´ ık, Jan Valentin, Jan Kr´ ol, Piotr Pokorski, V´ aclav Neˇzerka References [1] J. Valentin, J. Trejbal, V. Neˇzerka, T. Valentov´ a, P. Vackov´ a, P. 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