*3.3. Prediction Capability of the FNN–IWO Model*

In this section, the performance of FNN–IWO in predicting the P<sup>u</sup> of CFST is investigated. The predicted outputs versus the corresponding experimental results associated with the training, testing, and all datasets are presented in Figure 8. The fitted linear lines are also plotted (red lines) in each graph to show the performance of the algorithm. R<sup>2</sup> values with respect to the training, testing, and all datasets were estimated at 0.978, 0.979, and 0.978, respectively, showing an excellent prediction capability of FNN–IWO. Furthermore, three linear equations representing the relationships between actual and predicted data were also given in each graph, including the intercepts and slopes. It is observed that the FNN–IWO algorithm possessed a strong linear correlation between actual and predicted P<sup>u</sup> values.

The detailed performance of the proposed FNN–IWO algorithm is summarized in Table 5, including R<sup>2</sup> , RMSE, MAE, standard deviation error (ErrorStD), slope, and slope angle. Regarding the results of quality assessment and error analysis, FNN–IWO exhibited a strong capability in predicting the critical compression capacity of the rectangular section. *Materials* **2020**, *13*, x FOR PEER REVIEW 15 of 25

**Figure 8.** Comparison between actual and predicted data in regression scatter mode for (**a**) training data, (**b**) testing data, and (**c**) all data. **Figure 8.** Comparison between actual and predicted data in regression scatter mode for (**a**) training data, (**b**) testing data, and (**c**) all data.


**Table 5.** Performance indicators of the optimal FNN–IWO model. **Table 5.** Performance indicators of the optimal FNN–IWO model.

For further assessment of the performance of the FNN–IWO algorithm, comparison between the experimental and predicted results was performed at different quantile levels. For this purpose, quantiles from 10% to 90% were computed to track the behavior of the distribution of the data, with a focus on the most important statistical distribution. The results are presented (Figure 9a–c) for the training, testing, and all data, respectively, whereas the percentage of error (%) between the predicted and actual values at each quantile level is displayed in Figure 10. It is seen that, for the training dataset, the actual and predicted data were highly correlated, For further assessment of the performance of the FNN–IWO algorithm, comparison between the experimental and predicted results was performed at different quantile levels. For this purpose, quantiles from 10% to 90% were computed to track the behavior of the distribution of the data, with a focus on the most important statistical distribution. The results are presented (Figure 9a–c) for the training, testing, and all data, respectively, whereas the percentage of error (%) between the predicted and actual values at each quantile level is displayed in Figure 10.

whereas a small difference was observed at each level of quantile for the testing part. With respect to the whole dataset, the highest error ratio was observed at Q80, followed by Q90 and Q10. For the values of error, it was seen that the FNN–IWO model exhibited a strong efficiency in predicting P<sup>u</sup> within the Q10–Q70 range (error < 5%) and from Q80 to Q90 (with error in the 5%–10% range). It is seen that, for the training dataset, the actual and predicted data were highly correlated, whereas a small difference was observed at each level of quantile for the testing part. With respect to the whole dataset, the highest error ratio was observed at Q80, followed by Q90 and Q10. For the values of error, it was seen that the FNN–IWO model exhibited a strong efficiency in predicting P<sup>u</sup> within the Q10–Q70 range (error < 5%) and from Q80 to Q90 (with error in the 5%–10% range).

columns.

data, (**b**) testing data, and (**c**) all data.

and actual values at each quantile level is displayed in Figure 10.

within the Q10–Q70 range (error < 5%) and from Q80 to Q90 (with error in the 5%–10% range).

**Figure 8.** Comparison between actual and predicted data in regression scatter mode for (**a**) training

**Table 5.** Performance indicators of the optimal FNN–IWO model. **Indicator R<sup>2</sup> RMSE MAE ErrorStD Slope Slope Angle** Training part 0.978 0.039 0.024 0.039 0.976 44.296° Testing part 0.979 0.045 0.036 0.042 0.966 44.015° All data 0.978 0.042 0.029 0.041 0.969 44.101°

For further assessment of the performance of the FNN–IWO algorithm, comparison between the experimental and predicted results was performed at different quantile levels. For this purpose, quantiles from 10% to 90% were computed to track the behavior of the distribution of the data, with a focus on the most important statistical distribution. The results are presented (Figure 9a–c) for the training, testing, and all data, respectively, whereas the percentage of error (%) between the predicted

It is seen that, for the training dataset, the actual and predicted data were highly correlated, whereas a small difference was observed at each level of quantile for the testing part. With respect to the whole dataset, the highest error ratio was observed at Q80, followed by Q90 and Q10. For the

**Figure 9.** Comparison between actual and predicted data at different quantile levels of the distributions for (**a**) training data, (**b**) testing data, and (**c**) all data. distributions for (**a**) training data, (**b**) testing data, and (**c**) all data.

