# Results of Application of Artificial Neural Networks in Predicting Geo-Mechanical Properties of Stabilised Clays—A Review

^{*}

## Abstract

**:**

^{2}and low MAE, RMSE and MSE values. The Levenberg–Marquardt algorithm is effective in shortening the convergence time during model training.

## 1. Introduction

_{S}, etc. [4,5,6,7,8]. These parameters are most times influenced by several variables leading to a multi-variable problem for which researchers have used several statistical methods to analyse, investigate and model trends using laborious mathematical processes involving multivariable regression analyses.

#### 1.1. Artificial Neural Network

#### 1.2. Components of Artificial Neural Network

#### Neurons and Edges

## 2. Materials and Methods

_{j}) is the activation function where I

_{j}is the propagation function defined in Equation (1) [22,23]. The activation function could be any of binary step function (Equation (3)), logistic function (Equation (4)), hyperbolic function (Equation (5)), Gaussian function (6), etc. [12].

_{ji}, are continuously adjusted for each training iteration step using the expression below [24].

#### 2.1. ANN Architecture

#### 2.2. Feedforward and Recurring Networks

#### Data Preprocessing

_{1}(x

_{11}, x

_{21}, x

_{31,}…, x

_{n}

_{1}) and y

_{2}(x

_{12}, x

_{22}, x

_{32,}…, x

_{n}

_{2}) in a data space can be expressed as Equation (7). Therefore, in order to ensure that proper significance is attributed to the features, it is imperative to scale the features. Feature scaling can be achieved using methods such as standardization (Z-score normalization) or the max-min normalization method. For standardization, the feature ${X}_{i}$ is expressed in its standardised form as in Equations (7) and (8) below.

## 3. Selecting Design Parameters for ANN in Soil Stabilisation

#### 3.1. Training, Validation, and Testing

^{2}), the mean absolute error (MAE), the root mean square error (RMSE), the mean square error (MSE) and others. The R

^{2}, MAE, RMSE and MSE expressions are defined in Equations (9), (10), (11) and (12), respectively.

#### 3.2. Estimating the Amount of Training Data

## 4. Application of ANN in Predicting the Properties of Stabilised Clays

#### 4.1. Unconfined Compressive Strength

_{S}), coefficient of uniformity (C

_{U}), coefficient of gradation (C

_{C}), LL, PL, optimum moisture content (OMC), and maximum dry density (MDD) for three UCS outputs (7, 14 and 28 days). A total of 72 data sample data were utilised in model development. The ANN model topography consisted of one input layer with eight neurons and one output neuron (the UCS value). However, the number of hidden layers was varied in order to determine the optimum number of neurons in the hidden layer. Figure 6 shows the performance of the trial models.

^{2}. The optimum network model was taken as two hidden layers with eight neurons each. The result of the analysis showed a good understanding of the relationship and modelling of the increase factor with a high coefficient of determination and low error. As seen in Figure 9, there is a close approximation of the UCS by the ANN model. The prediction and experimental values are clustered around the line of equality for all three conditions of training, validation and testing.

#### 4.2. California Bearing Ratio

^{2}and MSE. The performance of the models based on the training algorithm is given in Table 4.

^{3}and 15%, respectively. The ANN models were developed using LL, PL, fly-ash content, OMC, MDD and the number of geotextile layers as input variables, while soaked CBR was the target variable in the output neuron. Different training algorithms were employed while varying the number of neurons in the hidden layer to select the best model architecture. The performance of the models is presented here, as shown in Table 5.

^{2}. The models performed well with minimum error in relation to a higher coefficient of determination. A regression analysis of the CBR of soils stabilised with lime and rice husk ash (RHA) was carried out by [62]. The LL and PL of the soil were reported as 34% and 20%, with clay and silt content of 20% and 71%. Additionally, the MDD and OMC of the soil were reported as 17.7 kN/m

^{3}and 14.7%, respectively. The CBR test was conducted in line with ASTM D1883 -07. Soaked CBR tests were conducted on 0, 7 and 28 days cured samples. The predictive variables considered in the study were the percentage of RHA, percentage of lime, curing duration, OMC and MDD. The topography of the ANN model was determined by trial and error by varying the number of neurons in the hidden layer. Optimum model topography was found to be one hidden layer and twelve neurons from the R value and MAE. The model performance as per the training data showed higher R and low MAE. The number of data utilised for training is deemed small. The performance of the model in generalizing new data may be affected due to overfitting. This again is a common factor confronting the application of ANN in soil stabilisation studies.

