# Bituminous Mixtures Experimental Data Modeling Using a Hyperparameters-Optimized Machine Learning Approach

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

^{2}, equal to 0.790) is less reliable than an ANN approach (R

^{2}= 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.

^{2}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

^{2}= 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.

## 2. Materials and Experimental Design

#### 2.1. Materials

#### 2.2. Experimental Design

## 3. Methodology

#### 3.1. Artificial Neural Networks

#### 3.2. ANN Optimization

^{®}ANN Toolbox [22,23,29,45,46,50].

#### 3.3. ANN Regularization

#### 3.4. k-Fold Cross Validation

#### 3.5. Bayesian Hyperparameters Optimization

- ${X}_{L}=\left\{1,\dots ,5\right\}$, for the number of hidden network layers L;
- ${X}_{N}=\left\{4,\dots ,64\right\}$, for the number of neurons $N$ for each hidden layer;
- ${X}_{act}=\left\{tanh,ReLU,ELU\right\}$ for the set of activation functions to be applied after each hidden layer;
- ${X}_{\alpha}=\left[{10}^{-6},{10}^{-2}\right]$ for the learning rate $\alpha $;
- ${X}_{\beta}=\left[{10}^{-6},{10}^{-2}\right]$ for the weight decay parameter $\beta $;
- ${X}_{E}=\left\{500,\dots ,5000\right\}$ for the number of learning process iterations.

#### 3.6. Implementation Details

## 4. Results and Discussion

_{CV}= 0.249 and then the maximum value of the Pearson coefficient R

_{CV}= 0.868) was found at iteration 54 (Figure 8). The hyperparameters discovered by the BO algorithm defined an ANN with $L$ = 5 layers, $N$ = 37 neurons, and hyperbolic tangent activation function ($tanh$), that was trained for $E$ = 3552 iterations, with a learning rate of $\alpha $ = 0.01 and weight decay $\beta $ = $1\xb7{10}^{-6}$. Table 5 shows the validation MSE (second column) and the R-score of the optimal model for each of the 5 folds (last column). In addition, the final average results for each mechanical characteristic and volumetric property are reported in the last row of Table 5. Figure 9 shows the relation between network output and experimental target for each fold.

_{4}= 0.829) or better (R

_{3}= 0.906) than the most likely situation represented by the k-fold CV (R

_{CV}= 0.868). Random and grid searches based on the prediction error by a fixed split training test may find solutions that are not optimal [49], due to fluctuations resulting from considering one partition rather than another. Figure 10 shows the comparison between experimental targets and predicted outputs, for the four parameters analyzed, as regards fold 4. Values calculated by the ANN model characterized by the highest prediction error (MSE

_{mean}= 0.346, Table 5) are very close in value to the experimental data, whatever variable is considered. This result is relevant from an engineering point of view, because it proves that ANNs can be an accurate method to model (even simultaneously) the mechanical response and physical properties of bituminous mixtures, also very different in terms of composition.

^{®}ANN Toolbox), as the above reported results suggest.

## 5. Conclusions

- To perform proper neural modeling, the evaluation of the several network structures resulting from the selection of different model hyperparameters values is required. The procedure developed in this article allowed the limitations of the most widely used ANN toolbox to be overcome.
- The proposed approach with the k-fold CV produces more reliable results in terms of model validation error, with respect to the standard grid search based on a random data set partition: in fact, if the procedure were based on a fixed random split of the available data set, different results are possible, worse (R
_{4}= 0.829) or better (R_{3}= 0.906) than the most likely situation represented by the k-fold CV (R_{CV}= 0.868), due to the different distribution of the training and validation data. - The BO algorithm has shown to be successful in solving the challenging problem of properly setting the model hyperparameters: it has identified the optimal solution, in terms of algorithmic and structural configuration of the ANN, in only 54 iterations. The hallmark of such a technique lies in the ability to take past evaluations into account so as to limit the loss function recalls. Nonetheless, the reader should be aware that the BO procedure results may be linked to the constraints set by the research engineer in terms of hyperparameters’ variability.
- In the current paper, Marshall parameters, ITSM, as well as AV content have been determined simultaneously by a single multi-output ANN, unlike previous studies; therefore, such approach represents an integrated predictive model of the selected mechanical and volumetric properties.
- The neural network structure best suited (MSE
_{CV}= 0.249, R_{CV}= 0.868) to model experimental mixtures data is defined by 5 layers, 37 neurons in each hidden layer and $tanh$ transfer function. A learning step size $\alpha $ equal to 0.01 and weight decay $\beta $ equal to $1\times {10}^{-6}$ are implemented in the Ranger training algorithm. - The algorithms applied and the analytical steps taken by the artificial networks have been illustrated in detail to make the procedure followed replicable to the reader. If it is desired to put the proposed model into service for the designed application (e.g., use in a laboratory or plant for estimates of mechanical parameters and volumetric properties of bituminous mixtures), then the optimized ANN must be trained with all available data.

