A Creep Model of Steel Slag–Asphalt Mixture Based on Neural Networks

: To characterize the complex creep behavior of steel slag–asphalt mixture in ﬂ uenced by both stress and temperature, predictive models employing Back Propagation (BP) and Long Short-Term Memory (LSTM) neural networks are described and compared in this paper. Multiple stress repeated creep recovery tests on AC-13 grade steel slag–asphalt mix samples were conducted at di ﬀ erent temperatures. The experimental results were processed into a group of independent creep recovery test results, then divided into training and testing datasets. The K-fold cross-validation was applied to the training datasets to ﬁ ne-tune the hyperparameters of the neural networks e ﬀ ectively. Compared with the experimental curves, both the e ﬀ ects of BP and LSTM models were investigated, and the broad applicability of the models was proven. The performance of the trained LSTM model was observed by a 95% con ﬁ dence interval around the ﬁ t errors, thereby the creep strain intervals for the testing dataset were obtained. The results suggest that the LSTM model had enhanced prediction compared the BP model for creep deformation trends of steel slag–asphalt mixture at various temperatures. Due to the potent generalization strength of arti ﬁ cial intelligence technology, the LSTM model can be further expanded for forecasting road ru tt ing deformations.


Introduction
The steel industry is fundamental pillar of the national economy in China.A large amount of steel slag is produced during the steel smelting process, but it is not utilized efficiently.However, there is strong demand for asphalt mixture composed of asphalt and conventional aggregates, namely granite and limestone, in constructing asphalt roads.Substituting a portion of these natural aggregates with steel slag not only improves resource utilization efficiency, but also promotes high-quality development within the steel industry.This approach also ensures the environment is protected and carbon emissions are reduced [1][2][3].Previous studies have revealed that steel slag in asphalt pavements effectively delays the onset of surface cracking and enhances resistance to skidding.Additionally, rutting defects on road surfaces are inevitably caused by heavy traffic.Ceep, alongside other mechanical properties of steel slag-asphalt mixture, has attracted much attention [4][5][6].
In order to describe the creep properties of asphalt mixture, researchers have established various differential or integral models [7][8][9].However, these mathematical models tend to be intricate, due to the complex determination process of model coefficients [10].More importantly, these models have limited predictive capability to describe creep behaviors under singular influences such as external load levels.Therefore, it is necessary to establish a creep model that takes into account both temperature and stress and has a higher predictive accuracy [11].Machine learning algorithms, especially artificial neural networks (ANN), have made profound advancements and are increasingly applied to tackle a spectrum of mechanical challenges [12][13][14].In the engineering field, the interrelations among variables can be extraordinarily complex, which often makes it impossible to achieve effective representation through traditional physical or empirical models.Machine learning methodologies are employed by approximating the highly nonlinear mappings that exist between these variables [15].Wang, J. (2021) [16] has employed machine learning to formulate a predictive model for the creep lifespan of Cr-Mo steels.Salari, S. (2022) [17] has innovatively merged machine learning with finite element analysis to investigate the evolution of contact areas and friction behaviors within nickel-based superalloys under elevated temperatures.Feng, J. (2022) [18,19] has studied the creep behavior of recycled aggregate concrete (RAC) using machine learning techniques.These scholars reveal the capability of machine learning in evaluating the complex mechanical properties of materials.
Using the nonlinear model and adaptive learning ability of ANN, experimental data can be processed efficiently and predict properties of materials effectively.Zhang, J. (2019) [20], Wan, P. (2020) [21], and Sun, Y. (2010) [22] have each implemented Back Propagation (BP) neural networks to delve into the constitutive relationships within alloy materials.Li, D. (2022) [23] utilized Long Short-Term Memory (LSTM) neural networks to effectively predict the mechanical dynamics of rubber.Nevertheless, the application of neural network models to the study of the nonlinear mechanical behavior exhibited by steel slagasphalt mixture remains an area that requires further extensive research.
Our study leverages the Long Short-Term Memory (LSTM) architecture, a sophisticated Recurrent Neural Network (RNN) capable of learning from sequences with longterm dependencies [24][25][26].The LSTM's design overcomes the gradient issues inherent in traditional RNNs by incorporating a memory cell with three critical components: the input, forget, and output gates.These gates regulate the flow of information, allowing the network to retain relevant data over time and forget outdated or less important information.The input gate controls the update of the cell state with new information, while the forget gate determines which information to discard from the cell state.The output gate then decides the extent to which the cell state should influence the next hidden state.This mechanism enables LSTMs to maintain a form of memory, making them particularly effective for tasks involving temporal dynamics.
In this paper, Back Propagation (BP) and Long Short-Term Memory (LSTM) networks are used to formulate creep models for steel slag-asphalt mixture.The experimental data are processed to generate training and test datasets.The neural networks are trained with these datasets, and the fit result analysis is conducted to establish a 95% confidence interval.
Section 2 describes creep recovery experiments that were conducted on AC-13 graded steel slag-asphalt mixture.Single creep recovery experimental data under different temperatures and stress levels are presented and used to establish the test set and the training set.In Section 3, Back Propagation (BP) and Long Short-Term Memory (LSTM) neural networks are introduced.In Section 4, the entire process of constructing creep models for asphalt mixture based on BP and LSTM neural networks is presented.The model structure and hyperparameter design are also investigated.In Section 5, the model's predictive results are compared with the experimental curves to confirm the validity of the LSTM model.A brief conclusion is provided in Section 6.

