Advanced Ai Tools for Predicting Mechanical Properties of Self-Compacting Concrete

Multi-Technique AI Approach: In the present study, multiple AI techniques, including GEP, ANFIS-GA, SVR, ANN, and MLR, are used. This diverse approach allowed for a thorough analysis and comparison of the predictive capabilities of different AI algorithms. Where SP is superplasticizer, PC is powder content that constitutes cement and fly ash, PPF is polypropylene fiber, SF is steel fiber, FA is fine aggregate, and CA1 and CA2 are coarse aggregates, respectively. In addition to constituents, the representation of test results is depicted as SFD for slump flow diameter, SFT for slump flow time, VF for V-funnel testing time, CS28 for compressive strength after 28 days, and TS28 for tensile strength after 28 days.

There are five essential phases to employing GEP [27]. The first and most critical stage in any modeling process is data preparation. The outlier approach may be used to assess data quality. If an outlier is discovered, it may be deleted. GEP necessitates specifying the predictors and outcomes for the modeling. In this study, predictors were the parameters influencing SCC strength (compressive and tensile), such as superplasticizer (kg), powder content (kg), polypropylene fibre (kg), steel fibre (kg), slump flow diameter (Dia.), slump flow time (sec), J-ring (mm), V-funnel (sec), and so on, and predictors were compressive and tensile strength [20]. The next step is to choose a fitness function and a starting population. The present research used multigenic chromosomes (composed of three genes). In the initial population, any number of population sizes can be used; it is inferred that chromosomes in the range of 30–100 have yielded promising results. Based on trials, a population size of 30 chromosomes was chosen as the optimal size and used in all GEP models.

For the present problem, the fitness fi of an individual program ‘i’ is evaluated using Eq. 1: 1fi=∑i=1ct(M−| Cij−Tj |){f_i} = \sum\nolimits_{i = 1}^{{c_t}} {\left( {M - \left| {{C_{ij}} - {T_j}} \right|} \right)} Where, M – data range, C_i,j – value returned by the individual chromosome “i” for fitness case “j”, and T_j – target value for fitness case j.

Other parameter descriptions for the model development are given in Table 1. The significant fourth step is to choose the linking function as multiplication. The last step is to select the set of genetic operators. A combination of all operators, i.e., mutation, transposition, and crossover, are used for this purpose, as shown in Table 2. Once all the parameters are added to the algorithm, the combination of different genes and chromosomes is processed to evaluate the data fit, and a model that satisfies the fitness criteria will be printed, followed by algorithm termination.

The ANFIS-GA algorithm starts with generating a fuzzy inference system using the Takagi-Sugeno-Kang (TSK) fuzzy model. The initial values of ANFIS-GA parameters are presented in Table 3. The fuzzy system involves a set of rules defining linguistic relationships between variables (input and output). Each rule consists of precursor and resulting parts.

The precursor part utilizes linguistic terms to represent the input variables’ fuzzy sets, while the consequent part comprises linear functions involving the output or resulting variables. Next, hybridization with neural networks takes place, where the parameters of the fuzzy inference system are tuned using a learning algorithm based on the backpropagation method. The neural network component must alter the membership functions and coefficients of linear functions to lessen the discrepancy between the predicted and actual SCC strengths [13, 28, 29]. The ANFIS network and its many layers are used for modeling and learning. The model can adjust its parameters based on the training data. A genetic algorithm is used to improve the ANFIS-GA model even more. The best individuals are chosen for reproduction by the genetic algorithm, which assesses the fitness of each candidate solution based on how well it predicts the training data, causing the population to evolve towards more ideal solutions. Once the ANFIS-GA model shown in Fig. 1 is trained and optimized, it can be used to predict the strength of SCC for the new input data. The model’s ability to capture complex nonlinear relationships and its adaptive nature provide accurate and reliable predictions for various SCC compositions and properties.

Figure 1.

ANFIS network designed for the SCC strength modeling [28]

The concept of SVM, developed by [30], involves using mathematical principles to create predictive models. [31] outlined the diverse applications, highlighting how SVM had been utilized to address challenges such as structural integrity assessment, vibration analysis, and load prediction refer to Fig. 2.

