Prediction of the Soil Permeability Coefficient of Reservoirs Using a Deep Neural Network Based on a Dendrite Concept

Abstract

Changes in the pore water pressure of soil are essential factors that affect the movement of structures during and after construction in terms of stability and safety. Soil permeability represents the quantity of water transferred using pore water pressure. However, these changes cannot be easily identified and require considerable time and money. This study predicted and evaluated the soil permeability coefficient using a multiple regression (MR) model, adaptive network-based fuzzy inference system (ANFIS), general deep neural network (DNN) model, and DNN using the dendrite concept (DNN−T, which was proposed in this study). The void ratio, unit weight, and particle size were obtained from 164 undisturbed samples collected from the embankments of reservoirs in South Korea as input variables for the aforementioned models. The data used in this study included seven input variables, and the ratios of the training data to the validation data were randomly extracted, such as 6:4, 7:3, and 8:2, and were used. The analysis results for each model showed a median correlation of r = 0.6 or less and a low model efficiency of Nash–Sutcliffe efficiency (NSE) = 0.35 or less as a result of predicting MR and ANFIS. The DNN and DNN−T both have good performance, with a strong correlation of r = 0.75 or higher. Evidently, the DNN−T performance in terms of r, NSE, and root mean square error (RMSE) improved more than that of the DNN. However, the difference between the mean absolute percent error (MAPE) of DNN−T and the DNN was that the error of the DNN was small (11%). Regarding the ratio of the training data to the verification data, 7:3 and 8:2 showed better results compared to 6:4 for indicators, such as r, NSE, RMSE, and MAPE. We assumed that this phenomenon was caused by the DNN−T thinking layer. This study shows that DNN−T, which changes the structure of the DNN, is an alternative for estimating the soil permeability coefficient in the safety inspection of construction sites and is an excellent methodology that can save time and budget.

1. Introduction

Changes in the pore water pressure of an earth dam play an important role in maintaining its stability. The pore water pressure inside the dam is altered by external loading conditions, such as the rapid drawdown of reservoir water and water table rise due to rainfall infiltration [1].

This pressure is important in determining the fluid flow characteristics of soil and depends on various factors, such as density, water content, porosity, and soil structure. Generally, the pore water pressure can be expressed as a soil permeability coefficient, which denotes the quantity of water transferred through interconnected pores over time [2,3]. Since the permeability coefficient is used to identify geotechnical challenges, such as structural destruction, infiltration, and leakage, several studies have investigated the causation factors for the soil permeability coefficient using empirical relationships [4,5]. However, this process is challenging and requires considerable time and resources.

Particle size is directly related to the soil permeability coefficient. Hazen proposed an empirical formula based on the particle size [6] and demonstrated that it is proportional to the square of the effective particle size of sand [7]. As previously mentioned, a soil permeability coefficient depends on several factors from a geotechnical perspective, such as soil density, water content, porosity, mineralogy, and soil structure [8]. Moreover, it is affected by the density, particle size, and particle shape [9,10] because the water flow in the ground is related to the size and connectivity of the pores [11].

However, the pore–size distribution cannot be measured directly. To address this issue, several studies have proposed methodologies to estimate the range of the shrinkage of the pore size distribution using the soil particle size distribution [12,13]. In addition, a methodology has been proposed in which the pore size distribution is used to calculate the saturated soil permeability [14,15] and unsaturated hydraulic permeability coefficients [16,17,18,19,20]. The soil permeability coefficient is influenced by the particle size of the soil, However, this does not apply to all soil types [2,17]. These methodologies are limited and not experimentally realizable because of the evaluation of intrinsic material properties as a technique for fluid flow in porous ground [20]. Even in the Kozeny–Carman equation, because the measurement of voids in the ground is challenging, knowledge of the characteristics of particle size and effective particle size is required [20]. Therefore, the demand for the image analysis of porous geometries and powerful new regression techniques, as well as the use of machine learning and computer vision tools to enable them, is increasing [21].

The soil permeability coefficient is crucial in describing fill dams. Alternative procedures are essential in the absence of sufficient data from soil permeability coefficient tests [22]. The prediction of the permeability is critical since significant changes in fill dams occur within the body [23]. If experimental data are unavailable, they can be predicted from the relationship between the test and log data using an expansion of statistical techniques [24]. In this context, neural networks (NN) have been proposed.

