1 Introduction

Soil compaction is critical in geotechnical engineering, particularly in construction projects like roads and embankments. Accurate prediction of the Maximum Dry Density (MDD) and Optimum Moisture Content (OMC) is essential to achieve the desired level of compaction. In recent years, researchers have increasingly turned to soft computing models and artificial neural networks (ANN) to improve the accuracy of such predictions. This literature review provides an overview of the existing research in this field.

Soil compaction plays a vital role in geotechnical engineering by enhancing the mechanical properties of soil. Proper compaction ensures structural integrity and prevents settlement and instability [18].

Historically, compaction characteristics were predicted using empirical and semi-empirical methods, such as the Proctor test. These methods, while practical, often lack the accuracy and flexibility required for various soil types and conditions [57].

Soft computing models, including artificial neural networks (ANN), fuzzy logic, and genetic algorithms, have gained popularity for predicting soil compaction characteristics. These models can handle complex, non-linear relationships within large datasets [116].

ANN is a popular choice for predicting soil properties due to its ability to learn from data and adapt to various soil types. Studies have successfully applied ANN to predict MDD and OMC from soil index properties [83].

Previous research has shown that the accuracy of predictions is highly dependent on the selection of input parameters. Parameters like gravel (G), sand (S), plastic limit (PL), liquid limit (LL), and plasticity index (PI) have been identified as significant predictors of compaction characteristics.

Researchers often evaluate the performance of predictive models using various statistical metrics, including R-squared (R2), root mean squared error (RMSE), mean absolute error (MAE), and scatter index (SI). These metrics help assess the accuracy and reliability of the models.

Recent studies have focused on enhancing the efficiency and accuracy of predictive models. Some have incorporated advanced optimization techniques and ensemble modeling to improve results.

The choice of an appropriate model is essential and varies based on the specific application. ANN may perform well in some cases, while linear or multilinear regression models may be more suitable in others. Understanding each model's strengths and limitations is crucial.

Accurate predictions of MDD and OMC from soil index properties have practical implications in geotechnical engineering, enabling engineers to optimize construction processes, reduce costs, and enhance the quality and durability of infrastructure projects. There are several methods to stabilize soils. One of the most significant methods to densify the soil is compaction. Soil compaction results in the decrease of the voids between soil particles. Consequently, soil shear strength observably increases along with the declining compressibility and pearmeability of the soil [33]. To determine both compaction characteristics, Maximum Dry Density (MDD) and Optimum Moisture Content(OMC), in the laboratory, either the Standard Proctor (SP) or Modified Proctor (MP) methods are used [26].

To assess the suitability of a soil for an earthwork, the compaction characteristics of the soil must be identified. For an earthwork project, large quantities of soil are necessitated. Obtaining the massive volume with a desired compaction characteristic from a single borrow source might be tedious and time-consuming. Compaction characteristics must be obtained from a laboratory compaction test to determine the suitability of soils collected from various borrows sources. Nevertheless, for laboratory compaction tests, sufficient effort and time are required. Consequently, for any such project, the suitability of the required soils is preliminarily assessed by establishing correlations between the compaction characteristics and simple physical properties obtained through simple index tests [104]. There have been several attempts to determine the compaction characteristics indirectly. For this reason, several correlations have been developed to estimate the compaction characteristics through soil index properties [7, 23, 28, 31, 44, 51, 52, 89, 104, 117] and because of soil fractions [45, 99]. Some studies indicated that an individual input parameter could not sufficiently predict the compaction characteristics. Therefore, multiple linear regression (MLR) models were utilized to predict the compaction characteristics considering different basic soil parameters [9, 24, 26, 45, 49, 67, 98]. In addition to this, the machine learning technique was used in some other studies to develop more accurate correlations [48, 79, 101, 110]. An early study employing artificial neural networks (ANN) by Shook and Fang [97] to predict the soil compaction characteristics indicated that the ANN technique outperforms the traditional statistical models, and a reliable prediction can be attained. Further, Sivrikaya and Soycan [100] utilized the ANNs for estimating the compaction characteristics of 108 fine-grained soil samples.

Sivrikaya [98] performed the multilinear regression model (MLR) to predict the compaction characteristics of fine-grained soils from soil index properties and soil gradations. The considered input parameters were combinations of gravel content (G), sand content (S), fine-grained content (F), plasticity index (PI), liquid limit (LL), and plastic limit (PL). The study concluded that the compaction characteristics could correlate with Plastic limit well compared to other index properties. [31] employed different techniques: simple—multiple analysis and artificial neural networks to forecast the compaction characteristics based on the soil gradations. The study highlights that reliable correlations (R2 = 0.70–0.95) for preliminary design can be achieved from both techniques. Furthermore, [67] developed multiple regression analysis models for experimental data collected from 110 sandy soils to predict the compaction characteristics in terms of the uniformity coefficient (Cu) and Compaction Energy (CE). Considering the liquid limit (LL), the plasticity index (PI), and compaction energy (CE). Tenpe and Kaur [108] Investigated artificial neural network (ANN) modeling performance for forecasting compaction parameters based on soil index properties. Moreover, Omar et al. [77] utilized advanced mathematical models and an innovative solution to predict the compaction characteristics of fine-grained soils from numerous physical properties. In this study, multiple linear regression (MLR), artificial neural networks (ANNs), and support vector regression (SVR) are employed. In addition, Farooq et al. [26] utilized a multiple regression model to forecast the optimum moisture content. Also, Saikia et al. [89] developed a set of regression models for predicting the compaction characteristics concerning the consistency limits.

In their study, Karimpour-Fard et al. [53] utilized artificial neural networks (ANNs) and multilinear regression (MLR) techniques across 728 datasets to develop predictive models for compaction characteristics. These models were designed to estimate compaction characteristics based on soil type, grain size distribution (including LL, PL), and specific gravity at varying energy levels. Their findings emphasized the significant impact of fine content on compaction characteristics compared to other factors. Additionally, they underscored the potential advantages of MLR models in predicting compaction characteristics despite the greater effectiveness of ANN models, attributing this to the inherent "black box" nature of ANN models. Taking into account input parameters such as compaction energy and plastic limit, these models were evaluated.