Selection of Materials and Fabrication of Specimens
Grade 70 road petroleum asphalt, limestone, and steel slag sourced from Hebei Shengli Steel were selected as materials for this study.These materials were tested in accordance with standards.In accordance with an asphalt-to-aggregate mass ratio of 5.3%, the standard Marshall cylinders specimens were fabricated by the compaction method, with approximate dimensions of 101.6 mm in diameter and 63.5 mm in height.Considering the specific gravity coefficient of 1.16 between steel slag and fine aggregate (limestone), a direct replacement of coarse aggregate with steel slag is not feasible.Consequently, based on the AC-13 grading scale, the mass proportion for each coarse aggregate size was enlarged in accordance with the specific gravity coefficient, and the mass proportion of the fine aggregate was recalculated.The compositions of aggregates in this paper are illustrated in Table 1.The standard AC-13 gradation curve and the AC-13 with steel slag gradation curve are shown in Figure 1.5.9 0.15~0.3 3.8 0.075~0.15 2.0 <0.075 6.9 The binder-to-aggregate ratio exerts a considerable influence on the quality of steel slag-asphalt mixture specimens.An insufficient binder-to-aggregate ratio leads to challenges in compacting the mixture, resulting in a dry surface and a tendency for aggregate particles to dislodge.In contrast, an overly abundant binder-to-aggregate ratio causes the asphalt mixture to exhibit oily efflorescence on its surface.This research adheres to a binder-to-aggregate ratio within the threshold that ensures optimal compaction, specifically between 4.2% and 6.6%.For the purposes of this paper, a ratio of 5.3% has been selected for its adherence to the established compaction standards.
The fabrication of specimens for the asphalt mixture is as follows: (1) The aggregates and asphalt are kept in a drying oven under 160 °C for 6 h, then all heated materials are mixed in a blender under 170 °C for 120 s. (2) According to the Standard Marshall Test, the mixed material is compacted by the Marshall equipment in the oiled cylindrical molds with two cylindrical plungers.(3) After cooling to room temperature, the modeled cylindrical specimens are removed from the mold, and the dimension and tolerance are controlled to (101.5  0.5) mm in diameter and (63.5  0.5) mm in height.

Experimental Equipment
Experiments were executed utilizing a WDW-100 model electronic universal testing machine, illustrated in Figure 2.This state-of-the-art apparatus, directed by an electronic computer program, is adept at conducting a spectrum of sophisticated stress and strain loading protocols, simultaneously capturing and archiving load and displacement data with temporal precision.The measurement precision for load and displacement are meticulously calibrated to 0.1 N and 1 µm, respectively.Adorned with an environmental temperature chamber, the machine ensures temperature regulation with an accuracy of 0.5 °C.It interfaces with an internal reverse loading apparatus through of its two levers.The test specimen is strategically positioned at the epicenter of the lower press plate of the reverse loading apparatus, where it is subjected to load via the upper press plate.To attenuate friction at the specimen-press plate interface, intercalated paper sheets imbued with lubricating oil are used.In preparation for loading, specimens are conditioned within the environmental chamber for a duration of 3 to 4 h to stabilize at the predetermined internal temperature.

