# Advanced Machine Learning Modeling Approach for Prediction of Compressive Strength of FRP Confined Concrete Using Multiphysics Genetic Expression Programming

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## Abstract

**:**

## 1. Introduction

## 2. Gene Expression Programming

## 3. Research Methods

#### Database Establishment and Division

## 4. Modeling Approach

#### 4.1. GEP Modelling

_{co}(MPa), and E (GPa), to build a prediction model for the f

_{cc}(MPa) of confined concrete columns through a reliable database. Dependably, GeneXpro Tools 5.0 is employed in this work to develop the GEP model.

_{cc}= A + B + C,

#### 4.2. Regression Models

^{2}is the coefficient of correlation, which is one of these. In fact, it cannot be enough to classify a model on its own because it does not forecast the outcome of a constant’s division or multiplication.

#### 4.2.1. Multiple Non-Linear Regression (MNLR)

#### 4.2.2. Multiple Linear Regression (MLR)

## 5. Results and Discussion

#### 5.1. Model Performance and Evaluation

_{i}and m

_{i}are the averages of the measured and estimated output for the ith indices, respectively. It is recommended that the model’s efficacy is tested on a variety of datasets to ascertain if it overfits. The R, RMSE, and MAE evaluation metrics could be used to accomplish this. Data sets with low levels of overfitting in the R, MAE, and RMSE on the train and test sections support this claim. [49,71,74]; however, as a way to showcase the GEP models’ capabilities, the anticipated versus actual values are shown in Figure 4, and it demonstrates the behavior among developed AI models against experimental output. It can be seen that at almost every data point, the GEP model effectively forecasts the corresponding output with maximum accuracy. Statistical metrics of the proposed model against actual values depict that the model possesses high generality with low discrepancy values when incorporating such a huge database. Moreover, it is demonstrated through Figure 4 that GEP-based f

_{cc}models which have high R and moderate RMSE and MAE indices are capable of predicting the output with adequate precision; however, near R

^{2}, RMSE, and MAE, values on the data sets indicate that it has both superior forecasting and heuristic capabilities, coupled with the fact that overfitting is nullified.

#### 5.2. External Validation

_{m}has also been developed by many scholars [49,73,74] to assess frameworks’ predictive power. The criterion is met if R

_{m}> 0.5. Both the squared coefficient of correlation (via the origin) among the forecasted and actual values (R

_{o}

^{2}), and the correlation among the actual and forecasted values (R

_{o}’

^{2}), must be near R

^{2}and 1. As shown in Table 2, o

_{i}and p

_{i}indicate the observed and anticipated results for the ith yield, accordingly, and n represents the total number of observations taken. To illustrate the validity of the validation criteria, Table 2 shows the findings produced by various approaches (AI and Regression models) for f

_{cc}(MPa).

_{cc}of CFRP confined concrete, indicates the resilience and competence of the generated models in conjunction with other well-known models, since the resulting models fulfill all of the requisite characteristics and dominate other forecasting frameworks significantly. It can be seen that some models did not even fulfill the requirements for external validation, as can be seen in the case of R

_{m}, which remains less than <0.5 for some models. In addition, it has been observed that merely using R values, or MAE and RMSE values, cannot even comprehend the model’s applicability and accuracy. These metrics significantly contribute towards the generalization and resilience of the model. As presented in the comparative graph, the GEP model provides a significantly improved performance, whereas other models even fail to fulfill some criteria for validation.

#### 5.3. Parametric Study

_{cc}for CFRP confined concrete has been assessed in this study; it is evident that f

_{cc}rises in accordance with f

_{co}, nt, h, and E, but it decreases as d increases. Parametric analysis is necessary to guarantee that the model outcomes are in agreement with actual outcomes, and to evaluate the resulting model from an engineering perspective. The capacity of analytical expression is to determine how well anticipated values coincide with the underlying computing behavior of the model [49,62,73].

_{cc}modeling to respective introductory variables with a certain variability. A predictor variable is changed at a time, whereas the rest of the parameters stay unchanged at the arithmetic mean in the process. Thus, the parametric analysis evaluation is done by creating a set of input data for each parameter based on its distribution in the database. The proposed model is provided with these values, and the f

_{cc}is determined. Each predictor variable’s behavior with regard to the model is then produced by repeating this process using another variable. There are several variables that influence the prediction models perspicuously, such as d, nt, E, and f

_{co}, except for h. Figure 7 shows the response of the f

_{cc}prediction models to the changes in these variables. These responses are shown to be in close agreement with the models developed previously in the literature by accurately anticipating the f

_{cc}based on their physical behavior.