A novel application of artificial neural network (ANN) was utilized in a recent study Verma and Kumar [110] to estimate fine-grained soil's modified Proctor compaction parameters. From a highway construction work site, from 532 in situ soil samples, several geotechnical parameters were obtained from the laboratory testing. Besides the index properties test, modified Proctor compaction tests were conducted on the collected soil samples. Python V3.7.9 platform was adopted to write the ANN algorithm code for the analysis. Such useful geotechnical parameters as gravel (%), sand (%), fine content (FC), and percent material retained on 2.0 mm (R2.0 mm), 0.425 mm (R0.425 mm), and 0.075 mm (R0.075 mm), coarse sand (CS), medium sand (MS), fine sand (FS), liquid limit (LL), plastic limit (PL), and plasticity index (PI) were considered as input parameters. Verma and Kumar [111] focused on developing a multi-layer perceptron neural network model to predict the modified compaction characteristics of coarse and fine-grained soils. Similarly, the Python V3.7.9 platform was adopted to write artificial neural network (ANN) algorithm code. One hundred seventy-nine datasets of coarse-grained and 69 datasets of fine-grained soils are examined, considering gravel (%), sand (%), FC (%), LL (%), PL (%), PI (%) as input parameters to predict the modified compaction characteristics. In the developed models, the high correlation coefficient (R was obtained; as the value was more than 0.80 and 0.90 for coarse-grained and fine-grained soil respectively.

In the context of existing literature, various statistical and machine learning models were employed to forecast compaction characteristics based on diverse soil properties, primarily for assessing soil suitability in earthworks. Statistical models offer ease of use, delivering output predictions in equations that can prove valuable for practical field applications. On the other hand, machine learning models excel in processing substantial datasets and identifying nuanced trends and patterns that might elude human perception. Machine learning algorithms exhibit exceptional capability in handling complex, multi-dimensional, and multifaceted data, even in dynamically changing and uncertain conditions, as demonstrated by studies conducted by Mahmood et al. [61], Piro et al. [82], Abdalla and Mohammed [2], and Hama Ali [34]. These models have been employed in a few research studies to predict compaction characteristics.

Furthermore, in rare studies, large quantities of data covering soil index properties and particle—size were considered. In this work, a large volume of the dataset (2162 datasets) has been compiled from previous studies. Four different models of Linear Regression, Nonlinear Regression (NL), Multilinear Regression (ML), and Artificial Neural Network (ANN) have also been utilized to predict the compaction characteristics (OMC, MDD) concerning soil index properties and soil particle—sizes. In addition to predicting the compaction characteristics, this work evaluates the performance of the utilized models.

1.1 Related studies

Using various modeling techniques, the literature review provides valuable insights into predicting soil compaction characteristics, specifically the Maximum Dry Density (MDD) and Optimum Moisture Content (OMC). Below are the essential findings and trends observed in the related studies: soil compaction is a critical aspect of geotechnical engineering, ensuring the structural integrity of construction projects such as roads and embankments. Accurate predictions of MDD and OMC are essential for achieving the desired level of compaction. Researchers have increasingly turned to soft computing models, including Artificial Neural Networks (ANN), to improve the accuracy of predictions. These models are valued for handling complex, non-linear relationships within large datasets. Traditional methods, such as the Proctor test, were historically used for predicting compaction characteristics. However, these methods often lack accuracy and flexibility across different soil types and conditions.The accuracy of predictions depends significantly on the selection of input parameters. Parameters like gravel content, sand content, plastic limit, liquid limit, and plasticity index have been identified as significant predictors of compaction characteristics. Researchers employ various statistical metrics, including R-squared (R2), root mean squared error (RMSE), mean absolute error (MAE), and scatter index (SI), to evaluate the performance of predictive models. These metrics help assess the accuracy and reliability of the models. Recent studies have focused on enhancing the efficiency and accuracy of predictive models. Some have incorporated advanced optimization techniques and ensemble modeling to improve results. Choosing the appropriate model is essential and depends on the specific application. While ANN performs well in some cases, linear or multilinear regression models may be more suitable in others. Understanding the strengths and limitations of each model is crucial. Accurate predictions of MDD and OMC from soil index properties have practical implications in geotechnical engineering, allowing engineers to optimize construction processes, reduce costs, and enhance the quality and durability of infrastructure projects. Some studies have employed machine learning techniques, such as ANNs, to predict soil compaction characteristics, with promising results regarding model performance and accuracy. Large datasets and model evaluation: the work under review stands out for its use of a large dataset (2162 datasets) and for applying four different models, including ANN, to predict compaction characteristics. The study not only predicts these characteristics but also rigorously evaluates the performance of the utilized models. Comparison of soft computing models and statistical models: the literature reveals a trend of comparing soft computing models (e.g., ANN) with traditional statistical models (e.g., linear regression) to determine which one offers better predictive capabilities for soil compaction characteristics. Several studies have indicated that relying on a single input parameter is insufficient for accurate predictions. Hence, models that consider multiple basic soil properties, especially those related to particle size distribution and consistency limits, have been developed and found to provide more reliable predictions. In summary, the reviewed literature emphasizes the importance of accurate predictions of MDD and OMC for soil compaction in geotechnical engineering. Soft computing models, particularly ANNs, have shown promise in improving the accuracy of predictions. The choice of input parameters, model evaluation metrics, and the practical implications of these predictions are significant considerations. Additionally, recent studies have incorporated advanced techniques to enhance the efficiency and accuracy of predictive models. The field is continually evolving, with an increasing focus on harnessing the potential of machine learning and large datasets for more precise predictions.

2 Scope of the work

In light of the literature, an individual input parameter is unlikely to be applicable to predict the compaction characteristics of different soil types. Therefore, numerous datasets from the literature were compiled to develop models to predict the compaction characteristics of soil by including simple soil properties such as the G, S, F, LL, PL%, and PI%. Subsequently, four different models were developed. This enables evaluating the models' performance and the identification of the effects of various soil properties. Consequently, the datasets in two groups of training and testing are examined in the models to reach the main objectives of the study:

  1. (i)

    The influence of the physical soil properties on the compaction characteristics will be investigated. The input parameter with the most influential role in measuring the value of the OMC and MDD will be assigned from a sensitivity analysis.

  2. (ii)

    Concerning the statistical evaluation tools, the model outperforming the prediction of the compaction characteristics of soil from basic soil properties will be determined and compared with the other models.

The major contributions and novelty in this work can be summarized as follows:

  • Multi-parameter approach: the study's innovation lies in adopting a multi-parameter approach to predict the compaction characteristics of soils. Instead of relying on a single input parameter, it incorporates six different parameters: gravel (G), sand (S), fine content (F), plastic limit (PL), liquid limit (LL), and plasticity index (PI). This approach recognizes that a single parameter may not be universally applicable across different soil types, and therefore, a more comprehensive set of input variables is considered.

  • Model Diversity: The work employs four distinct models: artificial neural network (ANN), nonlinear regression (NLR), linear regression (LR), and multilinear regression (MLR) models. This diversity in modeling techniques allows for a robust assessment of each model's predictive capabilities and performance, thus ensuring a comprehensive evaluation of the data.

