**1. Introduction**

The different types of bituminous mixtures (BMs) used for road pavements have to be properly designed as mixtures made of aggregates and bitumen, to withstand traffic loads and climate conditions. Unsuitable mechanical characteristics and volumetric properties of bituminous mixtures may lead to various types of distress in road pavements, generally comprising cracks due to fatigue or low temperature, permanent deformations, stripping, etc. Such failure modes decrease the service life of the pavement and represent serious safety issues for road users. As a result, it is important to properly characterize the mechanical behavior of mixes with respect to their composition to allow a performancebased optimization during the mix design phase [1–3]. Experimental methods, which require expensive laboratory tests and skilled technicians, are currently used to evaluate the bituminous mixtures' performance [4–9]. Consequently, any modification of the mixtures' composition, in terms of bitumen type or content, rather than of aggregate gradation, requires new laboratory tests with an increase in time and costs of the design process.

In recent years, many researchers have devoted their efforts to the problem of defining a mathematical or numerical model of BMs' mechanical behavior, which could quickly elaborate a reliable prediction of the bituminous mixture's response. To develop predictive equations, two main types of procedures can be used, namely, advanced constitutive modeling methods rather than statistical or data science approaches. Although the mechanistic

**Citation:** Miani, M.; Dunnhofer, M.; Rondinella, F.; Manthos, E.; Valentin, J.; Micheloni, C.; Baldo, N. Bituminous Mixtures Experimental Data Modeling Using a Hyperparameters-Optimized Machine Learning Approach. *Appl. Sci.* **2021**, *11*, 11710. https://doi.org/ 10.3390/app112411710

Academic Editor: Cesare Oliviero Rossi

Received: 10 November 2021 Accepted: 7 December 2021 Published: 9 December 2021

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

231

constitutive methods allow a rational and in-depth analysis of the material response to be performed [10–17], statistical or machine learning approaches are gaining considerable success in the academic community due to the good reliability of their predictions [18–21], even if they are not physically based. Nevertheless, it has also been reported that statistical regressions of experimental data can produce less accurate predictions than machine learning methods, specifically with regard to Artificial Neural Networks (ANNs) [22–27]. Recently, Lam et al. [28] have found that a multiple regression analysis (coefficient of determination, R2, equal to 0.790) is less reliable than an ANN approach (R<sup>2</sup> = 0.925) searching the analytical model to infer compressive strength of roller-compacted concrete pavement from steel slags aggregate and fly ash levels replacing cement.

The ANNs simulate simplified models of human brain: their computing power derive from the number of connections established between artificial neurons, fundamental computing units, and their main function is to map patterns between input and output from a representative experimental sample, mimicking the biological learning process. Such neural models are basically based on a nonlinear fitting approach, neither physical nor mechanistic, to correlate experimental data; the mathematical framework has already been widely discussed, e.g., by Baldo et al. [29]. In recent years, an increased number of researchers have used ANNs in many civil engineering applications, producing impressive results even with regards to the evaluation of road pavements' characteristics and performance. Tarefder et al. [30] developed a four-layer feed-forward neural network to correlate mix design parameters and BM samples performance in terms of permeability. Ozsahin and Oruc [31] constructed ANN-based models to determine the relationship between the resilient modulus of emulsified BMs and its affecting factors such as curing time, cement addition level, and residual bitumen content. Tapkın et al. [32] presented an application of ANN for the prediction of repeated creep test results for polypropylene-modified BMs. Accurate predictions (R<sup>2</sup> between 0.840 and 0.970) of the fatigue life of BMs under various loading and environmental conditions were also produced [33,34]. Ceylan et al. [35] discussed the accuracy and robustness of ANN-based models for estimating the dynamic modulus of hot mixes: such models exhibit significantly higher prediction accuracy (also at the input domain boundaries), less prediction bias and better understanding of the influences of temperature and mixture composition than their regression-based counterparts. Recently, Le et al. [36] developed an advanced hybrid model, as it is based both on ANNs and optimization technique, to accurately predict the dynamic modulus of Stone Mastic Asphalt (R<sup>2</sup> = 0.985); also, they use the proposed model to evaluate and discuss the effects of temperature and frequency on the mechanical parameter. Similarly, Ghorbani et al. [37] used a simple ANN approach for modeling experimental test results and examining the impact of different features on the properties of construction and demolition waste, such as the reclaimed asphalt pavement.