Figure 2.

Hyperplanes of SVR [30]

Saha P. et al. [23] discussed the SVR capabilities of the SCC. By accurately modeling the relationships between the parameters and the mechanical strengths, we aim to provide an efficient and reliable tool for estimating SCC’s compressive and tensile properties. A quadratic programming problem with linear constraints is used as the kernel function with SVM for defining network weights. This approach is an alternative training method to a radial basis function, multi-layer perceptron, and polynomial classifiers [20]. Mathematically, SVM can be expressed as given by Eq. 2. 2f(x,ai,ai*)=∑i=1n(ai−ai*)M(xi,xj)+bf\left( {x,{a_i},a_i^*} \right) = \sum\limits_{i = 1}^n {\left( {{a_i} - a_i^*} \right)} M\left( {{x_i},{x_j}} \right) + b Where, a_i and ai* a_i^* – positive Lagrange multipliers and M(x_i,x_j) – a non-linear transformation using linear kernel function of SVM, and b – “bias”. Radial Bias Function (RBF) kernels are well suited for the SCC modeling around a bridge pier [20]. The configuration set to predict the SCC model is shown in Table 4.

By utilizing multiple layers of interconnected nodes, ANNs, as shown in Fig. 3, are capable of discerning underlying relationships within the parameter space. This facilitates accurate predictions of compressive and tensile strength of SCC [10, 19, 20, 23].

Figure 3.

ANN architecture [23]

For the evaluation of model performances, evaluation criteria (EC) such as IOA, RMSE, Akaike information criterion (AIC), SS, and Symmetric Uncertainty (SU) were employed in the present study. The following section briefly describes EC.

In the realm of SCC, understanding the relationships between input parameters and vital mechanical properties like compressive and tensile strength is pivotal for optimizing material formulations. The correlation matrix gives the first insight into the selection parameters in AI modeling. Matrices 1 and 2 give the correlation between the selected parameters. When analyzing the correlation matrix 1 for CS28, a comprehensive picture emerges regarding the interplay between input parameters and the target strength. Each parameter’s correlation coefficient with compressive strength offers valuable information on its impact. The parameter SP demonstrates a positive correlation of 0.68, indicating that as the superplasticizer content increases, CS28 tends to rise. Similarly, PC boasts a correlation of 0.72, indicating that higher powder content is associated with enhanced compressive strength.

While PPF displays a weaker positive correlation of 0.16, SF exhibits a moderate correlation of 0.32, suggesting that both parameters contribute positively to compressive strength. SFD and SFT highlight correlations of 0.28 and -0.25, respectively, signifying that larger diameter slump flow values are linked to better compressive strength. In contrast, longer slump flow times tend to lower it slightly. VF shows a similar trend with a correlation of -0.26.

Turning attention to matrix 2 for TS28, the overarching pattern of positive correlations prevails. Most parameters exhibit positive relationships with tensile strength, signifying their potential to influence this mechanical property. SP and PC maintain their influence, with correlations of 0.71 and 0.69, respectively. PPF demonstrates a correlation of 0.21, indicating a moderate impact on tensile strength. SF presents a similar trend with a correlation of 0.29. Interestingly, SFD and SFT remain consistent with their compressive strength relationships, highlighting correlations of 0.26 and -0.23, respectively. Again, VF demonstrates a weak negative correlation of -0.25.

The results of the relative importance analysis through PMI for predicting the CS28 and TS28 of SCC using various input variables are shown in Fig. 4. Each row corresponds to a specific variable, and the “Rel_Importance_CS28” and “Rel_Importance_TS28” columns indicate the calculated relative importance values for CS28 and TS28, respectively. For CS28 prediction, the obtained relative importance values provide valuable insights into the significance of each variable. Among the variables, SFD exhibits higher importance with a relative importance value of 0.73. This implies that the SFD parameter has a substantial influence on predicting CS28. The VF parameter follows with a notable relative importance value of 0.68, emphasizing its meaningful contribution to CS28 estimation. Other variables, such as SF, SFT, and PPF, are essential, contributing 0.50, 0.58, and 0.42, respectively. Meanwhile, PC and SP have relatively lower importance values of 0.26 and 0.33, indicating a lesser impact on CS28 prediction.