  • Extensive dataset compilation: to support the development of these models, a comprehensive dataset of 2162 entries is compiled, encompassing various soil gradation and plasticity properties as input variables. This dataset's size and comprehensiveness contribute to the reliability and generalizability of the models.

  • Statistical evaluation tools: the study employs a range of statistical evaluation tools, including the coefficient of determination (R2), scatter index (SI), root mean squared error (RMSE), mean absolute error (MAE), a20-index, and Objective (OBJ) value. These tools provide a thorough assessment of the models' performance and ability to predict compaction characteristics accurately.

  • Identification of influential parameters: through sensitivity analyses, the study identifies the specific input parameter with the most significant influence on measuring the values of Optimum Moisture Content (OMC) and Maximum Dry Density (MDD). This insight can be crucial for understanding the key factors affecting soil compaction characteristics.

  • Comparative analysis: the research develops these models and rigorously compares their performance. It concludes that the ANN model excels in forecasting OMC, while for MDD prediction, the MLR and LR models outperform the ANN model. This comparative analysis aids in selecting the most appropriate model for specific applications also, Given the study's objective to streamline the assessment of soil compaction characteristics for constructing dams and roads, along with the use of multiple parameters and models, a novel method could involve the following approaches:

  • Hybrid model integration: combine the strengths of different models to improve prediction accuracy. For instance, a hybrid model that integrates the advantages of artificial neural networks (ANN) in capturing complex relationships with the interpretability of linear regression (LR) or multilinear regression (MLR) could be developed. This hybrid model could potentially enhance predictions by leveraging the ANN's ability to capture nonlinear patterns and the interpretability of LR/MLR for a better understanding of the impact of individual parameters.

  • Feature selection techniques: utilize advanced feature selection methods to identify the most influential parameters. Techniques like recursive feature elimination, feature importance ranking from tree-based models, or Lasso regularization can help narrow down the most critical parameters affecting soil compaction characteristics. This could streamline the assessment process by focusing on key parameters and reducing resource consumption.

  • Ensemble learning: implement ensemble techniques such as bagging, boosting, or stacking to combine predictions from multiple models. Ensemble methods often lead to better generalization and robustness by leveraging the strengths of different models. This approach could enhance prediction accuracy for Optimum Moisture Content (OMC) and Maximum Dry Density (MDD).

  • Sensitivity analysis optimization: utilize sensitivity analysis to identify influential parameters and optimize the assessment process. By understanding the impact of different parameters on OMC and MDD, the study can propose specific guidelines or thresholds that indicate when certain parameters have a significant effect. This could help streamline decision-making in soil compaction assessments.

  • Automated Data Processing: Implement automation in data processing and model selection. Leveraging machine learning pipelines or frameworks can streamline data preprocessing, feature engineering, model training, and evaluation, reducing the time and resources needed for analysis.

Combining the strengths of different modeling techniques, optimizing parameter selection, leveraging ensemble methods, and automating processes can streamline the assessment of soil compaction characteristics, making it more efficient and less resource-intensive while maintaining or improving prediction accuracy.

2.1 Components in the block diagram

  • Input data: this represents the dataset containing soil properties such as Gravel (G), Sand (S), Fine content (F), plastic limit (PL), liquid limit (LL), and plasticity index (PI).

  • Preprocessing module: this stage involves data cleaning, normalization, and feature scaling to prepare the input data for modeling.

  • Modeling phase:

  • LR model: linear regression model based on soil physical properties.

  • NLR model: nonlinear regression model.

  • MLR model: multilinear regression model.

  • ANN model: artificial neural network model.

  • Evaluation and comparison: this part includes the assessment of model performance using various metrics like R-squared (R2), Root Mean Squared Error (RMSE), Mean Absolute Error (MAE), Scatter Index (SI), and Objective (OBJ) values.

  • Results and analysis: this section interprets and analyzes the model outputs, compares predictions, and identifies influential parameters for OMC and MDD.

  • Sensitivity analysis module: this component pinpoints the significant parameters affecting OMC and MDD predictions.

  • Conclusion and recommendations: this stage presents conclusions drawn from the analysis, recommendations for model selection, and potential areas for further research.

3 Methodology

Overall, 2162 datasets from the literature were gathered. The data were randomly mixed and divided into two groups: training datasets and testing datasets. The training datasets comprised 70% of the dataset, while 30% was for the testing ones. The training data groups were used to develop the models. The foremost objective of the models was to predict the standard Proctor compaction characteristics (OMC and MDD). Then, using the testing data, the models were evaluated. Table 1 includes the number of used data from different studies and ranges of the input parameters: gravel content (G %), sand content (S %), fine content (F %), liquid limit (LL %), plastic limit (PL %) and plasticity index (PI %).

Table 1 The range and the number of the used datasets
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Further, in the table, the ranges of the measured values of optimum moisture content (OMC %) and maximum dry density (MDD kN/m3) are contained within, compared with the predicted values obtained from the models. These input parameters are used to develop the models and the actual values of output parameters. The procedure of the work is illustrated in a flowchart in Fig. 1 and summarized as:

Fig. 1
figure 1

The procedure of the study by a flow chart diagram

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In total, 2162 datasets were compiled from the available literature. These datasets were randomly combined and then divided into two sets: one for training the models, consisting of 70% of the data, and the other for testing the models, which accounted for the remaining 30%. The primary aim of these models was to forecast the standard Proctor compaction characteristics, specifically the Optimum Moisture Content (OMC) and Maximum Dry Density (MDD). Subsequently, the models' performance was assessed using the testing data. Table 1 presents the quantity of data sourced from various studies and the ranges of the input parameters, including gravel content (G %), sand content (S %), fine content (F %), liquid limit (LL %), plastic limit (PL %), and plasticity index (PI %).

Furthermore, Table 1 provides a rangeds of measured values of OMC (%) and MDD (kN/m3), with which the obtained values from the models will be compared later. These input parameters are the foundation for model development, while the actual output parameter values are also included. The workflow of the study is visually represented in Fig. 1 and can be summarized as follows:

  • Stage 1: Collecting data.

  • Stage2: Correlating input and output parameters.

  • Stage 3: Splitting data into two groups: 70% training and 30% testing.

  • Stage 4: Developing different models.

  • Stage 5: Evaluating the performance of the models.