Experiment Method
This study is designed to evaluate the strain of steel slag-asphalt mixtures under different stress conditions and temperatures to obtain curves that describe the relationship between strain and time.These curves facilitate the analysis of creep deformation in various scenarios, contributing to the experimental data to support the development and verification of a creep model.The axial deformation of the steel slag-asphalt mixture specimen, as shown in Figure 3, was recorded by an electronic universal testing machine.The strain of the steel slag-asphalt mixture specimen is determined by the application of the following formula: A series of creep recovery tests with gradual stress increments were conducted on cylindrical specimens using an electronic universal testing machine, equipped with a thermostatic chamber at temperatures of 25 °C, 35 °C, 45 °C, 55 °C, and 65 °C, as per the loading protocol shown in Table 2 and Figure 4.The experimental setup ensured the automatic collection of time and axial deformation metrics, with tests repeated three times at each temperature level to derive average values for comprehensive assessment.Three parallel tests were replicated at each temperature level to supply the stochastic samples.

Experimental Results and Data Processing
Following the earlier work of our team [27], this study hypothesizes that viscoelastic deformations fully recover, given sufficient recovery time.The starting times of each stress application were set as breakpoints, namely 0 s, 650 s, 1600 s, 2550 s, 3700 s, and 5250 s.After transformation and subtraction of the corresponding residual strain, six single creep recovery curves under different stresses were generated.Demonstrated by the incremental stress creep recovery trials conducted at 25 °C (as illustrated in Figure 5), the refined analysis provided a time-strain curve at 25 °C and 0.3 MPa, detailed in Figure 6.Applying consistent methodologies to data across various temperature conditions, a comprehensive dataset comprising 30 single-stress creep recovery curves at temperatures of 25 °C, 35 °C, 45 °C, 55 °C, and 65 °C, with stresses ranging from 0.05 MPa to 1.2 MPa was generated.The dataset included data points from five curves at 1.2 MPa, which were divided into the test set for this analysis.The data from the remaining 25 curves were divided into the training set, providing a robust foundation for the study's predictive modeling efforts.

Back Propagation Neural Network
The BP neural network represents a sophisticated multi-layer feedforward neural model suitable for mapping relationships between input and output data to facilitate tasks such as pattern recognition and data classification [28,29].This network enhances its learning through a process of error Back Propagation, methodically tuning parameters across its various layers.Including an input layer, multiple hidden layers, and an output layer, the learning dynamics of the BP neural network unfold across two pivotal stages: forward and backward propagation.Renowned for its straightforward design, ease of execution, and versatile adaptability, the BP neural network has gained extensive application across domains like function approximation, pattern recognition, and data classification [20,30].It is currently one of the most widely used neural networks.

Long Short-Term Memory Neural Network
Recurrent Neural Networks (RNNs) are utilized to capture the influence of past input data on present outputs, and to reveal the underlying temporal patterns in the data [23,24].The significant nonlinear viscoelastic and plastic behaviors of asphalt mixture mean that their creep responses are heavily influenced by the history of stress loading.Therefore, RNNs demonstrate superior performance in the development of complex models for the characterization of creep behavior of asphalt mixtures.
LSTM represents a sophisticated enhancement of the traditional RNN architecture.It introduces specialized "gate" mechanisms tasked with regulating the flow of information within the network's internal memory cells.These innovative gates allow for the selective preservation of crucial information and overcome the longstanding issue of long-term dependencies that plague standard RNNs [25,26].Illustrated in Figure 7, the LSTM neuron is characterized by three distinct gates.One is the forget gate, which purges extraneous data.One is the input gate, which safeguards essential information.The final is the output gate, which adjusts the output magnitude according to the neuron's present state.The mathematical expressions governing these gates are listed below. Forget Gate: where t f is the parameter for storing information,  is the activation function, t 1 h  is the output of the hidden layer from the previous moment, t x is the input, f W is the weight, and f b is the bias.The update of the memory cell is jointly determined by the forget gate and the input gate, as described by the following equation: In this equation, t C is the information stored in the memory cell, and t 1 C  is the information stored in the previous memory cell. Output Gate: where t o is the parameter for storing information,

Data Preprocessing
Stress, strain, time, and temperature each possess distinct dimensions and exhibit substantial differences in numerical magnitude.To ensure the predictive precision of the model and accelerate the convergence process, it becomes imperative to normalize the values of stress, strain, time, and temperature, ensuring they lie within the range from −1 to 1.This study adopts the min-max normalization method, which is formulated as follows: Due to the normalization processing applied to the data, it is essential to perform an anti-normalization process on the neural network's output values to obtain the strain values that have practical significance.The formula for the anti-normalization operation is as follows: where x is the experimental data, x  is the value after normalization, max x is the maximum value, and min x is the minimum value.