## Supplementary Materials

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 5.**Activation functions: exponential linear (ELU), hyperbolic tangent (TanH), rectified linear (ReLU).

**Figure 7.**Step-by-step procedure followed in this study: it starts with the mix design (left side) and testing processes (bottom side); these tests define the set of target variables (lower right side); the input-target fitting is performed by a neural network, whose structure and algorithmic functioning are searched by the Bayesian optimizer (upper side) comparing network outputs and experimental targets (right side).

Property | Aggregate Type | |
---|---|---|

Limestone | Diabase | |

Los Angeles coeff. (%), EN 1097-2 | 29 | 25 |

Polished stone value (%), EN 1097-8 | - | 55 to 60 |

Flakiness index (%), EN 933-3 | 23 | 18 |

Sand equivalent (%), EN 933-8 | 79 | 59 |

Methylene blue v. (mg/g), EN 933-9 | 3.3 | 8.3 |

Property | Bitumen Type | |
---|---|---|

50/70 | Modified | |

Penetration (0.1 × mm), EN1426 | 64 | 45 |

Softening point (°C), EN1427 | 45.6 | 78.8 |

Elastic recovery (%), EN 13398 | - | 97.5 |

Fraas breaking point (°C), EN 12593 | −7.0 | −15.0 |

Maximum Size (mm) | Aggregate Type | Bitumen Type | Production Site | Mixture ID | Specimens |
---|---|---|---|---|---|

12.5 | Limestone | 50/70 | Laboratory | M1 | 30 |

12.5 | Limestone | Modified | Laboratory | M2 | 30 |

12.5 | Diabase | 50/70 | Laboratory | M3 | 30 |

12.5 | Diabase | Modified | Laboratory | M4 | 30 |

20 | Limestone | 50/70 | Laboratory | M5 | 39 |

20 | Limestone | Modified | Laboratory | M6 | 30 |

20 | Limestone | Modified | Plant | M7 | 41 |

20 | Diabase | 50/70 | Laboratory | M8 | 30 |

20 | Diabase | Modified | Laboratory | M9 | 30 |

20 | Diabase | Modified | Plant | M10 | 30 |

Mixture ID | Parameter | Minimum Value | Maximum Value | Mean Value | Standard Deviation |
---|---|---|---|---|---|