4 Correlation between soil index properties and compaction parameters

The illustration demonstrates the validity of the connection between input parameters and compaction characteristics. Each input parameter was graphically depicted and scrutinized concerning OMC and MDD. Figure 2 presents a matrix plot elucidates the link between input parameters and compaction characteristics. The figures unmistakably reveal that, except for the correlation between PL and OMC, with an R-value of 0.82 (as shown in Fig. 3), there are generally insufficient correlations between compaction characteristics and the input parameters. This particular correlation can be considered robust but may apply primarily to fine-grained soil. In other words, PL may not accurately represent the soil's true properties for soils where the percentages of G (gravel) and S (sand) dominate. Moreover, it is worth noting that some types of purely coarse-grained soil may not possess a plastic limit (PL) at all.

Fig. 2
figure 2

Matrix Plot between the input parameters and the compaction characteristics

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Fig. 3
figure 3

Correlation Matrix between input and output parameter

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Additionally, Fig. 4 illustrates the typical distribution of OMC and MDD. Moreover, Table 2 presents various statistics, including minimum, maximum, mean, standard deviation, skewness, kurtosis, and variance. In the context of kurtosis, a strongly negative value suggests shorter tails compared to a normal distribution, while a positive value indicates longer tails. As for the skewness parameter, a negative value signifies an extended left tail, while a positive value signifies an extended right tail.

Fig. 4
figure 4

Normal distribution of the compaction characteristics

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Table 2 Statistical parameters for input and output variables
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5 Modeling

In the above figures, each parameter of G, S, F, LL PL, and PI is connected to the OMC and MDD so that a direct relationship between these parameters can be developed to predict the compaction characteristics from one of these parameters. However, because the R2 value is extremely low, a reliable direct correlation between the compaction characteristics and the input parameters cannot be attained, following the statistical analysis and Figs. 2, 3 and 4. Therefore, four separate models are developed considering the influence of soil gradation and index properties to overcome this shortcoming and establish a reliable correlation to forecast OMC and MDD from the input parameters.

In this work, the models are utilized to predict the OMC and MDD, and each model's performance is evaluated according to the original data. The following evaluation criteria are used to evaluate and compare the performance of the four models: For a model to be scientifically accurate, there should be a minor percentage error between observed and predicted data (higher a20 – index) and a higher R2 value with a lower RMSE, Objective (OBJ), MAE, and SI values (Figs. 5, 6, 7, 8, 9, 10, 11, 12).

Fig. 5
figure 5

Comparison between measured and predicted values of a OMC and b MDD for LR model: training data and testing data

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Fig. 6
figure 6

Comparison between measured and predicted values of a OMC and b MDD for NLR model: training data and testing data

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Fig. 7
figure 7

Comparison between measured and predicted values of a OMC and b MDD for MLR model: training and testing data

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Fig. 8
figure 8

The architecture of the used ANN models a OMC and b MDD

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Fig. 9
figure 9

Comparison between measured and predicted values of a OMC and b MDD for ANN model: training data and testing data

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Fig. 10
figure 10

The a20-index values of a OMC and b MDD using all data

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Fig. 11
figure 11

The OBJ values of a OMC, b MDD for all the models

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Fig. 12
figure 12

The OBJ values of a OMC and b MDD for all developed models

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Complexity analysis of Multiple Linear Regression (MLR) is relatively straightforward compared to more complex models like neural networks. MLR is a simple, linear modeling technique to establish relationships between dependent and multiple independent variables. Here are the key aspects to consider when analyzing the complexity of an MLR model:

5.1 Model complexity

MLR models are linear, assuming a linear relationship between the dependent and independent variables. This simplicity contributes to low model complexity.

The complexity of MLR, LR, and NLR models increases with the number of independent variables (features) included in the model. Adding more components can lead to a more complex model.

Each independent variable has a coefficient (weight) associated with it in an MLR model. The number of independent variables determines the total number of parameters in the model. As the number of independent variables grows, so does the number of model parameters.

MLR, LR, and NLR can become more complex if interactions (cross-product terms) or higher-order terms (quadratic or cubic terms) are included in the model. These additions increase the number of parameters and model complexity.

Increasing model complexity by adding more features or terms can lead to overfitting. Overfitting occurs when the model fits the noise in the data rather than the underlying patterns. Complexity should be carefully balanced to avoid overfitting. Proper variable selection is crucial to managing complexity. Including irrelevant or redundant variables can increase model complexity without improving predictive accuracy.

Techniques like L1 (Lasso) and L2 (Ridge) regularization can be applied to penalize large coefficient values to control model complexity and reduce overfitting.

Simplicity in an MLR model can enhance its interpretability. Complex models with numerous variables and interactions can be challenging to interpret.

MLR models are computationally efficient and require minimal memory and processing power resources, making them suitable for large datasets.

Model complexity should be balanced with model performance. Evaluation metrics like R-squared (R2), adjusted R-squared, and RMSE can help assess model fit and complexity.

Multiple Linear Regression is a relatively simple and interpretable modeling technique. Its complexity primarily depends on the number of features and the inclusion of interactions or higher-order terms. Proper variable selection and regularization techniques are essential for managing complexity and preventing overfitting. MLR suits situations where a straightforward linear relationship between variables is sufficient, and model interpretability is a priority.

MLR is straightforward, interpretable, and computationally efficient. It is well-suited when you clearly understand the relationships between variables and want a simple model that can provide insight into how each independent variable affects the dependent variable.

NLR is employed when the relationship between variables is nonlinear, meaning that the effect of an independent variable is not constant but varies across the range of the variable.

NLR allows more flexibility in modeling complex, nonlinear relationships. It can capture relationships that MLR cannot. NLR can provide more accurate predictions when the data suggests a nonlinear relationship.

LR is a simplified form of MLR with only one independent variable. It's used when you want to model the relationship between a dependent variable and a single independent variable linearly.

Why Use LR: LR is helpful when you're interested in understanding the impact of a single variable on the dependent variable. It's a basic and interpretable model for simple relationships.

ANNs are used when dealing with complex, nonlinear relationships involving many independent variables. ANNs can capture intricate patterns and interactions within the data.

Researchers and data scientists often start with simpler models like MLR or LR to understand the data and establish a baseline. If the relationships are nonlinear or involve complex interactions, NLR or ANN might be more appropriate. Model selection is often an iterative process involving testing multiple models to determine which best fits the data and produces accurate predictions.

