Integrated Machine Learning for Enhancing Footing Stability Prediction in Shallow Foundations

In a paper published in the journal Scientific Reports, researchers emphasized the significance of analyzing the stability of footings in civil/geotechnical engineering projects. The study proposed two novel predictive tools based on artificial neural networks (ANNs) to assess the bearing capacity of footings on a two-layered soil mass. The researchers employed a backtracking search algorithm (BSA) and equilibrium optimizer (EO) to train the ANN to approximate the stability value (SV) of the system.

Rotated factor loadings for PC 1 and PC 2. Image Credit:
Rotated factor loadings for PC 1 and PC 2. Image Credit:

Settlement values lower or higher than a specified threshold indicated system stability or failure, determined through finite element analyses. The results demonstrated the effectiveness of these algorithms in fulfilling the assigned task. The training error of the ANN decreased, and the prediction accuracy improved with the application of BSA and EO. Comparative analysis revealed EO's larger accuracy, while BSA proved to be a more time-effective optimizer. Lastly, an explicit mathematical formula derived from the EO-ANN model facilitated the convenient stability value (SV) prediction.

Related Work

Previous research has demonstrated the impact of emerging technologies on engineering fields, particularly in civil and geotechnical engineering. Techniques like image processing, advanced sensors, and artificial intelligence models have been instrumental.

Many researchers in geotechnical engineering use ANNs to predict parameters such as bearing capacity and settlement. Additionally, integrating metaheuristic algorithms with predictive models has shown promise in optimizing engineering tasks and improving accuracy and efficiency.

Algorithmic Optimization Strategies

The BSA algorithm, devised by Civicioglu, integrates local exploitation and global exploration strategies to search for solutions effectively. It progresses through five key stages: initialization, selection, mutation, crossover, and a final selection process based on fitness values.

Initialization uses a uniform distribution function to generate a random population over the search space. Selection and redesigning of old members occur through determining search directions and shuffling member orders. Mutation and crossover steps further refine the trial population, focusing on maintaining the problem space's boundaries.

The EO draws inspiration from physical laws rather than natural animal behaviors. The three main steps in EO—initialization, equilibrium pool formation, and concentration updating—aim to balance diversification and intensification. EO's initialization phase generates particles with response vectors to the given problem, balancing diversification and intensification.

Researchers establish equilibrium by selecting the most successful agents to form an equilibrium pool within the algorithm while incorporating an average agent to contribute to diversification and exploitation efforts. Concentration updating involves adjusting parameters to control diversification and intensification. Both algorithms offer efficient optimization strategies, with BSA focusing on global exploration and local exploitation, while EO seeks equilibrium through a balance between diversification and intensification.

Model Performance and Analysis

Researchers present the results and discussions derived from the study, which introduced two novel predictive models for approximating the failure or stability of soil-foundation systems through bearing capacity analysis. These models combine metaheuristic techniques with a neural computing tool to enhance prediction accuracy.

The basic model comprises a multilayer perceptron (MLP) neural network trained using the BP algorithm, featuring one hidden layer. Through experimentation, researchers determined that the optimal structure for the network is 7 × 6 × 1, with activation functions set to Tansig and Purelin for the hidden and output layers, respectively.

Following this, researchers describe the optimization process in which the BSA and EO actively train the MLP neural network. These metaheuristic algorithms iterate to minimize the error, evaluated using the root mean square error (RMSE) as the objective function.

Results indicate that the EO consistently outperforms the BSA in minimizing the objective function across different population sizes. Moreover, convergence curves demonstrate the effectiveness of both BSA-ANN and EO-ANN models in minimizing learning errors over iterations.

Researchers assessed the models' accuracy using statistical indices such as RMSE, mean absolute error (MAE), and the area under the receiver operating characteristics curve (AUROC). Both BSA-ANN and EO-ANN models exhibit superior performance compared to the basic BP-ANN model, as evidenced by reduced errors and higher AUROC values.

Further analysis of testing phase results confirms the enhanced prediction capability of the ensemble models compared to the basic ANN model. ROC curves also demonstrate the reliability of all three models in approximating the stability or failure of soil systems, with ensemble models slightly outperforming the basic ANN model.

Lastly, developing a predictive formula derived from the optimized EO-ANN model is discussed, offering a simplified approach for calculating stability values. Future work may involve expanding the scope of input parameters and employing additional statistical indicators to support this study's findings.


To sum up, the backtracking search algorithm and equilibrium optimizer were effectively applied to predict footing bearing capacity. BSA-ANN and EO-ANN models significantly improved prediction accuracy compared to a conventional ANN, reducing RMSE by 11.72% and 17.46%, respectively.

These models offered reliable early stability approximation for similar systems. Researchers also developed a simplified predictive formula based on the EO-ANN model for practical use. Future research should enhance model applicability and efficiency and compare metaheuristic algorithms for further improvements.

Journal reference:
Silpaja Chandrasekar

Written by

Silpaja Chandrasekar

Dr. Silpaja Chandrasekar has a Ph.D. in Computer Science from Anna University, Chennai. Her research expertise lies in analyzing traffic parameters under challenging environmental conditions. Additionally, she has gained valuable exposure to diverse research areas, such as detection, tracking, classification, medical image analysis, cancer cell detection, chemistry, and Hamiltonian walks.


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