M1 | ITSM (MPa) | 3756 | 5554 | 4556.43 | 567.93 |

MS (kN) | 7.71 | 12.17 | 9.93 | 1.09 | |

MF (mm) | 1.99 | 4.70 | 3.18 | 0.84 | |

AV (%) | 1.77 | 6.37 | 3.99 | 1.37 | |

M2 | ITSM (MPa) | 3628 | 5142 | 4345.50 | 486.50 |

MS (kN) | 8.74 | 14.00 | 11.35 | 1.73 | |

MF (mm) | 2.00 | 4.20 | 3.20 | 0.57 | |

AV (%) | 2.20 | 6.29 | 4.18 | 1.23 | |

M3 | ITSM (MPa) | 3812 | 5942 | 4804.10 | 725.03 |

MS (kN) | 10.30 | 15.20 | 12.88 | 1.53 | |

MF (mm) | 2.00 | 5.00 | 3.35 | 0.95 | |

AV (%) | 1.49 | 8.91 | 5.22 | 2.38 | |

M4 | ITSM (MPa) | 4035 | 6293 | 5076.17 | 759.06 |

MS (kN) | 11.60 | 16.43 | 13.62 | 1.42 | |

MF (mm) | 2.20 | 5.00 | 3.40 | 0.92 | |

AV (%) | 1.33 | 8.36 | 5.05 | 2.18 | |

M5 | ITSM (MPa) | 3215 | 4919 | 4252.26 | 502.24 |

MS (kN) | 8.91 | 14.86 | 11.37 | 1.51 | |

MF (mm) | 2.18 | 4.60 | 3.15 | 0.50 | |

AV (%) | 2.17 | 6.75 | 4.28 | 1.16 | |

M6 | ITSM (MPa) | 3907 | 6043 | 5243.10 | 538.97 |

MS (kN) | 10.40 | 13.99 | 11.81 | 1.21 | |

MF (mm) | 2.24 | 4.16 | 3.24 | 0.40 | |

AV (%) | 1.68 | 5.21 | 3.49 | 1.08 | |

M7 | ITSM (MPa) | 3103 | 6399 | 5065.34 | 906.93 |

MS (kN) | 6.60 | 14.75 | 9.86 | 2.20 | |

MF (mm) | 2.10 | 9.86 | 3.22 | 0.62 | |

AV (%) | 3.03 | 2.20 | 5.22 | 1.19 | |

M8 | ITSM (MPa) | 2304 | 4900 | 3829.63 | 783.23 |

MS (kN) | 10.45 | 15.48 | 13.05 | 1.36 | |

MF (mm) | 2.20 | 5.00 | 3.37 | 0.83 | |

AV (%) | 0.35 | 8.44 | 4.37 | 2.43 | |

M9 | ITSM (MPa) | 2930 | 5994 | 4911.30 | 851.15 |

MS (kN) | 8.92 | 15.48 | 12.22 | 2.03 | |

MF (mm) | 1.85 | 5.00 | 3.10 | 0.72 | |

AV (%) | 1.26 | 8.44 | 4.97 | 1.86 | |

M10 | ITSM (MPa) | 4049 | 5968 | 5309.27 | 565.88 |

MS (kN) | 7.80 | 16.55 | 11.98 | 2.15 | |

MF (mm) | 2.70 | 5.40 | 3.95 | 0.76 | |

AV (%) | 4.60 | 9.70 | 7.12 | 1.59 |

Fold | MSE_{mean} | R—Pearson Correlation Coefficient | R_{mean} | |||
---|---|---|---|---|---|---|

ITSM | MS | MF | AV | |||

0 | 0.219 | 0.837 | 0.866 | 0.842 | 0.964 | 0.877 |

1 | 0.203 | 0.963 | 0.835 | 0.725 | 0.973 | 0.874 |

2 | 0.254 | 0.872 | 0.836 | 0.799 | 0.917 | 0.856 |

3 | 0.223 | 0.918 | 0.826 | 0.917 | 0.956 | 0.905 |

4 | 0.346 | 0.841 | 0.731 | 0.834 | 0.912 | 0.829 |

CV_{result} | 0.249 | 0.886 | 0.819 | 0.824 | 0.944 | 0.868 |

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**MDPI and ACS Style**

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

**AMA Style**

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. *Applied Sciences*. 2021; 11(24):11710.
https://doi.org/10.3390/app112411710

**Chicago/Turabian Style**

Miani, Matteo, Matteo Dunnhofer, Fabio Rondinella, Evangelos Manthos, Jan Valentin, Christian Micheloni, and Nicola Baldo.
2021. "Bituminous Mixtures Experimental Data Modeling Using a Hyperparameters-Optimized Machine Learning Approach" *Applied Sciences* 11, no. 24: 11710.
https://doi.org/10.3390/app112411710