5.2 Linear regression model

The linear regression model (LR) [7, 31, 32, 34, 35, 44], as demonstrated in Eq. (1), is the most common technique for predicting the compaction characteristics of soils:

$${\text{OMC}},{\text{ MDD}} = {\text{a}} + {\text{bX}}$$
(1)

A and b are constants, and x might be one of the G, S, F, LL, PL, or PL. The previous formula does not include the combination of other variables that might affect OMC and MDD, including particle size and soil plasticity. Equation (2) is presented to combine all different parameters and factors that might influence OMC and MDD to obtain more reliable scientific findings.

$${\text{OMC}},{\text{ MDD}} = {\text{a}} + {\text{bG}} + {\text{cS}} + {\text{dF}} + {\text{eLL}} + {\text{fPL}} + {\text{gPI}}$$
(2)

where G is gravel content (%), S is sand content (%), F is fine content (%), LL is the liquid limit, PL is the plastic limit, and PI is the plasticity index.

Moreover, the model parameters are a, b, c, d, e, f, and g. All variables can be changed linearly. Hence Eq. (2) can be considered an expansion of Eq. (1). This combination might not always be the case as all factors are unlikely to affect the compaction characteristics and interact with one another. Consequently, the model should be updated often to accurately estimate the OMC and MDD [45, 76, 99].

5.3 Nonlinear regression model

Generally, to develop a nonlinear model, the following Eq. (3) can be applied [39]. To forecast the OMC and MDD, the connection among various variables in Eqs. (1) and (2) can be expressed in Eq. (3):

$${\text{OMC}},{\text{ MDD}} = {\text{a}} + {\text{b}}({\text{G}})^{{\text{c}}} + {\text{d}}({\text{S}})^{{\text{e}}} + {\text{f}}({\text{F}})^{{\text{g}}} + {\text{h}}({\text{LL}})^{{\text{i}}} + {\text{j}}({\text{PL}})^{{\text{k}}} + {\text{l}}({\text{PI}})^{{\text{m}}}$$
(3)

where G stands for gravel content (%), S stands for sand content (%), F stands for fine content (%), LL stands for liquid limit, PL stands for plastic limit, and PI stands for plasticity index. Moreover, the model parameters are a, b, c, d, e, f, g, h, I, j, k, l, and m, calculated based on the least square method.

5.4 Multiple linear regressions

The MLR, also a regression procedure, can be employed when the expected variable has a parameter greater than two stages. MLR is a statistical approach that is comparable to multiple linear regressions.

Equation (4) can be utilized to find the variance among predictable and independent variables.

$${\text{OMC}},{\text{ MDD}} = {\text{a}}({\text{G}})^{{\text{b}}} ({\text{S}})^{{\text{c}}} ({\text{F}})^{{\text{d}}} ({\text{LL}})^{{\text{e}}} ({\text{PL}})^{{\text{f}}} ({\text{PI}})^{{\text{g}}}$$
(4)

If soil is purely coarse-grained (F% = 0) or fine-grained (G% + S% = 0), Eq. (4) cannot be applied to forecast the compaction characteristics. Therefore, there is a limitation in applying this model as it cannot represent the real formation of the soil. The least-square method was implemented to find the parameters (a, b, c, d, e, f, and g) and model variables.

The datasets are divided into two groups for the above models: Training and Testing. 70% of the datasets are used for training, while 30% go to the testing phase.

5.5 ANN model

ANN is a robust simulation software developed for data analysis and computing to process and analyze information similarly to a human brain. Construction engineering frequently uses this machine learning method to forecast the future behavior of numerous numerical issues [110, 111]. The ANN model has three primary layers: input, hidden, and output. Depending on the intended problem, each input and output layer may consist of one or more layers. The hidden layer is often extended for two or more layers. The objective of the designed model and the data collected often decide the input and output layers, while the rating weight, transfer function, and the bias of each layer toward other layers determine the hidden layer. Based on a combination of proportions, weight/bias, and various parameters, including (G, S, F, LL, PL, and PI) as inputs, a multi-layer feed-forward network is constructed, and output ANN here is either the OMC or MDD. For designing the network architecture, no standard approach is available. The number of hidden layers and neurons is determined based on the trial and error test. The prime goal of the training process of the network is to achieve the optimized number of iterations (epochs) from which the minimum mean absolute error (MAE), root means square error (RMSE), and the high R2-value are provided. Several studies have examined the influence of iteration on reducing the MAE and RMSE. In order to prepare for the designed ANN, the obtained dataset (a total of 2162 data) has been separated into two groups. Approximately 70% of datasets were utilized as trained data for training the network. For the testing data, 30% of the overall data was used for the trained network [34]. The constructed ANN was trained and tested for several hidden layers to select the best network structure based on the compatibility of the predicted compaction characteristics with the obtained data, for predicting OMC, the best-trained network that offers the highest R2 and the lowest MAE and RMSE as the ANN structure with one hidden layer, eight neurons, and a hyperbolic tangent transfer function (as illustrated in Fig. 13, Table 3). As far as the prediction of MDD is concerned, the best-trained network was three hidden layers, five neurons, and a hyperbolic tangent transfer function.

Fig. 13
figure 13

The R2 values of a OMC and b MDD for all four models

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Table 3 The tested ANN architecture for OMC
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Analyzing the complexity of an Artificial Neural Network (ANN) involves considering several factors related to its architecture and operation. Below the following is the breakdown of the key aspects to consider when conducting a complexity analysis of an ANN:

  1. i.

    Number of layers: the number of layers in the network, including input, hidden, and output layers, significantly impacts complexity. Deeper networks are generally more complex.

  2. ii.

    Layer size: the number of neurons or units in each layer, especially in the hidden layers, affects model complexity.

  3. iii.

    Connectivity: the extent of connections between neurons, also known as the network's topology, plays a role in complexity.

  4. iv.

    Number of parameters: the number of weights and biases in the network. The total parameters in an ANN increase with the number of neurons and layers.

  5. v.

    Input features: the dimensionality of the input data contributes to the complexity. Higher-dimensional input data require larger networks.

  6. vi.

    Activation functions: each layer's choice of activation functions can influence complexity. Non-linear activation functions can add to the complexity of the model.

  7. vii.

    Training algorithm: the algorithm used for training the ANN can impact complexity. Some training algorithms, like backpropagation, require more iterations to converge.

  8. viii.

    Hyperparameters: the settings of hyperparameters, such as learning rate, batch size, and dropout rate, can impact training time and model complexity.

  9. ix.

    Regularization techniques: techniques like L1 and L2 regularization can add complexity to the model by penalizing large weights.

  10. x.

    Pruning: pruning techniques can reduce complexity by eliminating unnecessary connections and neurons after training.

  11. xi.

    Parallelization: distributed training across multiple GPUs or CPUs can change the complexity of the training process.

  12. xii.

    Complexity metrics: several complexity metrics, like FLOPs (floating-point operations), are used to quantitatively measure the computational complexity of ANN models.

  13. xiii.