_{S}), linear shrinkage (L

_{S}), coefficient of uniformity (C

_{U}), coefficient of gradation (C

_{C}), OMC and MDD, were utilised in developing 72 data samples for two ANN models; soaked and unsoaked CBR. The model architecture was determined by varying the number of neurons in the hidden layer from 1 to 20 while evaluating the MSE and R values. Figure 10 shows the performance of various model architectures.

#### 4.3. Permeability and Resilient Modulus

^{3}and 16.5%, respectively. The clays soil was stabilised with various combinations of lime and pozzolan. Permeability tests were conducted to develop a dataset of 69 samples for the regression analysis. A total of six independent variables were used in the development of the ANN model, including percentage passing 0.005 mm size sieve, PI, MDD, lime percentage, pozzolan percentage and curing time (C

_{d}). To develop the predictive model, ANNs with varying numbers of neurons in the hidden layer were tried. An optimum model with nine neurons in the hidden layer was selected based on low MSE and high R value as with other studies. In line with other studies, 70% of the data was used for training, 15% for testing, and 15% for validation. The performance of the model was satisfactory in terms of low MSE and high R values.

#### 4.4. Plasticity Index and Compaction Characteristics

_{S}), linear shrinkage (L

_{S}), free swell, grain sizes (D10, D30 and D60), coefficient of uniformity (C

_{U}), and coefficient of gradation (C

_{C}). Two separate ANN models were developed for MDD and OMC using five and seven neurons, respectively, in the hidden layers. The models were reported to have performed effectively and predicted the MDD and OMC with high correlation and low errors. Table 6 shows detailed results of different ANN models and their performances.

## 5. Discussions

^{2}and low MSE, MAE and RMSE of ANN regression analysis are pointers to its advantage. From the study, it was discovered that most of the ANN models utilized were backpropagation feed-forward networks with one hidden layer. Only two studies [56,58] used two hidden layers for the analysis. Other studies have also pointed out that most regression analyses in soil stabilisation problems are easily solved using a simple architecture of one hidden layer.

## 6. Conclusions

- The advantages of the artificial neural over traditional regression analysis as applied to stabilisation have been highlighted in the foregoing sections. In a typical field stabilisation project, in order to improve the properties of expansive clays, experimental data are usually generated from several field and laboratory tests to monitor and ascertain the progress made in terms of improvement. These procedures are expensive and time-consuming and may be reduced to a minimum using ANN to predict the field response of the soils. In summary, the following conclusions are made.
- An artificial neural network is reliable and can be employed in modelling various properties of stabilised clays for easy prediction of soil response while eliminating the need for extensive experimental procedures.
- Backpropagation feedforward networks are the most used models in dealing with the problem of regression analysis for stabilisation of clays.
- An artificial neural network should be developed with a relatively substantial dataset to regression models with good correlation. Many of the studies in regression analysis of stabilised clays have used relatively small data sets, although the models have performed well. The ability of the models to generalize can be improved with a larger dataset which fields a wide range of possible soil behaviour for proper training of the model.
- Shallow networks made up of one hidden layer are the most used ANN architecture in developing predictive models for the prediction of geotechnical characteristics of stabilised clays and in modelling the response of stabilised expansive clays. The Levenberg–Marquardt training algorithm has been reported to be the most used among the studies reviewed.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 5.**Experimental versus ANN-Predicted UCS values [50].