**Figure 10.** Percentage of error between quantile estimation for training, testing, and all data. **Figure 10.** Percentage of error between quantile estimation for training, testing, and all data.

#### *3.4. Prediction Accuracy in Function of Structural Parameters of FNN–IWO 3.4. Prediction Accuracy in Function of Structural Parameters of FNN–IWO*

*3.5. Comparison of the Hybrid Model of FNN–IWO and the Single FNN Model*

conjugate gradient (SCG)), FNN architecture, and dataset.

In this section, the prediction accuracy of FNN–IWO with respect to different ranges of structural parameters is presented. The actual and predicted P<sup>u</sup> in function of the depth /width ratio, t, fy, fc', and slenderness ratio are displayed in Figure 11a–e, respectively. Besides, error analysis in terms of R2 , RMSE, and MAE for several intervals of the depth/width ratio, t, fy, fc', and slenderness ratio, respectively, is also indicated in Table 6 and Figure 11, together with the associated number of data. In this section, the prediction accuracy of FNN–IWO with respect to different ranges of structural parameters is presented. The actual and predicted P<sup>u</sup> in function of the depth /width ratio, t, fy, fc', and slenderness ratio are displayed in Figure 11a–e, respectively. Besides, error analysis in terms of R 2 , RMSE, and MAE for several intervals of the depth/width ratio, t, fy, fc', and slenderness ratio, respectively, is also indicated in Table 6 and Figure 11, together with the associated number of data.

In the case of the depth/width ratio, 11 configurations were found between 1 and 1.2, exhibiting R<sup>2</sup> = 0.98, RMSE = 137.57 kN, and MAE = 95.25 kN; 22 configurations were found between 1.2 and 1.4, showing R2 = 0.98, RMSE = 71.07 kN, and MAE = 56.01 kN; 43 configurations were found between 1.4 and 1.6, exhibiting R<sup>2</sup> = 0.97, RMSE = 144.71 kN, and MAE = 109.65 kN; 11 configurations were found between 1.6 and 1.8, exhibiting R<sup>2</sup> = 0.89, RMSE = 56.16 kN, and MAE = 38.91 kN; and only 3 configurations were found between 1.8 and 2, exhibiting R<sup>2</sup> = 1.00, RMSE = 24.75 kN, and MAE = 21.81 kN. Such an analysis allowed confirming that the FNN–IWO model is efficient in predicting P<sup>u</sup> from nearly square to highly rectangular columns. In the case of the depth/width ratio, 11 configurations were found between 1 and 1.2, exhibiting R <sup>2</sup> = 0.98, RMSE = 137.57 kN, and MAE = 95.25 kN; 22 configurations were found between 1.2 and 1.4, showing R<sup>2</sup> = 0.98, RMSE = 71.07 kN, and MAE = 56.01 kN; 43 configurations were found between 1.4 and 1.6, exhibiting R<sup>2</sup> = 0.97, RMSE = 144.71 kN, and MAE = 109.65 kN; 11 configurations were found between 1.6 and 1.8, exhibiting R<sup>2</sup> = 0.89, RMSE = 56.16 kN, and MAE = 38.91 kN; and only 3 configurations were found between 1.8 and 2, exhibiting R<sup>2</sup> = 1.00, RMSE = 24.75 kN, and MAE = 21.81 kN. Such an analysis allowed confirming that the FNN–IWO model is efficient in predicting P<sup>u</sup> from nearly square to highly rectangular columns.

In the case of slenderness, 78 configurations were found between 0 and 20 of slenderness, exhibiting R<sup>2</sup> = 0.98, RMSE = 123.29 kN, and MAE = 86.64 kN; 6 configurations were found between 20 and 40 of slenderness, showing R<sup>2</sup> = 0.98, RMSE = 42.80 kN, and MAE = 32.09 kN; 13 configurations were found between 40 and 60 of slenderness, exhibiting R<sup>2</sup> = 0.99, RMSE = 72.25 kN, and MAE = 55.32 kN. Although the number of data is small for large slenderness, such an analysis allowed

In order to highlight the efficiency of the evolutionary IWO algorithm, comparisons between FNN–IWO and the individual FNN were performed, using a similar training algorithm (scaled