Although the documented research has attempted to introduce new approaches to an empirical–mechanical mix design, the Marshall approach is still widely adopted in many laboratories [38–44]. Tapkın et al. [45] have verified the possibility of applying ANNs for the prediction of Marshall test results of BMs. The proposed NN model uses the physical properties of standard Marshall specimens to predict the Marshall stability (MS), flow (MF) and Marshall Quotient values obtained at the end of mechanical tests. Ozgan [46] has studied the effects of varying temperatures and exposure times on the stability of BMs and modeled the test data by using a multilayer ANN. Conversely, Mirzahosseini et al. [23] have validated the efficiency of the multilayer perceptron ANNs for the assessment of the rutting potential of dense BMs: the flow number of Marshall specimens has been correlated to the aggregate and bitumen contents, percentage of voids in mineral aggregate, MS and MF. The mechanical characteristics of the bituminous mixtures depend on the volumetric properties as well as the bitumen content. Such parameters have to meet the limits, set by the current local specifications, for the pavement layer interested from the intervention. Nevertheless, voids in mineral aggregate, voids filled with bitumen and air voids (AV) are determined with a specific test (EN 12697-8), which requires additional

time and costs. Khuntia et al. [47] have proposed a neural network model that uses the quantities of bitumen and aggregate in Marshall specimens to predict the MS, MF value and AV obtained from the tests. Likewise, Zavrtanik et al. [48] have used ANNs to estimate air void content in several types of BMs produced according to EN 13108-1. Anyway, the literature presented has in common the need to provide the road engineer with an algorithm that can provide accurate predictions of empirical parameters related to BMs, without the need for sophisticated, time-consuming and expensive laboratory testing.

Despite the fact that ANNs have successfully provided predictive equations to quicken the empirical Marshall mix design, such computational models were usually based on a neural network structure set a priori by the research engineer and trained on a random subset of the available data sample. In case of a relatively small data set, such a practice may involve the risk of leaving out some relevant trends from the training set and leading to a variable prediction error, measured on the validation set, due to data sample variability and selected ANN architecture [49]. These issues can be avoided if an efficient model selection and appropriate data partition are performed. In particular, the search for the optimal network architecture, one of the most difficult tasks in ANN studies, consists of tuning the model settings, called hyperparameters, that yield the best performance score on the validation set. Applications of the trial-and-error procedure, as random or grid search, to find the optimal hyperparameters of a machine learning algorithm for a given predictive modeling problem can be found in the relevant literature [23,24]. Nevertheless, Baldo et al. [50,51] have highlighted the limits of such a time-consuming approach and applied a statistical technique of data partitioning, called k-fold Cross Validation, that allows a more accurate estimation of a model's performance.

An efficient hyperparameters tuning approach, in contrast to random or grid search, is the Bayesian optimization, which has become popular in recent years [52]. Given that evaluating the performance function score for different hyperparameters is extremely expensive, the Bayesian approach builds a probabilistic model, called "surrogate", mapping hyperparameters of past evaluations to a probability of a score on the performance function and uses such a model to find the next set of promising hyperparameters (i.e., that optimize the surrogate function) to evaluate the actual performance function [53,54].

This paper aimed to develop an autonomous and impartial procedure of neural model selection for predictive modeling problems of bituminous mixtures' mechanical behavior, using the Bayesian optimization method, that would replace the more costly trial-and-error procedure. In particular, the ANN approach was used to analyze stiffness modulus (ITSM), MS, MF and AV content of 320 Marshall specimens tested at the Highway Engineering Laboratory of the Thessaloniki Aristotle University. The experimental database includes different types of bitumen and aggregate and covers a wide range of bitumen contents and aggregate gradations. In addition, both laboratory- and plant-prepared mixtures were used and their production site was considered among the feature's variables of the proposed NN model; it correlates mechanical and volumetric properties, collected by means of laboratory tests, to fundamental characteristics of bituminous mixtures, such as bitumen content (% by weight of mix), filler-to-bitumen ratio (%), type of bitumen and aggregate as well as maximum nominal grain size.