Figure 4.

Relative importance of each parameter

Turning to TS28 estimation, Fig. 4 reveals the relative importance values of the variables for predicting tensile strength. Like CS28, SFD plays a pivotal role with a high relative importance value of 0.72, indicating its strong influence on TS28 estimation. The VF variable also maintains significance, as indicated by a relative importance value of 0.67. Moreover, SF demonstrates substantial importance with a relative value of 0.48, reinforcing its contribution to predicting TS28. Similarly, SFT and PPF exhibit notable importance values of 0.56 and 0.41, respectively. SP and PC show lower importance values of 0.34 and 0.27, suggesting their comparatively lesser roles in TS28 prediction.

The relative importance values in CS28 and TS28 prediction scenarios provide a clear hierarchy of the variables' contributions. These values guide the prioritization of variables while constructing accurate and efficient AI models for estimating SCC strength. The results underscore the key variables that heavily influence the prediction outcomes, facilitating better decision-making in selecting input parameters and enhancing the interpretability and performance of the AI models in SCC strength estimation.

GEP modeling was done using MATLAB GEP modeling for predicting CS28 and TS28 of SCC. This provides valuable insights into the accuracy and performance of the developed models Eq. 13 and Eq. 14, respectively. For CS28 prediction, the Coefficient of Variation (CV) value of 0.116 indicates the variability in the predicted values around the mean. A lower CV suggests that the predicted values are relatively consistent and closely aligned with the actual data points, indicating good model performance in terms of precision and reliability. The Normalized Mean Square Error (NMSE) of 0.517 reflects the extent of the model’s accuracy in predicting CS28. A lower NMSE value signifies a better fit between the predicted and actual values. In this case, the NMSE value indicates that the GEP model captures more than half of the variation in the data, implying that the model performs reasonably well in approximating the target CS28 values. The correlated values of the actual and prediction of 0.696 demonstrate the strength of the linear relationship between the predicted and actual CS28 values. A correlation value closer to “1” shows a higher degree of agreement between the model’s predictions and the actual data, suggesting that the GEP model accurately represents CS28.