With the development of technologies using artificial intelligence, artificial neural networks (ANNs) are used to predict soil compaction, shear, and bearing [8,25,26,27]. Several studies have been conducted on soil permeability. These include machine learning (ML) methods, such as hybrid optimization models of a deep neural network (DNN), an adaptive network-based fuzzy inference system (ANFIS), and a genetic algorithm-ANFIS (GA-ANFIS) [1,2,3,28,29,30]. Sinha and Wang (2008) presented an ANN model related to permeability, dry unit weight, and optimal water content as the classification characteristics and output variables of clay soil [2]. Sezer et al. (2009) concluded that the ANFIS algorithm is superior in estimating the permeability of coarse soil by considering the particle size distribution and particle shape [28].

The aforementioned studies have limitations as only soils with constant permeability were analyzed. Therefore, various soil ranges should be investigated by developing a model using experimental data from the fill dam body, reservoir, and coastal sites to predict the permeability coefficient applicable to the corresponding fields.

Old embankments can threaten the stability of levees due to uncontrolled water flow [31]. However, studies on embankments of agricultural reservoirs have focused on impermeability reinforcement. A study for the impermeability reinforcement of a reservoir can be done as a preventive measure; however, it is not a way to analyze the root cause. In the case of reservoirs, a method that uses neural networks is effective for adapting to high computational requirements, real-time applications, uncertain environments, and coping with unknown changes [32]. A single simulation is time consuming (usually several hours) because thousands of simulation runs are required to make a complete prediction [33]. However, DNNs can directly address complex problems in an end-to-end manner [34].

In this study, the permeability coefficient of soil is predicted and evaluated using multiple regression (MR), which is an ANFIS, DNN, and DNN model with thinking layers (DNN-T) using the physical properties of the soil collected from the embankment of the reservoir.

The physical properties of the soil are used as input parameters, and the results of the model are evaluated by dividing the training and validation data ratios into 6:4, 7:3, and 8:2. Through this study, the data on physical properties can contribute to the prediction of ground properties using neural networks. From the research results, we can confirm the excellent performance of both the DNN and DNN−T models.

2. Materials and Methods

2.1. Classification Method of Soil Property Data

The Korea Rural Community Corporation, a Korean institute for agricultural infrastructure management, conducts precision safety diagnoses every five years to check structural safety and minimize damage during emergencies, such as a reservoir collapse due to an earthquake or a natural disaster.

Four administrative districts in Korea were classified as research areas: Figure 1a Gyeonggi−do and Gangwon−do, Figure 1b Chungcheong−do, Figure 1c Gyeongsang−do, and Figure 1d Jeolla−do.

The analysis data are a summary of the results obtained by the authors in the laboratory for the physical properties and triaxial permeability tests. It was not possible to express all the data; thus, they were organized by region and a unified classification system (USCS).

The proportions of data analyzed for these districts were 11.9%, 19.0%, 26.2%, and 42.9%, respectively. This is because Korea has an east−west low−level topography, with mountains located in the northeast and well-developed basins in the southwest. The samples were collected from agricultural reservoirs located in four administrative districts. In this study, the physical and mechanical characteristics of the undisturbed samples collected during a precision safety examination of 42 reservoirs were investigated. The test results were classified into unit weights, void ratios, and particle sizes. Particle size analysis was performed in compliance with the KS F 2302, 2303, 2306, 2308, and 2309 standards, and the soil was classified based on KS F 2324. KS F refers to Korean Industrial Standards prepared by changing the technical content for provisions and content that cannot conform to international standards. The permeability coefficient was determined by referring to the soil laboratory testing manual of the Engineering Laboratory Equipment (ELE) Int’l. Ltd. as a triaxial permeability test [35]. Furthermore, the test results were classified according to the region, uniformity, and permeability.

We designed seven variables as input data (independent variables), including unit weight, void ratio, and five particle sizes (e.g., 4.76, 1.0, 0.42, 0.074, and 0.002 mm), using the methodology described in Section 2.1. Furthermore, soil permeability coefficients were used as target data (dependent variables).

2.2. Analysis of the Models

As mentioned in Section 1, this study employed MR, ANFIS, DNN, and DNN−T to predict and compare the soil permeability coefficients of reservoir embankments. The DNN−T is described in Section 2.2.5.