**Figure 6.**The selection of optimum number of neurons in hidden layer [52].

**Figure 7.**Observed Versus Predicted UCS [52].

**Figure 8.**The selection of the optimum number of neurons in hidden layer [55].

**Figure 9.**Predicted and experimental observations [56].

**Figure 10.**Mean square error versus number of neurons in the hidden layer [63].

**Figure 11.**The experimental and predicted resilient modulus [65].

**Table 1.**Statistical Evaluation of the Performance of ANN and MVR models [50].

Model | Dataset | Statistical Parameter | ||
---|---|---|---|---|

R^{2} | MSE | MAE (%) | ||

ANN | Training data | 0.992 | 0.34 | 3.65 |

Testing data | 0.964 | 1.50 | 8.34 | |

MVR | Training data | 0.828 | 7.24 | 19.20 |

Testing data | 0.808 | 8.04 | 19.26 |

**Table 2.**Classification of shrink-swell clays [51].

Modified Plasticity Index ${\mathit{I}}_{\mathit{P}}^{\prime}$ | Volume Change Potential (VCP) |
---|---|

>60 | Very high |

40–60 | High |

20–40 | Medium |

<20 | Low |

**Table 3.**Statistical evaluation of the performance of ANN and MVR models [55].

Model | Dataset | Statistical Parameter | ||
---|---|---|---|---|

R^{2} | MSE | RSME | ||

ANN | Training data | 0.9813 | 0.0395 | 0.1987 |

Testing data | 0.9714 | 51.34 | 7.1651 | |

MVR | 0.8870 | 68.7603 | 8.2921 |

**Table 4.**ANN performance using various training algorithms [60].

Training Algorithm | R | MSE |
---|---|---|

Conscience bias learning function | 0.7693 | 7.08 |

Gradient descent weight and bias learning function | 0.8991 | 3.98 |

Gradient descent with momentum weight | 0.9163 | 6.66 |

Levenberg–Marquardt function | 0.94317 | 0.49 |

Hebb weight learning rule | 0.8761 | 2.4 |

**Table 5.**ANN performance using various training algorithms [61].

Training Algorithm | R | MSE |
---|---|---|

Quasi-Newton back propagation | 0.88712 | 1.083 × 10^{−4} |

Bayesian regularisation back propagation | 0.85190 | 4.983 × 10^{−5} |

Conjugate gradient back propagation with Powell–Beale restarts | 0.94122 | 3.776 × 10^{−7} |

Conjugate gradient back propagation with Fletcher–Reeves updates | 0.81167 | 7.339 × 10^{−6} |

Conjugate gradient back propagation with Polak–Ribiére updates | 0.85819 | 2.964 × 10^{−9} |

Gradient descent back propagation | 0.94862 | 9.985 × 10^{−9} |

Levenberg–Marquardt back propagation | 0.98695 | 8.0242 × 10^{−11} |

One-step secant back propagation | 0.92335 | 1.388 × 10^{10} |

Scaled conjugate gradient back propagation | 0.96904 | 1.946 × 10^{−6} |

ANN Data | Network Architecture | Model Performance | References | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Soil Parameter Investigated | ANN Input Variables | No. of Data Used | No. of Training Data Used | No. of Testing Data Used | No. of Validation Data | No. of Hidden Layers | No. of Neurons in Hidden Layer | Training Algorithm Utilised | Training | Validation | Testing | |

UCS | 8 Inputs. LL, PI, GGBS content, PFA content, Molarity, activator/binder ratio, Na/Al ratio, Si/Al ratio | 283 | 70% | 15% | 15% | 1 | 9 | Bayesian regularization | R = 0.996, MSE = 0.34, MAPE (%) = 3.65 | R = 0.982, MSE = 1.50, MAPE (%) = 8.34 | [50] | |

UCS (7 days) | Gs, L_{S}, C_{U}, C_{C}, LL, PL, OMC, MDD | 72 | 70% | 15% | 15% | 1 | 11 | - | R = 0.9782 | R = 0.996 | R = 0.9711 | [52] |

UCS (14 days) | Gs, L_{S}, C_{U}, C_{C}, LL, PL, OMC, MDD | 72 | 70% | 15% | 15% | 1 | 11 | - | R = 0.9824 | R = 0.9843 | R = 0.9615 | [52] |

UCS (28 days) | Gs, L_{S}, C_{U}, C_{C}, LL, PL, OMC, MDD | 72 | 70% | 15% | 15% | 1 | 11 | - | R = 0.9946 | R = 0.9992 | R = 0.9929 | [52] |