Similarly, for TS28 prediction, the CV value of 0.112 suggests consistent and reliable predictions with relatively low variability around the mean. The lower CV indicates stable model predictions for TS28. The NMSE of 0.376 reflects the accuracy of the TS28 prediction model. With a value less than 0.5, the NMSE indicates a relatively good fit between the predicted and actual TS28 values, signifying the model's capability to capture a substantial portion of the data variation. The correlation of 0.797 between the actual and predicted TS28 values illustrates a robust linear relationship between the two sets of values. This high correlation value implies that the GEP model for TS28 prediction aligns closely with the actual data, indicating the model’s ability to estimate TS28 with a high degree of accuracy. 13CS28=( (FA+((CA1+SF)*(−0.89)))/((FA+SP)+(−4.84−CA2)))CS28=CS28+(FA−((((SP+Water )−CA2)/(FA/29.71))−(−4.69)))CS28=CS28+((((CA1−SP)−78.44)+(SF*10.50))/(11.79−(SF−CA1)))CS28=CS28+((CA1/CA2)+(((SF+ Water )/17.16)−(FA−8.20)))CS28=CS28+((SF/((SF*4.96)−12.78))+((PC/ Water )−(−9.60+(−9.60))))\matrix{ {{\rm{CS}}28 = \left( {\left( {{\rm{FA}} + \left( {{{({\rm{CA}}1 + {\rm{SF}})}^*}( - 0.89)} \right)} \right)/(({\rm{FA}} + {\rm{SP}})} \right. + ( - 4.84 - {\rm{CA}}2)))} \hfill \cr {{\rm{CS}}28 = {\rm{CS}}28 + ({\rm{FA}} - (((({\rm{SP}} + {\rm{Water }}) - {\rm{CA}}2)/({\rm{FA}}/29.71)) - ( - 4.69)))} \hfill \cr {{\rm{CS}}28 = {\rm{CS}}28 + (((({\rm{CA}}1 - {\rm{SP}}) - 78.44) + ({\rm{SF}}*10.50))/(11.79 - ({\rm{SF}} - {\rm{CA1}})))} \hfill \cr {{\rm{CS}}28 = {\rm{CS}}28 + (({\rm{CA}}1/{\rm{CA}}2) + ((({\rm{SF}} + {\rm{ Water }})/17.16) - ({\rm{FA}} - 8.20)))} \hfill \cr {{\rm{CS}}28 = {\rm{CS}}28 + (({\rm{SF}}/(({\rm{SF}}*4.96) - 12.78)) + (({\rm{PC}}/{\rm{ Water }}) - ( - 9.60 + ( - 9.60))))} \hfill \cr } 14TS28=((CA2/(((−0.75+SF)*CA2)−(PPF*FA)))−4.94)TS28=TS28+( ( (CA2+(SF*−4.22))/((SF−Water )*SP))+6.59)TS28 = TS28 +( Water /((−3.66−((3.88−Water )/4.00))− PPF ))\matrix{ {{\rm{TS}}28 = (({\rm{CA}}2/((( - 0.75 + {\rm{SF}})*{\rm{CA}}2) - ({\rm{PPF}}\;*\;{\rm{FA}}))) - 4.94)} \hfill \cr {{\rm{TS}}28 = {\rm{TS}}28 + \left( {\left( {\left( {{\rm{CA}}2 + \left( {{\rm{S}}{{\rm{F}}^*} - 4.22} \right)} \right)/(({\rm{SF}} - } \right.} \right.{\rm{Water }})*{\rm{SP}})) + 6.59)} \hfill \cr {{\rm{TS28 }} = {\rm{ TS28 }} + ({\rm{ Water }}/(( - 3.66 - ((3.88 - {\rm{Water }})/4.00)) - {\rm{ PPF }}))} \hfill \cr }

This innovative model integration aimed to harness the adaptive learning capacity of neuro-fuzzy systems developed in MATLAB, while optimizing the model's parameters using genetic algorithms. Results are presented in Fig. 5 to Fig. 8. During the model development phase, the MSE and RMSE were calculated as 30.8 and 5.53, respectively, for CS28. These numbers express the difference between the strength values that the model predicts and the training data-based strength values. A favorable level of accuracy in reproducing the relationships between the input variables and CS28 was indicated by the relatively low MSE and RMSE values. The CS28 model produced a higher MSE and RMSE of 42.2 and 8.5 when tested against fresh, unexplored data. This result indicates that although the model was proficient at identifying patterns in the training data, it struggled when presented with data outside the parameters of its initial training. The potential overfitting of the model to the training data or the inherent complexity of real-world data may be to blame for this discrepancy. With MSE and RMSE of 0.4 and 0.6, respectively, the model showed even lower errors during model development when focusing on TS28. This encouraging outcome highlights the model's potential to predict tensile strength during its learning phase accurately. However, the MSE and RMSE of the TS28 model increased to 5.4 and 2.4, respectively, during validation. These values remained low, but the difference in performance between development and validation performance points to the need for further validation and fine-tuning to increase the model's ability to generalize to new data.

Figure 5.

Training phase results of ANFIS for CS28

Figure 6.

Testing phase results of ANFIS for CS28

Figure 7.

Training phase results of ANFIS for TS28

Figure 8.

Testing phase results of ANFIS for TS28

The model’s training phase revealed a moderate linear association between the predicted and actual CS28 values, as indicated by the correlation coefficient for CS28 of 0.77. This demonstrated that the model captured a sizable portion of the underlying patterns in the training data. The model successfully reproduced the variations in CS28 based on the input variables as it learned from the training data. After testing the ANN model’s performance, the enhanced correlation coefficient of 0.93 revealed a more vital agreement between the predicted and actual CS28 values. With this improvement, the model could generalize beyond its training set and predict CS28 with accuracy for brand-new, untested inputs. The higher correlation during testing demonstrated the ANN model’s robustness and generalization abilities, which showed that the ANN model could successfully extrapolate its learning to novel scenarios.