Dataset Division and K-Fold Cross-Validation
Due to the distinctive characteristics of the experimental data and the variations in the levels of applied stress, the dataset was meticulously categorized into six groups, which are outlined in Table 3. Groups 1 to 5 were designated as the training set, which was deployed for both training and assessing the model, accounting for 83.3% of the entire dataset.Group 6 was assigned as the test set, employed to evaluate the model's generalization capabilities, and constituted 16.7% of the total data.A K-fold cross-validation strategy was implemented to diminish the randomness in predictive outcomes (illustrated in Figure 8), with K set at five.This strategy involved training the model using four of the groups, while the fifth group was used for validation in each iteration.This process was repeated across five cross-validation cycles.The mean error from these evaluations was then calculated and used as the benchmark to gauge the model's efficacy.

Sliding Window Parameters for Data Preprocessing
Since the creep behavior of asphalt mixture is significantly influenced by the stress loading history, this study adopted a sliding window method to preprocess the stress data.This advanced preprocessing enhances the neural network's capability to discern the intricate relationship between the stress inputs and the resultant strain outputs.The schematic of the employed sliding window is shown in Figure 9, characterized by a length of eight and a step size of one.The data for a given moment, t, encompasses the current stress, and the stress from the preceding seven moments.These eight sequential stress values served as the input variables for predicting the corresponding strain at the same instant, t.  10, the configuration of the LSTM network was designed in accordance with the dimensions delineated by the sliding window mechanism.The input layer was equipped with ten neurons: eight dedicated to processing the stress values σ1 through σ8 via the sliding window, one representing time, t, and another representing temperature, T. The network contained two hidden layers, with 32 neurons for each one.The output layer comprised a single neuron to generate the LSTM's predicted outputs, which were subsequently normalized to compute the predicted strain values.

BP
From a mechanics-based viewpoint, the integral of stress over time illuminates the sustained impact that stress exerts on an object during a designated period.The structural design of the Back Propagation (BP) network struggled to describe the impact of past data inputs on current outputs.To overcome this limitation, this paper introduces a new data preprocessing method, outlined as follows: (1) Calculate the time integral of the stress data.
(2) Process these integrated data using the sliding window technique.
(3) Expand the input layer of the BP network by adding eight neurons to accommodate the processed integral data.Align the input configuration of the LSTM network with the input layer of the BP network.

Prediction Process
Figure 11 presents the creep prediction methodology for steel slag-asphalt mixture as implemented in this research.

Selection of Evaluation Metrics
In order to evaluate the model's prediction performance, the root-mean-square error (RMSE) and the mean absolute error (MAE) were utilized as validation metrics.Furthermore, the correlation coefficient (R) was employed to evaluate the degree of deviation between the predicted strain curves and the actual strain curves. Root-mean-square error (RMSE):  Mean absolute error (MAE): where n is the number of data points, i y is the actual strain value, y is the average value of the actual strain, ' i y is the predicted strain value, and ' y is the average value of the predicted strain.

Cross-Validation Results Discussion
The averages derived from five predictions were employed to evaluate the outcomes of cross-validation.In the BP neural network, the RMSE registers at 1.22 × 10 −3 , with the MAE at 1.12 × 10 −3 .For the LSTM network, these errors are measured at 7.29 × 10 −4 for RMSE and 6.51 × 10 −4 for MAE, respectively.In the training set, the strain amplitude during the creep phase is 1.11 × 10 −2 .The error of the predictions made by the BP neural network is roughly 10% of the strain amplitude, while the LSTM neural network's predictions have an error margin of approximately 6% of the strain amplitude.Consequently, the prediction results are deemed to be relatively accurate.The detailed error statistics are presented in Table 4.

Validation of Sliding Window Effectiveness
To substantiate the efficacy of integrating historical stress within the sliding window framework to augment the precision of predictive outcomes, a comparative analysis was conducted between two neural network forecasts: one incorporating historical stress, and the other excluding it.As illustrated in Figure 12, the neural network forecast that leveraged the sliding window approach exhibited a closer alignment with the experimental data.