2.2.1. Multiple Regression

Statistical techniques of the traditional linear time-series model were used. However, their practical application is limited [36]. Particularly, MR analysis is a tool used to determine the cause−effect relationship between two or more independent variables and dependent variables. This model uses Equation (1) and determines constant ${\mathsf{\beta }}_0$ and regression coefficients ${\mathsf{\beta }}_1$, ${\mathsf{\beta }}_2$, …, ${\mathsf{\beta }}_{\mathrm{k}}$ [37].

A general MR model formula can be expressed using Equation (1).

$${\mathrm{Y}}_{\mathrm{t}}={\mathsf{\beta }}_1+{\mathsf{\beta }}_2{\mathsf{\beta }\mathrm{x}}_{\mathrm{t}2}+{\mathsf{\beta }}_3{\mathsf{\beta }\mathrm{x}}_{\mathrm{t}3}+\cdots +{\mathsf{\beta }}_{\mathrm{k}}{\mathsf{\beta }\mathrm{x}}_{\mathrm{tk}}+{\mathsf{ϵ}}_{\mathrm{t}}$$

(1)

where k is the number of parameters (i.e., ${\mathsf{\beta }}_1$…${\mathsf{\beta }}_{\mathrm{k}}$) and ${\mathsf{\beta }}_1$ is the intercept (constant) term.

In this study, the MR analysis was conducted using IBM SPSS Statistics 26. The permeability coefficient of the independent variable was determined using the seven input variables as dependent variables.

2.2.2. Adaptive Network-Based Fuzzy Inference System

A network structural design [38] that includes variable selection, rule−based reduction, and fuzzy set number optimization is required to implement a fuzzy NN [39]. This algorithm automatically adjusts the membership function and control rule to control the target according to the input/output information obtained from the control environment using the structure and learning ability of an NN in the combined form of a neural network and a fuzzy network [40]. In general, the ANFIS construction method repeatedly divides the input variable space based on a certain rule and assigns a corresponding conclusion rule to each divided space to express it as a fuzzy characteristic [41]. In the schematic of the ANFIS model (Figure 2), the fixed nodes are represented as circles, and the adaptive nodes are represented as squares.

2.2.3. Procedure of the Deep Neural Network

The analysis method using DNNs is shown in Figure 3. The permeability coefficient was determined using a triaxial permeability test in the laboratory. The permeability coefficient results were collected and organized based on the region and soil type. Normalized data were input as .csv file in the order of training and validation by the neural network. The dense layer of the neural network was modified to process the input data. It prevents overfitting of the training and validation data because of the changes in the dense layer. In addition, we predicted the permeability coefficients by stopping the training before overfitting began. As shown in Figure 3, the .csv file is entered as training data and validation data, divided into 6:4, 7:3, and 8:2. If the loss rate and accuracy of the input data are satisfied, the error is evaluated by comparing it with the prediction result. If the input data does not satisfy the loss rate and accuracy, the above training is repeated by adjusting the hyper−parameters.

The effect of the ratio of training to validation data on the input variables used in the DNN analysis on the prediction results was identified. One hundred and sixty-four collected data points were analyzed by dividing them into training and verification data in the ratios of 6:4, 7:3, and 8:2.

2.2.4. Deep Neural Network

ANNs are developed based on mathematical models that mimic biological NNs as information processing systems [43,44]. They are the most promising candidates for mimicking the activities and abilities of the human brain and nervous system [2,28,29]. Information processing in an ANN is performed using processors with simple individual neurons. When an input is implemented as a neuron, the weight associated with each connection is multiplied, biased toward the next neuron, and added to the sum of the weighted inputs. Each neuron generates an output value by applying a linear or nonlinear transfer function [43]. ANNs have been widely used to solve numerous problems in geotechnical engineering and have evolved into DNNs. A DNN has a deeper structure than an ANN and is commonly used to determine the hidden layers between outputs and inputs in both linear and nonlinear patterns.

This study employed a DNN model with a simple structure to predict the soil permeability coefficient and compared the results with those of other models (Figure 4). This model comprises two dense layers: a batch normalization layer and an output layer. The first and second dense layers had 4095 and 2048 nodes, respectively. A batch normalization layer was adopted to prevent overfitting of the model, and the output layer had one node, which was used to denote the soil permeability coefficient.

2.2.5. DNN-T

The current deep-learning structure employs a concept wherein the perceptron invented by Frank Rosenblatt has multiple layers known as the multilayer perceptron (MLP).

