The performance of the models was evaluated using conventional coefficients of determination such as M.A.E., RMSE, and R²values, which were validated using test data³⁸. Tables 3, 4 and 5 show the values that were addressed for Nu, f and η. Compared to the performances of the G.L.R., R.T., and SVM models, the ANFIS model has the lowest error and the highest R² value. The ANFIS model’s MMRE, RMSE, and R² values were 0.3429, 0.0805, and 0.9990 for Nu, 0.2368, 0.0004 and 0.9942 for f, and 0.3242, 0.4534, and 0.9985 for η, respectively.

Due to its lowest error and highest R² value, which is extremely near to unity and demonstrates the model’s correctness, the ANFIS model is superior to the G.L.R., R.T., and SVM models in light of the above information. After ANFIS, the G.L.R. model outperforms R.T. and SVM.

Figure 7a and b illustrate the predicted and experimental nusselt values for each iteration of the training and testing sets. It is evident that compared to the other three models, the projected outcomes of ANFIS are more closely matched with the regression line. The performance was good for ANFIS, with an R² score of 0.999 and 0.9976 for training and testing data iteration, respectively. But for training data iteration, G.L.R., R.T. and SVM models give the R² value of 0.9961, 0.9562 and 0.928, and for testing data iteration G.L.R., R.T. and SVM models give the R² value of 0.9975, 0.9155 and 0.951 respectively.

Figure 8a and b illustrate the predicted and experimental FFs for every iteration of the training and testing sets. The ANFIS model’s anticipated outcomes exhibit a higher degree of alignment with the regression line compared to the other three models. ANFIS had strong performance, as evidenced by an R² value of 0.9966 for the training data iteration and 0.9816 for the testing data iteration. The G.L.R., R.T., and SVM models provide R² values of 0.9683, 0.981, and 0.956, respectively, when applied to the training data. For the testing data, the corresponding R² values are 0.9556, 0.9278, and 0.944. The aforementioned result indicated that ANFIS had accurately predicted the FF.

Figure 9a and b demonstrate the predicted and experimental η for each iteration of the training and testing sets. Compared to the predictions of the other three models, those of ANFIS are more closely matched to the regression line. ANFIS worked well, with an R² value of 0.9997 for the training data iteration and 0.9943 for the testing data iteration. But for iterating on training data, G.L.R., R.T., and SVM models offer R² values of 0.9976, 0.9845, and 0.9614, respectively, and 0.9936, 0.9437, and 0.9773 for iterating on testing data.

The above outcome demonstrated that ANFIS had more precisely forecast. In this situation, ANFIS has predicted the Nu and η more accurately than the f since the f outcomes are more distributed than the Nu and η.

The residual errors obtained while predicting the Nu by all the training models are presented in Fig. 10. In that, it can be observed that the least deviation of the residuals obtained from the zero-axis line is for ANFIS, followed by G.L.R., while the most deviation was while predicting with SVM followed by R.T. This sums up the poor performance of the latter two models while predicting the Nu from training data. The maximum relative error ANFIS, G.L.R., R.T. and SVM models were ± 1.99%, ± 3.53%, ± 11% and ± 16.42%, respectively. It is evident that among ANFIS, G.L.R., R.T. and SVM models, the average maximum relative error of the ANFIS model has the lowest value. Compared to G.L.R., R.T., and SVM models, the ANFIS model predicts the outcomes more precisely.

A similar trend can also be observed in Fig. 11, representing the residuals obtained while predicting the FF for training data. The maximum relative error ANFIS, G.L.R., R.T. and SVM models were found to be ± 0.991%, ± 2.014%, ± 1.584% and ± 2.51%, respectively. While predicting FF, though ANFIS still performed the best, its performance cannot be compared to predicting the other two output variables.

Figure 12 illustrates the residuals found while estimating the η using training data. It was found that the maximum relative error for the ANFIS, G.L.R., R.T., and SVM models, respectively, was ± 1.88%, ± 4.05%, ± 6.9%, and ± 15.53%. While ANFIS outperformed all other models in forecasting η, its performance cannot be compared to the other two output variables.

The residuals obtained are unmatchable to that of the training data. However, suppose only the testing data is considered. In that case, the residuals obtained by ANFIS for Nu and η are the best among all the models, almost similar. They have not deviated much from the zero-axis line. The G.L.R. model is next best to ANFIS, followed by R.T., while SVM is the worst-performing model. In predicting FF, almost all the models have generated substantial residuals compared to Nu and η. In this case, also, ANFIS is the best-performing model.

By giving testing data as inputs to the models for prediction, graphs were also produced along comparable axes. Figures 13, 14, and 15 show the experimental and predicted outcomes for the Nu, FF and η, for testing. ANFIS performs better than G.L.R., SVM and R.T. models with a minimum error of about ± 2.86%, ± 1.42%, and ± 6.32% for Nu, FF and η, respectively. It is not the case for SVM and R.T., for which the values typically coincide with the trend line, indicating its poor performance, with SVM being the worst-performing model.

The results mentioned above demonstrate that all four models considered in the current study can accurately predict Nu, FF, and η; however, the ANFIS model is the most suited since it makes predictions with less error than the other models. So, with a confidence level above 95%, the ANFIS model accurately predicts the data.