The innovative aspect of the presented research is the application of state-of-the-art procedures in the machine learning domain (namely, k-fold Cross Validation and Bayesian optimization) that allow researchers and engineers to solve the problems of classical neural network applications in bituminous mixtures' behavior modeling. However, the procedure is not intended to replace the experimental method for mixture characterization, but to integrate it with a predictive algorithm that allows the road engineer to improve the mix design process, reducing time and operational costs. The major drawback of the proposed approach is that its proper implementation requires human resources with specific skills, such as machine learning expertise, and large training data sets covering the diversity of BMs materials.

### **2. Materials and Experimental Design**

Note that 320 Marshall specimens, having a diameter of 100 mm and an average thickness of 63.7 mm, were produced, both in laboratory and in plant, according to the impact compactor method of test EN 12697-30. These mixtures, designed as part of a research project carried out at the Aristotle University of Thessaloniki, were characterized by different contents of bitumen and aggregate gradations. Aggregates represent the lithic skeleton of a bituminous mixture, while the bitumen is the component binding the aggregate grains together.

### *2.1. Materials*

The aggregates employed were limestone- or diabase-type crushed stones with maximum nominal size of 20 mm or 12.5 mm: the calcareous sedimentary aggregate came from the same Greek quarry, while the mafic igneous one from three different local quarries. To control the physical properties of the aggregates, several tests were conducted. The obtained results are presented in Table 1.

**Table 1.** Aggregate properties.


The bituminous mixtures composed of aggregates with maximum nominal size of 12.5 mm (BM12.5) belong to the category "binder course", while the ones characterized by maximum nominal size of the aggregates equal to 20 mm (BM20) to the "base course" category. In this research, 27 aggregate gradations were considered to meet the gradation limits for binder and base course, set by the current Greek specifications. In each mix category, there are various types of compositions related to the aggregate's maximum nominal size: for lab-prepared mixtures, 4 types of gradations were used to fit the limits for BM12.5 and 4 types for the BM20. The remaining ones concern BM20 mixtures prepared in plant and correspond to different production days. Figures 1–3 show the grading curves involved.

**Figure 1.** Gradation curves of lab-prepared BM12.5.

**Figure 2.** Gradation curves of lab-prepared BM20.

**Figure 3.** Gradation curves of plant-prepared BM20 (gradations of different production dates).

The standard 50/70 penetration bitumen was used in the preparation of 129 Marshall specimens, while the remaining 191 were produced utilizing a bitumen modified in the laboratory with styrene–butadiene–styrene polymers (SBS). The two types of bitumen were tested to ensure that their physical properties were compliant with specific acceptance requisites. The characteristics of bituminous binders are reported in Table 2. No aging process was performed on bituminous mixtures.



### *2.2. Experimental Design*

The Marshall samples were produced with a bitumen percentage between 3.8% and 6.0% (by weight of mix), and in number equal to three for each bitumen content adopted. Table 3 summarizes the number of specimens produced for each combination of bitumen and aggregate; abbreviations coding the mixtures are also reported.

**Table 3.** Number and codes of Marshall specimens.


Among the mechanical parameters of bituminous mixtures, the ITSM allows a rational performance-based characterization of the mixes to be performed [4,55]. Therefore, the ITSM test (Figure 4) was executed on all BM samples using the standard testing conditions, defined by EN 12697-26 (temperature of 20 ◦C, target deformation fixed at 5 μm, and rise time equal to 124 ms). Subsequently, considering that the Marshall parameters are still widely used in road pavement design [38–45], MS and MF were evaluated for the bituminous mixtures produced, according to EN 12697-34. Finally, the specimens' volumetric characteristics have been determined applying EN 12697-8. The test results are reported in Table S1; such experimental data have been already discussed in previous papers [29,50]. Table 4 shows some statistical information (minimum and maximum values along with the mean value and its standard deviation) about mechanical characteristics and volumetric property of the BMs.

**Table 4.** Statistical information about Marshall specimens.



**Table 4.** *Cont.*

**Figure 4.** ITSM test setup.