#### 5.4. Sensitivity Analysis

_{max}and f

_{min}are the ith input variable’s maxima and minima, other input parameters would remain fixed at respective arithmetic mean, and n represents the number of input parameters. Results from this study’s sensitivity analysis are demonstrated by a pie chart, as shown in Figure 8, for the confinement model of CFRP based confined concrete.

_{cc}is mostly more responsive to f

_{co}, nt, and E

_{f}, than to other variables. These metrics signify that the model’s output is governed by the abovementioned parameters. In addition, the SI values for all other models are based on their dependency on the system parameters. SA is a useful tool for future research, including identifying the most vulnerable input factors for a realistic model’s employment, future trials, and the development of new models; however, it should be noted that each model is distinctive for simulation purposes, irrespective of how accurately it can forecast the target value. Thus, engineers must be intimately conversant with the sensitivity and significance of every parameter in the model they employ for simulation purposes. Thus, the GEP approaches for f

_{cc}are distinguished from other models in that all parameters are effectively implicated in forecasting the outcome, and therefore, the design result.

## 6. Conclusions

_{cc}, for CFRP based confined concrete columns, which are fundamental for the development and construction of concrete structures. As part of an extensive data analysis to date, a new GEP model has been developed. The latest formulation has been presented to forecast f

_{cc}, taking into account the most influential parameters, as per the theoretical and experimental literature. The developed GEP models were evaluated using a variety of statistical measures and other investigations. Considering substantial input variables while modeling the behavior, the model can effectively capture the results that are more accurate and closer to the actual response.

_{co}are 41%, which means that the GEP model is more responsive toward the strength of unconfined concrete than other parameters, which has a significant impact on the design results. Parametric analysis is also considered to analyze the GEP model from an engineering and materials perspective, and the findings were confirmed via laboratory studies in the literature. When it comes to an indirect assessment of the f

_{cc}of enclosed CFRP columns, the presented findings for assessment and validation show that GEP models outperformed previous frameworks in terms of scientific perspectives and performance capability when forecasting the target. In addition to further signifying the accuracy of the GEP model, a comparison is also drawn between the developed regression models and GEP. For this purpose, MNLR model statistics were evaluated in contrast to the developed AI model, and GEP dominates the former one in terms of accuracy statistics. This study shows that the GEP technique can be used as a dependable and robust replacement for conventional procedures for highly nonlinear and complex engineering issues.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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---|---|---|

1 | - ➢
- Chromosome number
| 120 |

2 | - ➢
- Gene number
| 3 |

3 | - ➢
- Size of head
| 8 |

4 | - ➢
- Genes’ linkage function
| Addition |

5 | - ➢
- Set of functions
| +, /, −, ×, 3√ |

Constants Configurations | ||

6 | - ➢
- Constants per gene
| 10 |

7 | - ➢
- Data type
| Floating |

8 | - ➢
- Bound range
| −10 to +10 |

GEP Operators | ||

9 | - ➢
- Mutation:
| 0.00138 |

10 | - ➢
- Function Insertion:
| 0.00206 |

11 | - ➢
- Permutation:
| 0.00476 |

12 | - ➢
- IS Transposition:
| 0.00548 |

13 | - ➢
- RIS Transposition:
| 0.00496 |

14 | - ➢
- Inversion:
| 0.00548 |

15 | - ➢
- Gene Transposition:
| 0.00157 |

16 | - ➢
- Random Chromosomes:
| 0.0026 |

17 | - ➢
- Constant Insertion:
| 0.00123 |

Recombination Rates | ||

18 | - ➢
- Uniform
| 0.00755 |

19 | - ➢
- One-Point
| 0.00277 |

20 | - ➢
- Two-Point
| 0.00189 |

21 | - ➢
- Gene
| 0.00277 |

Sr. #. | Equation | Range | Model | Output | Reference |
---|---|---|---|---|---|