Considering RMSE, MAE, and standard deviation error (ErrorStD), Figure 12 identifies the values of the two algorithms for the training part (Figure 12a) and testing part (Figure 12b). It can be clearly seen that FNN–IWO is more accurate than the single FNN, represented by a reduction of error for RMSE (2 times), MAE (3 times), or ErrorStD (2 times). Improvement of the accuracy is more pronounced in the training part than the testing part. Considering R<sup>2</sup> and slope as error criteria, FNN–

In the case of slenderness, 78 configurations were found between 0 and 20 of slenderness, exhibiting R <sup>2</sup> = 0.98, RMSE = 123.29 kN, and MAE = 86.64 kN; 6 configurations were found between 20 and 40 of slenderness, showing R<sup>2</sup> = 0.98, RMSE = 42.80 kN, and MAE = 32.09 kN; 13 configurations were found between 40 and 60 of slenderness, exhibiting R<sup>2</sup> = 0.99, RMSE = 72.25 kN, and MAE = 55.32 kN. Although the number of data is small for large slenderness, such an analysis allowed remarking that the FNN–IWO model is efficient in predicting P<sup>u</sup> for short, medium, and long columns. IWO also exhibited an advantage compared with FNN without optimization, for both the training and testing datasets (Figure 11c,d). For the sake of comparison, Table 7 indicates the exact values and gains (in %) while using FNN– IWO with FNN for five error criteria. With a focus on the testing part, the gains reached 47.9%, 49.2%, 41.3%, 6.5%, and 1.5% for RMSE, MAE, ErrorStD, R<sup>2</sup> , and slope, respectively. As a conclusion, using IWO to tune the weights and bias of FNN strongly enhanced the accuracy in predicting Pu.

*Materials* **2020**, *13*, x FOR PEER REVIEW 17 of 25

**Figure 11.** Evaluation of axial capacity in function of the (**a**) depth/width ratio, (**b**) thickness, (**c**) yield stress, (**d**) compressive strength, and (**e**) slenderness ratio. **Figure 11.** Evaluation of axial capacity in function of the (**a**) depth/width ratio, (**b**) thickness, (**c**) yield stress, (**d**) compressive strength, and (**e**) slenderness ratio.


**Table 6.** Error analysis of prediction performance with respect to different ranges of values of structural variables.

#### *3.5. Comparison of the Hybrid Model of FNN–IWO and the Single FNN Model*

In order to highlight the efficiency of the evolutionary IWO algorithm, comparisons between FNN–IWO and the individual FNN were performed, using a similar training algorithm (scaled conjugate gradient (SCG)), FNN architecture, and dataset.

Considering RMSE, MAE, and standard deviation error (ErrorStD), Figure 12 identifies the values of the two algorithms for the training part (Figure 12a) and testing part (Figure 12b). It can be clearly seen that FNN–IWO is more accurate than the single FNN, represented by a reduction of error for RMSE (2 times), MAE (3 times), or ErrorStD (2 times). Improvement of the accuracy is more pronounced in the training part than the testing part. Considering R<sup>2</sup> and slope as error criteria, FNN–IWO also exhibited an advantage compared with FNN without optimization, for both the training and testing datasets (Figure 11c,d).

For the sake of comparison, Table 7 indicates the exact values and gains (in %) while using FNN–IWO with FNN for five error criteria. With a focus on the testing part, the gains reached 47.9%, 49.2%, 41.3%, 6.5%, and 1.5% for RMSE, MAE, ErrorStD, R<sup>2</sup> , and slope, respectively. As a conclusion, using IWO to tune the weights and bias of FNN strongly enhanced the accuracy in predicting Pu.

structural variables.

Depth/width ratio (-)

Thickness of steel tube (mm)

Yield stress of steel (MPa)

Compressive strength of concrete (MPa)

**Table 6.** Error analysis of prediction performance with respect to different ranges of values of

**Structural Parameter Lower Bound Upper Bound Number of Data R<sup>2</sup> RMSE (kN) MAE (kN)**

1 1.2 11 0.98 137.57 95.25 1.2 1.4 22 0.98 71.07 56.01 1.4 1.6 43 0.97 144.71 109.65 1.6 1.8 11 0.89 56.16 38.91 1.8 2 3 1.00 24.75 21.81

 2 4 0.91 87.07 70.74 4 52 0.91 118.58 80.36 6 29 0.97 85.70 61.60 8 8 0.91 178.54 143.84 10 6 0.89 92.27 72.82

