Estimating Brazilian Tensile Strength of Granite Rocks Using Metaheuristic Algorithms-Based Self-Organizing Neural Networks

Abstract

This article explores the use of different metaheuristic optimization algorithms for training a self-organizing neural network (SONN) model to estimate Brazilian tensile strength (BTS) in granite rocks, including brainstorm optimization (BSO), coronavirus herd immunity optimization (CHIO), and social ski-driver optimization (SSDO). The optimization processes of the algorithms were similar, and the same population size and maximum iterations were used based on 284 granite samples collected. The Schimidt hammer test (R_n), point load test (IS₅₀), and dry density test (DD) based on the ISRM standards were considered to predict BTS. Accordingly, 80% of the whole dataset (approximately, 228 samples) was used to train the intelligence models, while the remaining 20% was used for testing the trained performance of the models. Tenfold cross-validation with three repeats was applied to avoid the overfitting phenomena. While the BSO–SONN demonstrated excellent performance on both training and testing datasets with low MAE and RMSE values and a high R² (MAE = 0.164, RMSE = 0.201, R² = 0.974 on the training dataset, MAE = 0.179, RMSE = 0.231, R² = 0.970 on the testing dataset), the CHIO–SONN also performed well, but it had higher MAE and RMSE values compared to BSO–SONN. However, it still has a good R² (MAE = 4.212, RMSE = 4.555, R² = 0.897 on the training dataset, MAE = 3.984, RMSE = 4.379, R² = 0.904 on the testing dataset), and the SSDO–SONN showed lower performance, especially in terms of R² (MAE = 1.295, RMSE = 1.545, R² = 0.261 on the training dataset, MAE = 1.325, RMSE = 1.574, R² = 0.156 on the testing dataset), indicating that the model might not be capturing the variation in the data well. Another conventional machine learning model, namely support vector machine (SVM), was also applied and compared to the metaheuristic algorithm-optimized models. The SVM’s results showed that it performs reasonably well, with good R² values (MAE = 0.183, RMSE = 0.357, R² = 0.907 on the training dataset, MAE = 0.292, RMSE = 0.646, R² = 0.761 on the testing dataset). However, its MAE and RMSE are slightly higher compared to BSO–SONN. As a result, the BSO–SONN model still outperformed all other models in both training and testing datasets, providing the most accurate and reliable predictions of BTS values with the accuracy approximately 97% in practical engineering. The SSDO–SONN model lags behind in terms of R², suggesting that it may need further improvement. Sensitivity analysis results also suggested that R_n has the highest importance for the BSO–SONN model in predicting BTS.

Highlights

• Self-organizing neural network is a promising technique to estimate Brazilian tensile strength of rock mass.

• The Schimidt hammer test, point load test and dry density test based on the ISRM standards were considered to predict Brazilian tensile strength of rock mass.

• Three hybrid intelligence models were developed for rapid estimating Brazilian tensile strength of rock mass in practical engineering.

• Brainstorm optimization algorithm provided a robust efficiency in improving the accuracy of the self-organizing neural network with an accuracy of 97%.