The perceptron mimicked a neuron, as shown in Figure 5a. With advances in the field of brain science, various types of neurons have been discovered, including sensory neurons (afferent neurons), motor neurons (efferent neurons), and interneurons. Interneurons constitute most of the central nervous system of the brain and spinal cord and exhibit characteristics, such as short dendrites, no myelinated axons, and various forms. They analyze signals from sensory neurons, integrate them to form commands for appropriate responses, and send them to motor neurons [45]. This implies that the brain receives a signal through a synapse and receives information from an interneuron to process the signal, as shown in Figure 5b. Interneurons stabilize the circuit by inhibiting feedback [46]. Moreover, it can reduce the prediction errors of machines [47].

In this paper, a DNN−T was proposed to predict the soil permeability coefficient. This is a modification of the structure of a general DNN, which employs the functionality and structure of an interneuron. This hidden layer was termed the “thinking layer.” It considered input variables based on the change in the weighted value of the thinking layer just before the input data were injected into the neural network. That is, it mimicked the position of an interneuron in the cranial nerve and the stabilization function of the neural circuit. The structure of the DNN−T is characterized by placing an independent hidden layer on the path through which the input data enter just before they are injected into a dense layer (Figure 6).

2.3. DNN and DNN−T Configuration

The data used in this study included seven input variables, and the ratios of training data to validation data were randomly extracted, such as 6:4, 7:3, and 8:2, and were used. A ReLu function was used as the activation function in all dense layers of the DNN and DNN-T. This is because the commonly used sigmoid activation function has a value between zero and one. Therefore, a vanishing gradient may have occurred. Adam was used as the optimizer. The number of epochs was set to 10,000, and the learning rate was set to 0.0001. To verify whether the learning state of the learning rate used was efficient, the performance was evaluated using the mean square error (MSE) as a loss function.

2.4. Evaluation of Analysis Methods

We used four methods, namely, the Pearson correlation coefficient (r), Nash–Sutcliffe efficiency (NSE), root mean square error (RMSE), and mean absolute percent error (MAPE), to evaluate the performance of the models in this study. The parameter r ranges from −1 to 1. This was calculated using Equation (2) by examining the linearity between the measured and predicted values. The closer the r value is to 1, the stronger the linearity.

$$\mathsf{\gamma }=\frac{{{\displaystyle \sum }}_{\mathrm{i}=1}^{\mathrm{n}}({\mathrm{y}}_{\mathrm{i}}-\overline{\mathrm{y}})\left({\mathrm{y}}_{\mathrm{i}}^{\prime }-\overline{{\mathrm{y}}_{\mathrm{i}}^{\prime }}\right)}{\sqrt{{{\displaystyle \sum }}_{\mathrm{i}=1}^{\mathrm{n}}{({\mathrm{y}}_{\mathrm{i}}-\overline{\mathrm{y}})}^2{\left({\mathrm{y}}_{\mathrm{i}}^{\prime }-\overline{{\mathrm{y}}_{\mathrm{i}}^{\prime }}\right)}^2}}$$

(2)

The NSE is the normalization of the model performance to an interpretable scale, which is calculated using Equation (3). An NSE of one implies that the model result is in perfect agreement with the measurements.

$$\mathrm{NSE}=1-\frac{{{\displaystyle \sum }}_{\mathrm{i}=1}^{\mathrm{n}}{\left({\mathrm{y}}_{\mathrm{i}}-{\mathrm{y}}_{\mathrm{i}}^{\prime }\right)}^2}{{{\displaystyle \sum }}_{\mathrm{i}=1}^{\mathrm{n}}{\left({\mathrm{y}}_{\mathrm{i}}-\overline{\mathrm{y}}\right)}^2}$$

(3)

Because the RMSE is rooted in the MSE, it is less sensitive to large difference errors. Moreover, size-dependent errors may occur since it depends on the size of the actual value. Equation (4) can be used to quantitatively verify whether the model’s estimate is close to the actual value. The closer it is to 0, the more consistent the model’s results are with the actual values.

$$\mathrm{RMSE}=\sqrt{\frac{1}{\mathrm{n}}{\displaystyle \sum }_{\mathrm{i}=1}^{\mathrm{n}}{\left({\mathrm{y}}_{\mathrm{i}}-{\mathrm{y}}_{\mathrm{i}}^{\prime }\right)}^2}$$

(4)