 260 26 0.97 64.45 50.11 320 6 0.97 146.68 99.40 380 50 0.91 129.64 92.09 440 8 0.99 137.34 104.47 515 9 0.99 76.28 55.16

 20 25 0.90 157.25 113.71 30 22 0.93 120.57 84.45 40 24 0.99 87.44 67.86 50 28 0.99 75.40 53.80

 20 78 0.98 123.29 86.64 40 6 0.98 42.80 32.09 60 13 0.99 72.25 55.32 80 1 - 116.74 116.74 100 1 - 49.65 49.65

**Figure 12.** Comparison of performance indicators between the individual FNN and FNN–IWO model: (**a**) RMSE, MAE, and ErrorStD for training data; (**b**) RMSE, MAE, and ErrorStD for testing data; (**c**) R<sup>2</sup> and slope for training data; and (**d**) R<sup>2</sup> and slope for testing data.


**Table 7.** Comparison of performance indicators between FNN–IWO and individual FNN.

#### **4. Conclusions and Outlook**

Even though many studies attempted to predict the P<sup>u</sup> of CFST with different AI algorithms, the accuracy and robustness of these algorithms still need further comprehensive investigation. In this study, a novel hybrid approach of FNN–IWO was proposed and improved for the prediction of P<sup>u</sup> of CFST, of which IWO was used for tuning and optimizing the FNN weights and biases to improve the prediction performance.

The results showed that the FNN–IWO algorithm is an excellent predictor of Pu, with a value of R <sup>2</sup> of up to 0.979. The performance of FNN–IWO in predicting P<sup>u</sup> function of structural parameters such as depth/width ratio, thickness of steel tube, yield stress of steel, concrete compressive strength, and slenderness ratio was investigated and the results showed that FNN–IWO is efficient in predicting P<sup>u</sup> from nearly square to highly rectangular columns, as well as for short, medium, and long columns. Better performance of FNN–IWO was also pointed out with the gains in accuracy of 47.9%, 49.2%, and 6.5% for RMSE, MAE, and R<sup>2</sup> , respectively, compared with the simulation using the single FNN. This study may help in quick and accurate prediction of P<sup>u</sup> of CFST for better practice purposes.

In general, the main advantage of AI-based methods is its efficient capability to model the macroscopic mechanical behavior of the structural members without any prior assumptions or constraints. Therefore, the developed AI model in this study could be applied to the pre-design phase of the design process. Indeed, such quick numerical estimation is helpful to explore some initial evaluations of the outcome before conducting any extensive laboratory experiments. To this aim, a graphical user interface application should be compiled for facilitating the application by engineers/researchers.

On the other hand, empirical formulae should be derived based on the "black-box" AI-based model developed in this study for estimating the axial behavior of rectangular CFST columns. In addition, the performance of such empirical formulae should be compared with other existing equations in the literature such as Ding et al. [98], Wang et al. [125], and Han et al. [126]. Besides, numerical finite element scheme should also be studied, especially for investigating the mechanical behaviors of composite columns at both the micro and macro levels. Finally, improvement for current designs (such as Eurocode-4 [127], AISC [128], and ACI [129]), if it exists, should be proposed.

The axial behavior of CFST composite columns is a complex problem, involving various variables such as geometry and mechanical properties of constituent materials. Consequently, experimental databases are crucial for studying this problem. In further studies, a larger database should be considered, in order to cover more material strengths and geometric dimension ranges.

The methodology modeling of this work could be extended for predicting other macroscopic properties such as bending, compression, or tension strength of not only composite members, but also members made of a single material (i.e., concrete or steel members). Besides, an investigation based on homogenization and de-homogenization approaches [130–134] could also be useful for studying structural members under different boundary conditions and loadings. Such a framework, including the finite element scheme, could also be coupled with AI-based prediction in order to better understand the micro and macro behaviors of structural members.

**Author Contributions:** Conceptualization, H.Q.N., T.-T.L. and H.-B.L.; methodology, H.-B.L., T.-T.L., and B.T.P.; validation, H.-B.L., H.Q.N., and B.T.P.; formal analysis, H.Q.N., T.-A.N, T.-T.L., V.Q.T. and H.-B.L.; data curation, V.Q.T. and T.-A.N.; writing—original draft preparation, all authors; writing—review and editing, H.-B.L., T.-T.L., and B.T.P.; project administration, H.-B.L.; funding acquisition, H.Q.N. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Conflicts of Interest:** The authors declare no conflict of interest.