Insights from the Test Set Predictions
Based on the abovementioned hyperparameters, the model underwent comprehensive training with the entire dataset from the training set.The strain in the test set was subsequently predicted.Figure 13 demonstrates the juxtaposition of experimental values with the predictions made by two distinct creep models, with Table 5 enumerating the evaluative metrics across different temperatures.Notably, the LSTM creep model surpasses the BP model in predictive efficacy, registering RMSE and MAE that are merely 45.26% and 45.09%, respectively, of the BP model's figures.Optimal predictive performance is noted at 25 °C, beyond which, as temperatures escalate, prediction errors incrementally rise.However, the LSTM model consistently correlates strongly with the actual strain trajectory, evidenced by an almost perfect average correlation coefficient of 0.991.Furthermore, it is noted that prediction errors increase as temperatures rise, a consequence of the relatively poor thermal stability of asphalt materials.The softened point of asphalt in this paper is about 46 °C.Beyond this temperature, the mechanical characteristics of the steel slag-asphalt mixture underwent significant transformations.

Constructing the 95% Confidence Interval
In the process of training the LSTM network, an analysis of the fitting errors from 25 creep recovery curves across 6800 data points was performed.The errors were approximately normally distributed, with an average of 4.42 × 10 −5 and a standard deviation of 3.67 × 10 −4 .Therefore, a 95% confidence interval was constructed.The calculations involved adjusting the mean by subtracting and adding twice the standard deviation, resulting in lower and upper limits of −6.91 × 10 −4 and 7.91 × 10 −4 , respectively.Thus, the 95% confidence interval was precisely defined as spanning from −6.91 × 10 −4 to 7.91 × 10 −4 .The statistical chart of strain error data is shown in Figure 14.The 95% confidence intervals were utilized for predictions of the test dataset in the LSTM.A range for the strain predictions during the creep recovery stages was determined by setting the boundaries for the predicted values.As seen in Figure 15, the majority of the experimental data were contained in the 95% confidence intervals.Therefore, the model presented in this study has significant robustness and reliability to predict the performance of road with steel slag-asphalt mixtures.

Conclusions
1.A novel LSTM model is proposed in this paper to characterize the creep recovery behavior of steel slag-asphalt mixtures.By using ANN technology, only the stress, time, and temperature data were trained.The pre-formed hypothesis of traditional mechanical modeling methods was avoided, and the application of the creep model was simplified.Employing a sliding window method, the stress data preprocess markedly boosted the model's efficacy in capturing the effects of historical stress loading on strain.Even more importantly, the precision of the proposed model's prediction was optimized again.2. Based on limited test data, both BP and LSTM models were validated to solve creep recovery problems relating to steel slag-asphalt mixtures.The prediction model accuracy was as follows: LSTM > BP, and in the BP neural network, the RMSE registers at 1.22 × 10 −3 , with the MAE at 1.12 × 10 −3 , in the LSTM network.These errors were measured at 7.29 × 10 −4 for RMSE and 6.51 × 10 −4 for MAE, respectively.3. The 95% confidence interval was formulated and applied to the test dataset.Compared with the experimental results, the LSTM creep model's robust predictive accuracy was evidenced again.4.However, due to the noticeable acceleration in the specimen's shear flow when it exceeded the softening point of the asphalt, the predicted deviation from experimental values enlarged.The range of the LSTM model is limited to below the softening point of the asphalt.Obtaining the effects of shear flow on the creep deformation of asphalt mixture will be necessary in further studies.
In this equation, t i and t C  are the parameters for storing information, i W and C W are the weights, and i b and C b are the biases.

oW
is the weight, o b is the bias, and t h is the output of the hidden layer.

Figure 14 .
Figure 14.Statistical chart of strain error data.

Figure 15 .
Figure 15.The LSTM prediction values and the upper and lower bounds of the 95% confidence interval for different temperatures in the test set: ((a) 25 °C (b) 35 °C (c) 45 °C (d) 55 °C (e) 65 °C).

Table 1 .
Composition of aggregate qualities in steel slag-asphalt mixture.

Table 2 .
Multiple-stress repeated creep recovery experimental plan.

Table 3 .
Division of experimental data.

Table 5 .
Evaluation indicators for the test set.