The ANN model’s training correlation coefficient of 0.7 for the TS28 prediction indicated a moderate relationship between predicted and actual values. This initial correlation showed that, despite some variability, the model could capture trends in the training data.

The increased correlation coefficient of 0.9 during testing demonstrates the ANN model’s improved capability to predict TS28 values correctly. This higher correlation demonstrates the model’s ability to handle unknown input variables by showing that it's learning from training translated well to new data. The visual representations of observed versus predicted values for CS28 and TS28 highlighted the model's ability to predict values. Fig. 9 and Fig. 10 demonstrate how the predicted and actual strengths lined up, highlighting the model’s effectiveness in estimating SCC properties. Overall, the results of the ANN model’s development and testing showed how adept it was at capturing the intricate connections between the input parameters and the corresponding benefits of SCC. The improved correlations during testing for CS28 and TS28 highlighted the model’s capability for accurate predictions and strong generalization to new data. This demonstrates the effectiveness of ANN models in calculating SCC strength and their capability to support precise and successful concrete property prediction in practical contexts.

Figure 9.

CS28 results from prediction models

Figure 10.

TS28 results from prediction models

The SVM model performed admirably during the CS28 prediction task’s training and testing phases. The CS28 values anticipated and genuine during preparation showed a healthy level of understanding (0.77-relationship coefficient), demonstrating that the model effectively extricated basic examples from the preparation information. This demonstrated that the SVM model correctly identified and replicated the relationships between the input variables and CS28. The SVM model’s solid prescient capacities during testing were shown by the relationship coefficient of 0.93, which showed a more significant level of concordance among anticipated and genuine CS28 values. This improvement in connection featured the model’s better capacity to sum up new, untested information, as well as its constancy in assessing CS28. The testing predominant connection featured the SVM model’s capacity to adjust to various information mixes and make precise forecasts in real situations. The presentation of the SVM model was also empowering for TS28 expectations. The system’s ability to identify relationships and trends, although with some variability, was demonstrated by the training data’s 0.7 correlation coefficient. The model could construct an establishment for foreseeing TS28 because of the information factors gained from the preparation information. The correlation coefficient of 0.9 indicates that the SVM model performed significantly better during testing, demonstrating its strong generalizability to new data and its capacity to predict TS28 values accurately. This higher connection coefficient featured the model’s actual capacity as an exact indicator of TS28 by showing its capacity to extrapolate its learned connections.

The after-effects of the MLR demonstrating the CS28 and TS28 of SCC are shown in Table 4 and Table 5, respectively. These outcomes offer significant bits of knowledge into the accuracy and trustworthiness of the models. The various connection coefficients (Numerous R) for the CS28 expectation created by the MLR model were 0.711514. This worth proposes that the indicator factors (input boundaries) and the CS28 yield have a tolerably certain direct relationship. As per the coefficient of assurance R2 of 0.506253, the model's indicator factors can represent around 50.6% of the variety in CS28. This indicates that the MLR model captures a substantial portion of the variability in CS28, but other factors also contribute to the variation. The adjusted R2 value of 0.452249 reflects the model'saccuracy after considering the number of predictors and degrees of freedom, adjusting for potential overfitting. The standard error 6.190593 represents the average absolute difference between the observed and predicted CS28 values Eq. 15, indicating the model's precision. With 72 observations (Table 1), the model’s performance is based on a reasonably sized dataset. 15CS28=92.13+(0.38*SP)+(−0.12*PC)+(1.42*PPF)+(0.13*SF)+(0.16*Water )+(−0.04*FA)+(−0.01*CA1)+(0.04*CA2){\rm{CS}}28 = 92.13 + (0.38\;*\;{\rm{SP}}) + ( - 0.12\;*\;{\rm{PC}}) + (1.42\;*\;{\rm{PPF}}) + (0.13\;*\;{\rm{SF}}) + (0.16\;*{\rm{Water }}) + ( - 0.04\;*\;{\rm{FA}}) + ( - 0.01\;*\;{\rm{CA}}1) + (0.04\;*\;{\rm{CA}}2)