1 | $R=\frac{{{\displaystyle \sum}}_{i-1}^{n}\left({e}_{i}-\overline{e}i\right)\left({m}_{i-}\overline{m}i\right)}{\sqrt{{{\displaystyle \sum}}_{i-1}^{n}\left({e}_{i}-\overline{e}i\right)\xb2{{\displaystyle \sum}}_{i-1}^{n}({m}_{i}-\overline{m}i)\xb2}}$ | R > 0.8 | GEP | 0.917 | |

MLR | 0.788 | ||||

MNLR | 0.856 | ||||

2 | ${R}_{m}={R}^{2}\times \left(1-\sqrt{\left|{R}^{2}-{R}_{o}^{2}\right|}\right)$ | ${R}_{m}>0.5$ | GEP | 0.528 | (Roy and Roy, 2008) [87] |

MLR | 0.244 | ||||

MNLR | 0.398 | ||||

where ${R}_{o}^{2}=1-\frac{{{\displaystyle \sum}}_{i=1}^{n}{\left(o-{p}_{i}^{o}\right)}^{2}}{{{\displaystyle \sum}}_{i=1}^{n}{\left({o}_{i}-{\underset{\_}{\overline{p}}}_{i}^{o}\right)}^{2}},{o}_{i}^{o}=k\times {p}_{i}$ | ${R}_{o}^{2}\cong 1$ | GEP | 0.980 | ||

MLR | 0.987 | ||||

MNLR | 0.977 | ||||

$R{\prime}_{o}^{2}=1-\frac{{{\displaystyle \sum}}_{i=1}^{n}{\left({o}_{i}-{p}_{i}^{o}\right)}^{2}}{{{\displaystyle \sum}}_{i=1}^{n}{\left({o}_{i}-{\underset{\_}{\overline{o}}}_{i}^{o}\right)}^{2}},{p}_{i}^{o}=k\prime \times {o}_{i}$ | $R{\prime}_{o}^{2}\cong 1$ | GEP | 0.998 | ||

MLR | 0.997 | ||||

MNLR | 0.988 | ||||

3 | $k=\frac{{{\displaystyle \sum}}_{i=1}^{n}\left({o}_{i}\times {p}_{i}\right)}{{{\displaystyle \sum}}_{i=1}^{n}{o}_{i}^{2}}$ | $0.85<k<1.15$ | GEP | 0.934 | (Golbraikh and Tropsha, 2002) [88] |

MLR | 0.965 | ||||

MNLR | 0.952 | ||||

4 | $k\prime =\frac{{{\displaystyle \sum}}_{i=1}^{n}\left({o}_{i}\times {p}_{i}\right)}{{{\displaystyle \sum}}_{i=1}^{n}{p}_{i}^{2}}$ | $0.85<k\prime <1.15$ | GEP | 1.019 | |

MLR | 0.980 | ||||

MNLR | 1.014 |

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## Share and Cite

**MDPI and ACS Style**

Ilyas, I.; Zafar, A.; Afzal, M.T.; Javed, M.F.; Alrowais, R.; Althoey, F.; Mohamed, A.M.; Mohamed, A.; Vatin, N.I.
Advanced Machine Learning Modeling Approach for Prediction of Compressive Strength of FRP Confined Concrete Using Multiphysics Genetic Expression Programming. *Polymers* **2022**, *14*, 1789.
https://doi.org/10.3390/polym14091789

**AMA Style**

Ilyas I, Zafar A, Afzal MT, Javed MF, Alrowais R, Althoey F, Mohamed AM, Mohamed A, Vatin NI.
Advanced Machine Learning Modeling Approach for Prediction of Compressive Strength of FRP Confined Concrete Using Multiphysics Genetic Expression Programming. *Polymers*. 2022; 14(9):1789.
https://doi.org/10.3390/polym14091789

**Chicago/Turabian Style**

Ilyas, Israr, Adeel Zafar, Muhammad Talal Afzal, Muhammad Faisal Javed, Raid Alrowais, Fadi Althoey, Abdeliazim Mustafa Mohamed, Abdullah Mohamed, and Nikolai Ivanovich Vatin.
2022. "Advanced Machine Learning Modeling Approach for Prediction of Compressive Strength of FRP Confined Concrete Using Multiphysics Genetic Expression Programming" *Polymers* 14, no. 9: 1789.
https://doi.org/10.3390/polym14091789