UCS | Percentage of Clay, percentage of RHA, percentage of cement, percentage of PFA, curing duration(days) | 129 | 70% | 15% | 15% | 1 | 10 | Levenberg–Marquardt | R^{2} = 0.9813, MSE = 0.0395, RMSE = 0.1987 | R^{2} = 0.9714, MSE = 51.34, RMSE = 7.1651 | [55] | |

UCS | Curing duration, compaction energy. | 80 | 70% | 15% | 15% | 2 | 8 in each layer | R^{2} = 0.924 | R^{2} = 0.908 | R^{2} = 0.889 | [56] | |

UCS | S_{T}, MC, W_{e}, C_{M}, D_{I}, L, C_{A}, S_{V}, D, M_{S}, D_{S}, C_{D}, curing time (days), C_{T} | 216 | 80% | 20% | 1 | 50 | Adam | R = 0.980 (Average), MAE = 115.29, RMSE = 231.2 | R = 0.925 (Average), MAE = 292.2, RMSE = 419.82 | [57] | ||

UCS | Moisture content, cement (percentage), air foam(percentage) and waste fishing net content (percentage) | 51 | 70% | 30% | 2 | 12 and 10 neurons | Levenberg–Marquardt | R = 0.95, MAE = 3.9001, RMSE = 9.1948 | R = 0.94, MAE = 8.6535, RMSE = 10.3390 | [58] | ||

CBR (28 days) | LL, PL, OMC and MDD | 49 | 70% | 30% | - | - | - | Differential evolution | R = 0.98 | R= 0.86 | [59] | |

CBR (28 days) | LL, PL, OMC and MDD | 49 | 70% | 30% | - | - | - | Levenberg–Marquardt | R = 0.96 | R= 0.93 | [59] | |

CBR (28 days) | LL, PL, OMC and MDD | 49 | 70% | 30% | - | - | - | Bayesian regularization | R = 0.96 | R= 0.93 | [59] | |

CBR | Type of ash, mix proportion (percentage), LL, PL, MDD, OMC and number of geogrid layers | 210 | - | - | - | 1 | 7 | Levenberg–Marquardt | R = 0.94472 | R = 0.93327 | R = 0.94685 | [60] |

CBR (28 days Soaked) | LL, PI, percentage of PFA, OMC, MDD and no. of geotextile layers | - | - | - | - | 1 | - | Levenberg–Marquardt | R = 0.99846 | R = 0.98508 | R = 0.92149 | [61] |

CBR (28 days Soaked) | Percentage of RHA, percentage of lime, curing time (days), OMC and MDD | 48 | 70% | 15% | 15% | 1 | 12 | - | R = 0.9948 | R = 0.98909 | R = 0.98895 | [62] |

CBR (Soaked) | PL, LL, G_{S}, L_{S}, C_{U}, C_{C}, OMC and MDD | 72 | 70% | 15% | 15% | 1 | 8 | - | R = 0.9988 | R = 0.9996 | R = 0.9976 | [63] |

CBR (Unsoaked) | PL, LL, G_{S}, L_{S}, C_{U}, C_{C}, OMC and MDD | 72 | 70% | 15% | 15% | 1 | 17 | - | R = 0.9912 | R = 0.9993 | R = 0.9806 | [63] |

Coefficient of permeability (K) | percentage passing 0.005 mm, PI, MDD, lime percentage, pozzolan percentage, C_{d} | 69 | 70% | 15% | 15% | 1 | 9 | R = 0.9968 | R = 0.98883 | R = 0.99405 | [64] | |

Resilient Modulus (Mr) | Percentage of cement, percentage of lime, PI, percentage of silt, percentage of PFA, OMC, MC and clay | 125 | - | - | - | 1 | 9 | - | R = 0.9517 | - | R = 0.9467 | [65] |

MDD | LL, PI, LS, clay-silt ratio, sand content, lime content, cement content, asphalt content in percentage | 192 | 52% | 24% | 24% | 1 | 18 | Gradient descent momentum | R^{2} = 0.9183, MSE = 0.28%, MAE = 4.44% | R^{2} = 0.9101, MSE = 0.26%, MAE = 4.24% | [66] | |

