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The scatter plots in Fig.7 illustrate the relationship between experimental and predicted outcomes for various ML models applied to training and testing datasets for columns with different cross-section shapes. It can be observed that the data points tightly gather around the diagonal line for most of ML models, signifying a strong alignment between the model predictions and experimental results. This alignment signifies the reliability and prediction accuracy of the developed models. Table 4 introduces evolution metrics to assess the performance of the established ML models, including the mean (), coefficient of variance (CoV), coefficient of determination (R2), root mean squared error (RMSE), the mean absolute percentage error (MAPE), and a20-index, defined as follows:

$$begin{aligned} & mu = frac{1}{n}mathop sum limits_{i = 1}^{n} frac{{y_{i} }}{{hat{y}_{i} }}, ;;R^{2} = 1 - frac{{mathop sum nolimits_{i = 1}^{n} left( {hat{y}_{i} - y_{i} } right)^{2} }}{{mathop sum nolimits_{i = 1}^{n} left( {y_{i} - overline{y}} right)^{2} }} RMSE = sqrt {frac{1}{n}mathop sum limits_{i = 1}^{n} left( {hat{y}_{i} - y_{i} } right)^{2} } , \ & MAPE = frac{1}{n}mathop sum limits_{i = 1}^{n} left| {frac{{hat{y}_{i} }}{{y_{i} }} - 1} right| times 100% \ end{aligned}$$

(12)

where ({y}_{i}) and ({widehat{y}}_{i}) are the actual and predicted output values of the i-th sample, respectively, (overline{y }) is the mean value of experimental observations, and n is the number of specimens in the database. The a20-index16,38 measures the percentage of samples with actual to prediction ratio, ({widehat{y}}_{i}/{y}_{i}), falling within the range of 0.801.20.All data generatedand algorithms introduced in this study are included in thesupplementary file.

Comparison between proposed equations and ML models for training and testing datasets.

As shown in Table 4, all introduced ML models display mean , R2, and a20-index values close to 1.0 and small values for CoV, MAPE%, and RMSE for different cross sections. The prediction results of all introduced models exhibit CoV less than 0.076, MAPE% lower than 6%, and RMSE less than 552kN, indicating minimized scattering in the prediction results compared to the experimental results. Table 4 reveals that the CATB, GPR, and XGB models introduce the best evaluation metrics for the testing subsets, with MAPE% values equal to 1.394%, 1.518%, and 2.135% for CCFST, RCFST, and CFDST column datasets, respectively. In addition, PSVR can accurately predict the capacity of stub CFST columns with MAPE% values equal to 2.497 and 5.151 for CCFST and RCFST columns, respectively. The superior predictive capability of PSVR demonstrates that the SVR model, incorporating the metaheuristic optimization methods39 like the PSO algorithm, can significantly enhance the performance of the SVR model.

Furthermore, the evolution metrics of the testing resemble those of the training set, except for the GPR and CATB models. However, the performance of GPR and CATB models in the testing set is comparable to that of the remaining data-driven models and even better than that of other ML models. In addition, when examining the R2 value and a20-index for the entire dataset, it was found that they are nearly identical to those of the test and training subdatasets. Such robust and stable alignment between the performance of sub-datasets signifies a minimal occurrence of overfitting during the training process of the models.

Although the GPR, CATB, XGB models stands out with significantly superior results compared to other models, extracting an explicit design formula from these models is a challenging task. In contrast, the proposed equations extracted from the SR algorithm offer a distinct advantage by providing simple and practical explicit design formulas, making them more accessible and easier to interpret, even with slightly lower accuracy than the introduced ML models. Although ANN could provide accurate and explicit formulas for strength prediction, utilizing the network in engineering design might not be practical due to the lengthy formulas of the ANN model.

The compressive strength predictions of CFST columns by the proposed equations were compared with the existing code formulas, including EC430 and AISC36029 for different types of columns. As observed in Table 4, for all types of CFST columns, the proposed equations attain a mean, R2, and a20-index nearly equal to 1.0 with CoV less than 0.076 and MAPE% less than 5.9, while EC4 and AISC360 show CoV larger than 0.091, 0.168 with MAPE% larger than 7.1% and 15%, respectively. In addition, the AISC360 predictions, compared to EC4 predictions, appear to overestimate the axial capacity for different cross sections with a higher mean approaching 1.20, lower a20-index, and relatively high error indices. The RMSE and MAPE of AISC36029 predictions are approximately two to six times those of EC430, indicating the better performance of EC4 compared to AISC360. In addition, AISC360 introduces an a20-index with a value approximately 50% lower than that obtained from the EC4 results. This discrepancy could stem from the absence of confinement effect calculations in AISC36029. Although all cited codes standards display a safe design, the error indices introduced by the ML models and proposed equation are significantly small compared to these standards. Specifically, the proposed equations demonstrate superior performance compared to these standards across all evaluation criteria.

Figure8 displays the prediction errors of the design standards and the developed ML models for different cross sections. It indicates that most of the introduced ML models are more accurate than the design standards, especially for the GPR, CATB, and PSVR models, implying the superiority of these ML methods in estimating the axial capacity of stub CFST columns. In the case of CCFST columns, the CATB, GPR, and PSVR models display more than 95% of test samples within the 10% error range, while the proposed equation, EC4, and AISC360 show 83%, 75%, and 7% of test samples, respectively, within the same range. For RCFST columns, all ML models exhibit accuracy, with over 75% of test samples falling within a 10% error range, while the corresponding proportions for the proposed equation, EC4, and AISC360 are 85%, 73%, and 42%, respectively. Regarding CFDST columns, all ML models, excluding the RF and LGBM models, correctly predict 90% of the specimens within a 10% range error, while the proposed equation, EC4, and AISC360 attain nearly 83%, 68%, and 17% accuracy for the test samples, respectively, within the same error range. Thus, the introduced ML models can be considered valuable tools alongside the design standards in estimating the axial capacity of stub CFST columns.

Prediction errors of design standards and established ML models.

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