For the TS28 prediction, the MLR model demonstrated a higher multiple R2 value of 0.816217, implying a stronger linear relationship between the predictor variables and TS28. The R2 value of 0.66621 indicates that approximately 66.6% of the variability in TS28 can be attributed to the predictor variables employed in the model. This higher R2 suggests that the MLR model is more effective in explaining the variance in TS28 than in CS28. The adjusted R2 value of 0.629702 further validates the model's predictive capability while adjusting for the complexity of the model. The standard error of 0.689796 indicates the average absolute difference between the observed and predicted TS28 values calculated using Eq. 16, indicating the model’s precision in predicting TS28. Like the CS28 model, the analysis is based on a dataset containing 72 observations (Table 1). 16TS28=5.1557−(0.0299*SP)−(0.0060*PC)+(0.2266 * PPF )+(0.0160 * SF )+(0.0325 * Water )−(0.0009 * FA )−(0.0037 * CA1 )−(0.0037 * CA 2){\rm{TS}}28 = 5.1557 - (0.0299*{\rm{SP}}) - (0.0060*{\rm{PC}}) + (0.2266{\rm{ * PPF }}) + (0.0160{\rm{ * SF }}) + (0.0325{\rm{ * Water }}) - (0.0009{\rm{ * FA }}) - (0.0037{\rm{ * CA1 }}) - (0.0037{\rm{ * CA }}2)

ANFIS performs moderately with an IOA of 0.50, indicating moderate agreement between observed and predicted values. The AIC of 256.34 suggests that the model may have some complexity. The SS of 0.50 indicates reasonable predictive skill. The SU of 0.01 suggests low uncertainty in the model’s predictions.

ANN exhibits a positive IOA of 0.64, indicating a relatively good relation between observed and predicted CS28 values. The AIC of 232.34 indicates a less complex model than ANFIS. The SS of 0.64 suggests that this model has good predictive skills. The SU of 0.29 shows a moderate level of uncertainty in predictions.

SVM outperforms other models with an IOA of 0.96, indicating excellent agreement. The AIC of 68.33 signifies a simpler model. The SS of 0.96 shows strong predictive skill. The SU of 0.93 indicates very low uncertainty.

MLR exhibits moderate agreement with an IOA of 0.51 and an AIC of 256.04. The SS is 0.51, suggesting reasonable predictive skill. The SU is 0.01, indicating low uncertainty.

GEP shows an IOA of 0.62, indicating moderate agreement. The AIC of 159.31 suggests a relatively less complex model. The SS is 0.58, which is reasonable, and the SU is 0.73, indicating moderate uncertainty.

ANFIS performs well with an IOA of 0.67, indicating good agreement. The negative AIC (-59.96) suggests a relatively simple model. The SS of 0.67 indicates vital predictive skill, while the SU of 0.33 signifies moderate uncertainty.

ANN exhibits negative IOA (-0.54), indicating poor agreement. The positive AIC (50.01) suggests model complexity. The negative SS (-0.54) indicates poor predictive skill, and the SU of -2.07 suggests high uncertainty.

SVM shows good performance with an IOA of 0.81, indicating substantial agreement. The negative AIC (-99.25) suggests model simplicity. The SS of 0.81 reflects strong predictive skill, and the SU of 0.61 indicates moderate uncertainty.

MLR performs well with an IOA of 0.87, indicating excellent agreement. The negative AIC (-127.64) suggests a simple model. The SS of 0.87 indicates strong predictive skill, and the SU of 0.74 signifies moderate uncertainty.

GEP achieves an IOA of 0.62, indicating moderate agreement. The negative AIC (-51.32) suggests a relatively simple model. The SS is 0.62, which is reasonable, and the SU of 0.25 indicates moderate uncertainty.

SVM consistently performs well for CS28 and TS28, with high agreement, strong predictive skills, and low uncertainty. MLR also performs well for both properties. ANN shows good performance for CS28 but struggles with TS28, while ANFIS exhibits good performance for TS28 but moderate performance for CS28. GEP's performance is moderate for both properties.