OMC | LL, PI, LS, clay-silt ratio, sand content, lime content, cement content, asphalt content in percentage | 192 | 52% | 24% | 24% | 1 | 15 | Gradient descent momentum | R^{2} = 0.9025, MSE = 88.21%, MAE = 118.37% | R2 = 0.8916, MSE = 89.57%, MAE = 113.03% | [66] | |

MDD | G_{S}, L_{S}, free swell, D_{10}, D_{30}, D_{60,} C_{U}, C_{C}, LL, PL | 90 | 70% | 15% | 15% | 1 | 7 | R = 0.9946 | R = 0.9715 | R = 0.9754 | [67] | |

OMC | G_{S}, L_{S}, free swell, D_{10}, D_{30}, D_{60,} C_{U}, C_{C}, LL, PL | 90 | 70% | 15% | 15% | 1 | 5 | - | R = 0.9977 | R = 0.9779 | R = 0.8855 | [67] |

Network Architecture | ANN Data | References | |||||
---|---|---|---|---|---|---|---|

No of Hidden Layers | No of Neurons in Hidden Layer | ANN Input Variables | No. of Network Parameters | No. of Data Used | No. of Training Data Used | Remarks | |

1 | 9 | 8 | 91 | 283 | 198 | Sufficient | [50] |

1 | 11 | 8 | 111 | 72 | 50 | May overfit | [52] |

1 | 11 | 8 | 111 | 72 | 50 | May overfit | [52] |

1 | 11 | 8 | 111 | 72 | 50 | May overfit | [52] |

1 | 10 | 5 | 71 | 129 | 90 | Sufficient | [55] |

2 | 8 in each layer | 2 | 105 | 80 | 56 | May overfit | [56] |

1 | 50 | 14 | 801 | 216 | 172 | May overfit | [57] |

2 | 12 and 10 | 4 | 201 | 51 | 36 | May overfit | [58] |

- | - | 4 | - | 49 | 34 | No Remark | [59] |

- | - | 4 | - | 49 | 34 | No Remark | [59] |

- | - | 4 | - | 49 | 34 | No Remark | [59] |

1 | 7 | 7 | - | 210 | - | No Remark | [60] |

1 | - | 6 | - | - | - | No Remark | [61] |

1 | 12 | 5 | 85 | 48 | 34 | May overfit | [62] |

1 | 8 | 8 | 81 | 72 | 50 | May overfit | [63] |

1 | 17 | 8 | 171 | 72 | 50 | May overfit | [63] |

1 | 9 | 6 | 73 | 69 | 48 | May overfit | [64] |

1 | 9 | 8 | - | 125 | - | No Remark | [65] |

1 | 18 | 8 | 181 | 192 | 100 | May overfit | [66] |

1 | 15 | 8 | 151 | 192 | 100 | May overfit | [66] |

1 | 7 | 10 | 85 | 90 | 63 | May overfit | [67] |

1 | 5 | 10 | 61 | 90 | 63 | Sufficient | [67] |

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**MDPI and ACS Style**

Jeremiah, J.J.; Abbey, S.J.; Booth, C.A.; Kashyap, A.
Results of Application of Artificial Neural Networks in Predicting Geo-Mechanical Properties of Stabilised Clays—A Review. *Geotechnics* **2021**, *1*, 147-171.
https://doi.org/10.3390/geotechnics1010008

**AMA Style**

Jeremiah JJ, Abbey SJ, Booth CA, Kashyap A.
Results of Application of Artificial Neural Networks in Predicting Geo-Mechanical Properties of Stabilised Clays—A Review. *Geotechnics*. 2021; 1(1):147-171.
https://doi.org/10.3390/geotechnics1010008

**Chicago/Turabian Style**

Jeremiah, Jeremiah J., Samuel J. Abbey, Colin A. Booth, and Anil Kashyap.
2021. "Results of Application of Artificial Neural Networks in Predicting Geo-Mechanical Properties of Stabilised Clays—A Review" *Geotechnics* 1, no. 1: 147-171.
https://doi.org/10.3390/geotechnics1010008