A MAPE value close to zero implies that the model performs well. However, no clear criterion for the model performance was determined (Equation (5)). Furthermore, if the true value is 0, it is not divisible and can only be applied to values for which the ratio analysis is meaningful.

$$\mathrm{MAPE}=\frac{100}{\mathrm{n}}{\displaystyle \sum }_{\mathrm{i}=1}^{\mathrm{n}}\left|\frac{{\mathrm{y}}_{\mathrm{i}}-{\mathrm{y}}_{\mathrm{i}}^{\prime }}{{\mathrm{y}}_{\mathrm{i}}}\right|,$$

(5)

where ${\mathrm{y}}_{\mathrm{i}}$ and ${\mathrm{y}}_{\mathrm{i}}^{\prime }$ are the measured and predicted values, respectively, and $\overline{\mathrm{y}}$ and $\overline{\mathrm{y}\prime }$ are the means of the measured and predicted values, respectively.

3. Results and Discussions

3.1. Soil Analysis Result

This section presents the results of the soil tests described in Section 2.1. The test results were classified into unit weight, void ratio (e), particle size, and permeability (

Table 1. Detailed information on the permeability coefficient and soil parameters.

	e	γt 1 (kN/m3)	Grain Size Distribution (%)					K 2 (cm/s)
			4.76 mm	2.0 mm	0.425 mm	0.075 mm	0.002 mm
Min	0.36	12.97	62.4	46.5	16.0	1.0	2.0	1.43 × 10−8
Max	1.09	22.32	100.0	100.0	99.5	98.9	40.4	5.02 × 10−4

¹γ_t: Wet unit weight; ² K: Permeability coefficient.

). The results were classified by region, and the uniform classification and permeability coefficients were determined (Figure 7). In the unified soil classification system (USCS), SM and SC of sandy soil are composed of sandy soil, whereas CL and CH comprise fine soil types. The ranges of the permeability coefficients by soil type were 5.02 × 10⁻⁴ to 2.48 × 10⁻⁷ for SM, 7.11 × 10⁻⁵ to 1.43 × 10⁻⁸ for SC, 9.19 × 10⁻⁶ to 2.82 × 10⁻⁸ for CL, and 1.84 × 10⁻⁶ to 5.99 × 10⁻⁷ for CH. The S−M or C soil has a considerable amount of sand and a high permeability coefficient. Conversely, the C−L or H soil has a high clay composition and a low permeability coefficient. The S−series soil had a large peak with a high permeability coefficient, whereas the C−series soil had a distribution close to zero. The data used for the analysis were from undisturbed soil collected from the field, and the results were obtained through direct testing.

3.2. MR Result

The results predicted by the MR analysis of the training data showed a coefficient of determination (R²) of 0.327, Adj.R² = 0.349, and the explanatory power of the prediction result of the permeability coefficient was rather low. Moreover, based on a time-series analysis of the training and prediction data, the prediction data exhibited numerous errors in the test results. The errors in the MR analysis were MSE (0.011) and MAPE (7973.6%), confirming that the errors in the predicted data were significant. Figure 8 shows the test results and the permeability coefficients predicted using the MR model.

3.3. ANFIS Result

The ANFIS prediction results for the training data exhibited a coefficient of determination (R²) of 0.348. Notably, the explanatory power of the prediction results for the permeability coefficient was low. Moreover, according to the time−series analysis of the training and prediction data, the prediction data exhibited several errors in the test results. The errors in the training results were MSE (0.01) and MAPE (6952.4%), indicating considerable errors in the prediction data. The ANFIS is not suitable for high-dimensional data [48] and is tolerant of the number of input variables in the face of computational complexity [49]. The input variables used in this study had large deviations based on soil type, and the number of input variables was not sufficient to apply the ANFIS. Figure 9 shows the test results and permeability coefficients predicted by the ANFIS.

3.4. DNN Result

3.4.1. Training Result of the DNN

Figure 9 shows the results of training the input variables with the general DNN model according to the ratio of the input parameters. To determine whether the training was appropriate, the training results were the Loss, MAE, Val_Loss, and VAL_MAE. Figure 10 illustrates the evolution of the training results for the training and validation datasets for the ratios of the input variables in this study. None of the matrices of the DNN model represented ideal training conditions because the final analysis exhibited large amplitudes in all graphs. However, it converged to approximately zero as the number of epochs increased. Moreover, the matrices in the validation dataset are higher than those in the training dataset. Figure 10 shows that the model trains and generalizes sufficiently well because overfitting does not occur for any training.