    Overfitting: the potential for overfitting increases with the complexity of the model. Complexity can lead to the model fitting noise in the data rather than the underlying patterns.

Equations (5)–(7) display the ANN model's General Equation.

From linear node 0:

$${\text{OMC}},{\text{ MDD}} = {\text{Threshold}} + \left( {\frac{{{\text{Node }}1}}{{1 + {\text{e}}^{{ - {\text{B}}1}} }}} \right) + \left( {\frac{{{\text{Node }}2}}{{1 + {\text{e}}^{{ - {\text{B}}2}} }}} \right) + \cdots$$
(5)

From sigmoid node 1:

$${\text{B}}1 = {\text{Threshold}} + \sum \left( {\text{Attribute*Variables}} \right)$$
(6)

From sigmoid node 2:

$${\text{B}}2 = {\text{Threshold}} + \sum \left( {\text{Attribute*Variables}} \right)$$
(7)

5.6 Pseudocode/algorithm

figure a

6 Model assessment tools

The suggested models were evaluated using a variety of metrics, including coefficient of determination (R2), a20-index, scatter index (SI), Objective (OBJ), root mean squared error (RMSE), and mean absolute error (MAE), which may be calculated using the formulas below:

$${\text{R}}^{2} = \left( {\frac{{\mathop \sum \nolimits_{{{\text{p}} = 1}}^{{\text{p}}} \left( {{\text{yi}} - {\text{yi}}^{\prime}} \right)\left( {{\text{yp}} - {\text{yp}}^{\prime}} \right)}}{{\sqrt {\left[ {\mathop \sum \nolimits_{{{\text{p}} = 1}}^{{\text{p}}} \left( {{\text{yi}} - {\text{yi}}^{\prime}} \right)^{2} } \right]\left[ {\mathop \sum \nolimits_{{{\text{p}} = 1}}^{{\text{p}}} \left( {{\text{yp}} - {\text{yp}}^{\prime}} \right)^{2} } \right]} }}} \right)^{2}$$
(8)
$${\text{RMSE}} = \sqrt {\frac{{\mathop \sum \nolimits_{{{\text{p}} = 1}}^{{\text{p}}} \left( {{\text{yp}} - {\text{yi}}} \right)^{2} }}{{\text{p}}}}$$
(9)
$${\text{MAE}} = \frac{{\mathop \sum \nolimits_{{{\text{p}} = 1}}^{{\text{p}}} \left| {{\text{yp}} - {\text{yi}}} \right|}}{{\text{p}}}$$
(10)
$${\text{a}}20 - {\text{index}} = \frac{{{\text{m}}20}}{{\text{N}}}$$
(11)
$${\text{SI}} = \frac{{{\text{RMSE}}}}{{{\text{y}}^{\prime}}}$$
(12)
$$\begin{aligned} {\text{OBJ}} & = \left( {\frac{{{\text{n}}_{{{\text{tr}}}} }}{{{\text{n}}_{{{\text{all}}}} }}{*}\frac{{{\text{RMSE}}_{{{\text{tr}}}} + {\text{MAE}}_{{{\text{tr}}}} }}{{{\text{R}}_{{{\text{tr}}}}^{2} + 1}}} \right) + \left( {\frac{{{\text{n}}_{{{\text{tst}}}} }}{{{\text{n}}_{{{\text{all}}}} }}{*}\frac{{{\text{RMSE}}_{{{\text{tst}}}} + {\text{MAE}}_{{{\text{tst}}}} }}{{{\text{R}}_{{{\text{tst}}}}^{2} + 1}}} \right) \\ \quad + \left( {\frac{{{\text{n}}_{{{\text{val}}}} }}{{{\text{n}}_{{{\text{all}}}} }}{*}\frac{{{\text{RMSE}}_{{{\text{val}}}} + {\text{MAE}}_{{{\text{val}}}} }}{{{\text{R}}_{{{\text{val}}}}^{2} + 1}}} \right) \\ \end{aligned}$$
(13)

From the formulas above, yp and yi are the expected and actual values of the path pattern, and yp’ and yi’ are the averages of the actual and forecasted values. Training, testing, and validating datasets are tr, tst, and val, respectively; the number of patterns (collected data) in the associated dataset is denoted as n. In contrast to R2, which has an optimal value of one, the other evaluating factors have optimal values of zero. About the SI parameter, a model performs poorly when it is [0.3, fairly when it is between 0.2 and 0.3, well when it is between 0.1 and 0.2, and very well when it is < 0.1. To assess the effectiveness of the recommended models, the OBJ parameter was also used as an integrated performance parameter in Eq. (12). m20 is the ratio of experimental to the predicted value, which varies between 0.8 and 1.2, and H is the total number of data samples. The prime benefit of the a20-index is that the model predicts values with a deviation of ± 20% compared to actual values. In order to graphically show how each model overestimates and underestimates the expected results of OMC and MDD compared to the actual values from the experiments, positive and negative error margin lines were added to the model findings. A positive value is represented for the overestimated percentage of OMC and MDD, while a negative value means the underestimated percentage of OMC and MDD.

7 Results and analysis

7.1 The LR model

Figure 5 exhibits the relation between actual and estimated OMC and MDD for all training and testing datasets. In order to develop the model, 2162 data sets have been utilized. Concerning the developed equations, the two plasticity parameters of LL and PI play a substantial role in determining OMC and MDD. By optimizing the sum of error squares and the least square approach, which was performed in Excel using Solver to obtain the ideal value for the Equation, the present model’s weight of each parameter on the OMC and MDD was found. The following are the Equations for the LR model with various weight parameters [Eqs. (13) and (14)]:

$${\text{OMC}} = 0.96 - 0.05{\text{G}} - 0.04{\text{S}} + 0.03{\text{F}} + 0.72{\text{LL}} + 0.07{\text{PL}} - 0.74{\text{PI}}$$
(14)
$${\text{MDD}} = 27.13 - 0.025{\text{G}} - 0.045{\text{S}} - 0.073{\text{F}} - 0.215{\text{LL}} + 0.04{\text{PL}} + 0.216{\text{PI}}$$
(15)

As seen in the above Equations, among other input parameters, PI can have more influence on lowering the value of OMC; on the contrary, MDD value declines with the increase of LL. This seems to be consistent with the experimental findings reported in the literature that soil with higher LL has a lower MDD value. The obtained R2, RMSE, and MAE assessment parameters for the OMC are 0.82, 3.99, and 2.98%, respectively. For the MDD, R2, RMSE, and MAE are 0.76, 1.05, and 0.84 kN/m3, respectively. Furthermore, as shown in Figs. 11 and 12, the current model’s OBJ and SI values for OMC for the training dataset are 3.69 and 0.23, respectively, while these values are 1.07 and 0.062, correspondingly, for MDD.

7.2 NLR model

The NLR model is another model that might be interesting to examine for its performance among other models. Figure 6a, b depict the compaction characteristics values for predicted and real data obtained from the literature for training and testing datasets. According to the developed model, F plays a considerable role in changing the value of both OMC and MDD. The following are the suggested formulas for the NLR model with various variable parameters [Eqs. (15) and (16)]:

$$\begin{aligned} {\text{OMC}} & = 777.3 + 2.13({\text{G}})^{ - 0.213} + 0.000028({\text{S}})^{ - 6.31} - 784.6\left( {\text{F}} \right)^{ - 0.003} \\ \quad - 0.137\left( {{\text{LL}}} \right)^{0.93} + 0.144({\text{PL}})^{1.38} + 2\left( {{\text{PI}}} \right)^{0.47} \\ \end{aligned}$$
(16)
$$\begin{aligned} {\text{MDD}} & = 35.7 - 4({\text{G}})^{ - 0.046} - 94.4({\text{S}})^{ - 0.0013} + 60.8\left( {\text{F}} \right)^{ - 0.012} \\ \quad - 0.63\left( {{\text{LL}}} \right)^{0.83} + 23.68({\text{PL}})^{0.077} + 0.35\left( {{\text{PI}}} \right)^{0.92} \\ \end{aligned}$$
(17)

It should be noted that this equation cannot be used for pure fine-grained soil since it provides zero OMC and MDD. In order to overcome this issue, the value of G and S can be approximated to zero (not zero).

For OMC, this model's R2, RMSE, and MAE assessment parameters are 0.79, 4.22%, and 2.72%, respectively, whereas these values for MDD are 0.69, 1.062, and 0.84 kN/m3 correspondingly. Furthermore, the models’ OBJ for OMC and MDD is 3.38% and 1.09 kN/m3, respectively. Moreover, the SI values for OMC and MDD are 0.214 and 0.055, respectively.

7.3 MLR model

The comparison between the real compaction characteristics values from literature and predicted values are illustrated in Fig. 7a, b for both phases (training and testing datasets). The following equation shows that PL is the most influential parameter affecting the OMC. Changing the PL value can substantially change the value of OMC. However, both G and S contents have more influence on MDD. Equations (18) and (19) can be used to forecast the OMC and MDD, respectively, for the MLR model with various variable parameters:

$${\text{OMC}} = 0.632{*}({\text{G}})^{ - 0.0314} ({\text{S}})^{ - 0.0142} \left( {\text{F}} \right)^{0.08} \left( {{\text{LL}}} \right)^{0.04} ({\text{PL}})^{0.955} \left( {{\text{PI}}} \right)^{ - 0.0153}$$
(18)
$${\text{MDD}} = 39.74{*}({\text{G}})^{0.01} ({\text{S}})^{0.01} \left( {\text{F}} \right)^{ - 0.03} \left( {{\text{LL}}} \right)^{ - 0.12} ({\text{PL}})^{ - 0.12} \left( {{\text{PI}}} \right)^{0.02}$$
(19)

The R2, RMSE, and MAE assessment parameters for this model are 0.81, 4.098%, and 3.02%, respectively, for OMC, while, for MDD, these are 0.66, 1 kN/m3, and 0.8, respectively. Furthermore, the current model’s OBJ for OMC and MDD are 3.78 and 1.15, respectively. In addition, the SI values of the training dataset for both OMC and MDD are 0.239 and 0.054, respectively.

7.4 ANN model

The authors investigated several hidden layers, neurons, momentum, learning rate, and iterations to achieve high ANN efficiency. Tables 3 and 4 indicate several ANN architectures examined to obtain the most optimum ANN model for OMC and MDD. Consequently, it was detected that when the ANN contains one hidden layer, eight neurons on the left side (as shown in Fig. 8a), 0.1 momentum, 0.2 learning rate, and 2000 iterations, the OMC is best predicted. However, MDD was best predicted with three hidden layers and five neurons on each side (Fig. 8b). The comparison of both predicted and real values of the compaction characteristics are demonstrated in Fig. 9a, b for both phases (training and testing datasets). The ANN model performs better in predicting OMC value than other models. Nevertheless, compared to the other models, the ANN model only performed better in providing R2 value in predicting MDD. Regarding the OMC, the ANN model's R2, RMSE, and MAE assessment parameters are 0.92, 3.51, and 2.574%, respectively. However, for predicting MDD, R2, RMSE, and MAE values are 0.86, 1.164 kN/m3, and 2.31 kN/m3, respectively. Moreover, the present model’s OBJ for OMC and MDD are 3.29 and 2.03, respectively. Further, SI values for the training dataset for both OMC and MDD are 0.202 and 0.068, respectively.

Table 4 The tested ANN architecture for MDD
Full size table

8 Model comparisons

In order to categorize the effectiveness of the constructed models, six different quantitative tools, MAE, R2, RMSE, OBJ, a20-index, and SI, were used. Figures 13, 14, and 15 illustrate R2, RMSE, and MAE, respectively.

Fig. 14
figure 14

The RMSE values of a OMC and b MDD for all four models

Full size image
Fig. 15
figure 15

The MAE value of a OMC and b MDD for all the four models

Full size image

As far as the OMC is concerned, according to the figures, the ANN outperforms LR, NLR, and MLR models as it has a higher R2 and lowers RMSE and MAE values. To assess the reliability of the models, a new engineering index, the a20—index, was used. The advantage of the proposed a20 – index is that it states the number of samples that match expected values with a variance of less than 20% from experimental values [11]. Concerning this index, Fig. 10a compares the predicted and measured OMC of the models. The figure indicated that NLR provided a higher a20- index value of 0.73, and it was 0.7 for the ANN model, which means that 73% of samples with the predicted values have deviated 20% from the actual values of OMC. Thus, the NLR model predicts OMC with the least percentage of error. The OBJ for all the models is shown in Fig. 11. The OBJ values of LR, NLR, MLR, and ANN models are 3.69, 3.38, 3.78, and 3.29, respectively. Even though the OBJ values for all the models are very close, the ANN model has the least OBJ value compared to the others. Concerning all the assessment tools, the ANN approach provides more reliable outcomes and accurately forecasts OMC.

The SI values are exhibited in Fig. 12 for the training and testing phases. Figure 12a shows that the testing phases for LR, NLR, and MLR are between 0.2 and 0.3, indicating fair performance. Nevertheless, the SI value for the ANN model is on a merge of good performance. This contradicts the other phases, as LR, NLR, and MLR perform well. In light of these analyses, although all models can, to some extent, be used to forecast the OMC from physical soil properties, the ANN models are proven to have a distinct performance.

Concerning the prediction of MDD, in Fig. 10b, all the models provide a distinction prediction of MDD, and more than 99% of the predicted samples were within \(\pm\) 20% deviation from the actual data. ANN model provided a slightly higher a20-index. Regarding Fig. 11b, the OBJ values of LR, NLR, MLR, and ANN are 1.07, 1.09, 1.15, and 2.03, respectively. LR model has the least OBJ compared to the other models. In this regard, the LR, MLR, and ANN somewhat perform well in predicting MDD. According to Fig. 12b, the SI values for LR, NLR, MLR, and ANN models are 0.062, 0.055, 0.057, and 0.068, respectively. These values show that all the models are excellent at predicting MDD.

Regarding all the statistical assessment criteria, all the statistical assessment criteria, all the models perform well in predicting MDD, and the percentages of errors are less than 20%. However, the ANN model provides a higher R2 value, and LR can still be a reliable model to predict MDD with an R2 value of 0.76.

9 Sensitivity analysis

In order to obtain the evident influence of each parameter on the OMC and MDD, sensitivity analyses were performed for the models. The MLR model was selected since the value of all input parameters is greater than zero; this can provide the real contribution of each parameter in the model and the influence of each parameter on OMC and MDD. All training data were combined throughout these analyses, and a single input variable was excluded each time. The RMSE, R2, and MAE were individually determined so that the influence of each parameter could be observed. Tables 5 and 6 indicate the variation of evaluation criteria for predicting OMC and MDD, respectively.

Table 5 Sensitivity analysis for MLR model on OMC
Full size table
Table 6 Sensitivity analysis for MLR model on MDD
Full size table

The most efficient input parameter is the one that can remarkably change the values of statistical tools (increasing RMSE, MAE, and decreasing R2). Table 5 indicates the outcomes of the sensitivity study on predicting OMC. The findings show that PL is the most dominant parameter influencing the value of OMC. The R2 value of the model changed from 0.808 to 0.615 when excluding PL. On the contrary, the value of RMSE has increased from 4.098 to 5.8.06. Table 6 clarifies the influence of a single input parameter on MDD. When removing G, the R2 value declined from 0.66 to 0.62; however, the values of RMSE and MAE increased to 1.052 and 0.83, respectively. Therefore, PL can be considered a dominant input parameter that significantly influences the value of OMC, while, for quantifying MDD, G plays a remarkable role.

10 Recomandations for future work

  1. i.

    Refinement of model integration: explore methods to integrate different models (ANN, NLR, LR, MLR) to harness the strengths of each model in a combined predictive approach. Ensemble techniques or hybrid models could potentially yield more accurate and robust predictions by leveraging the unique capabilities of individual models.

  2. ii.

    Additional input parameters: consider expanding the input parameters beyond the six currently utilized factors. Incorporating other relevant soil properties or environmental factors that might influence compaction characteristics could enhance the models' predictive capabilities.

  3. iii.

    Data augmentation and validation: continuously collect new data and augment the existing dataset with additional samples to improve the models' training and validation. This can help ensure the models remain accurate and effective when applied to a wider range of soil variations.

  4. iv.

    Regional variability analysis: investigate how the predictive models perform across regions with varying soil compositions and climates. Understanding how these models generalize to geographical areas can enhance their applicability and reliability in diverse construction contexts.

  5. v.

    Dynamic modelling for real-time applications: develop dynamic models that can continuously adapt and predict compaction characteristics based on real-time data from sensors embedded in construction equipment. This could facilitate on-site decision-making and optimize the compaction process as it progresses.

  6. vi.

    Incorporation of advanced ai techniques: explore the integration of advanced artificial intelligence techniques, such as deep learning or reinforcement learning, to further improve the predictive accuracy of the models. These techniques might capture more complex nonlinear relationships within the dataset.

  7. vii.

    Field validation and case studies: conduct field validation studies where predictions from the models are compared against actual on-site compaction results. Case studies in real construction scenarios can provide practical insights into the models' effectiveness and help refine their application in practical settings.

  8. viii.

    Optimization algorithms: develop optimization algorithms that utilize the predictive models to suggest the most efficient and effective combinations of soil properties to achieve the desired compaction characteristics, considering both resource utilization and construction timelines.

11 Conclusions

Developing a predictive model to anticipate Optimum Moisture Content (OMC) and Maximum Dry Density (MDD) is crucial for leveraging fundamental soil characteristics such as soil gradation and plasticity. These parameters are pivotal in gauging soil suitability, simplifying the selection process from various soil sources, and circumventing the requirement for standard Proctor tests. An analysis of 2162 datasets, encompassing Gravel (G), Sand (S), Fine content (F), Plastic Limit (PL), Liquid Limit (LL), and Plasticity Index (PI) parameters, unveiled significant revelations:

Several models—Linear Regression (LR), Nonlinear Regression (NLR), Multilinear Regression (MLR), and Artificial Neural Networks (ANN)—were devised to predict OMC and MDD. Evaluation using diverse metrics highlighted the superior performance of the ANN model. When it came to predicting OMC, it had higher R-squared (R2) values and lower Objective (OBJ), Root Mean Squared Error (RMSE), Scatter Index (SI), and Mean Absolute Error (MAE) values.

All models demonstrated robust predictive accuracy for OMC and MDD, with SI values below 0.2.

The ANN model particularly excelled in predicting OMC, displaying an OBJ value approximately 12% lower than LR, 2.7% lower than NLR, and 15% lower than MLR. Conversely, for MDD prediction, the ANN model presented a higher OBJ value compared to LR, NLR, and MLR.

Although the ANN model showcased the highest R2 value, LR, NLR, and MLR models also exhibited commendable R2 values, hovering around 0.8, signifying reliability in predicting OMC based on parameters G, S, F, PL, LL, and PI. The LR model's R2 value (0.78) for MDD implies relative reliability in this context.

The model equations underscore the importance of Plastic Limit (PL) and Liquid Limit (LL) in projecting OMC, while the proportions of Gravel (G) and Sand (S) emerge as pivotal factors in determining MDD.

Sensitivity analyses identified Plastic Limit (PL) as the primary influencer in predicting OMC, while Gravel (G) holds substantial sway in quantifying MDD.

While the ANN model outshines others in OMC and MDD prediction, LR models might offer more practicality owing to their transparent equations grounded in soil physical properties. This contrasts with the ANN's inherently complex "black box" approach.