3.4.2. Prediction of DNN

Figure 11, Figure 12 and Figure 13 show the DNN analysis results for the ratios of the input parameters. According to the ratio of the input parameters in the analysis results, the coefficient of determination of the predicted result was (R²) = 0.60 at the input parameter ratio of 6:4, (R²) = 0.56 at the input parameter ratio of 7:3, and (R²) = 0.66 at the input parameter ratio of 8:2. The correlation of R² did not exhibit a significant difference as the ratio of the input parameters increased. In addition, the correlation between the prediction results increased compared to the coefficient of determination of the results predicted by MR and ANFIS.

3.5. DNN−T Result

3.5.1. Training Result of DNN−T

Figure 14 shows the change in the training results for the training and validation datasets as a function of the input parameter ratio. None of the matrices of the DNN−T model represented ideal training conditions because all graphs exhibited large amplitudes in the final analysis. Moreover, the matrices in the validation dataset are higher than those in the training dataset. DNN−T confirms that the training and generalization of the model have sufficiently progressed because overfitting does not occur.

3.5.2. Prediction of DNN-T

The DNN analysis results for the ratios of the input parameters are shown in Figure 15, Figure 16 and Figure 17. Based on the ratio of the input parameters in the analysis results, a coefficient of determination of the predicted result showed (R²) of 0.49 at the input parameter ratio of 6:4, (R²) of 0.74 at the input parameter ratio of 7:3, and (R²) of 0.63 at the input parameter ratio of 8:2. A comparison of the coefficients of determination of the results predicted by MR and ANFIS demonstrated that the errors in the prediction results were significantly reduced. As a result of the ratio of 7:3 input parameters of DNN−T, the error of the prediction result is significantly reduced compared to the coefficient of the determination of the result predicted by MR, ANFIS, and DNN models.

3.6. Estimation and Comparison of the Models

The data used as input variables in this study were the void ratio, unit weight, and particle size of various types of soils. Therefore, necessary data, such as those of the research results of previous studies, were used to predict and evaluate the permeability coefficient. The results obtained using various analytical methods were evaluated.

In this study, the performances of the MR, ANFIS, DNN, and DNN−T models were evaluated using the r, NSE, RMSE, and MAPE methods (

Table 2. Evaluation results of the analysis model.

	Target Data	r	NSE	RMSE	MAPE (%)
MR	All data	0.572	0.298	0.104	7973.6
ANFIS	All data	0.590	0.343	0.101	6952.4
DNN (6:4 ratio)	All data	0.774	0.479	0.091	898.87
DNN (7:3 ratio)	All data	0.745	0.477	0.090	162.88
DNN (8:2 ratio)	All data	0.813	0.512	0.105	157.83
DNN−T (6:4 ratio)	All data	0.703	0.431	0.095	391.13
DNN−T (7:3 ratio)	All data	0.859	0.542	0.084	173.89
DNN−T (8:2 ratio)	All data	0.793	0.562	0.099	215.23

). When evaluating a model using r, the value of r is considered weak within the range of 0.1–0.3, moderate in the range of 0.4–0.6, strong in the range of 0.7–0.9, and perfect when it is 1 [50]. Our results showed that the MR and ANFIS exhibited a moderate correlation with r less than 0.6. However, in the DNN model, r exceeds 0.813, indicating a strong correlation. The results predicted by the DNN−T model exhibited the strongest relationship, with r = 0.859. An NSE of 1 indicates that the simulation is a perfect match, and an NSE of 0 indicates that the model simulation has the same explanatory power as the mean of the test results. If the NSE is less than 0, the model is a worse predictor than the mean of the observations [34]. In this study, the NSE approached 1 in the order of MR, ANFIS, DNN, and DNN−T. The RMSE exhibited smaller values of the same order as that of the NSE. It should be noted that the performance of the DNN−T model is superior. For the MAPE, the DNN model exhibited a lower error rate than the DNN−T model.

RMSE and MAPE did not have any noticeable evaluation criteria. However, the closer the value is to 0, the better the performance of the model. Therefore, the permeability coefficient was predicted using MR analysis, ANFIS, and DNN, and the results were compared. The “thinking layer” model of DNN−T will be further investigated for its application in various sectors of geotechnical engineering.

Figure 18 shows the comparison of the training results of DNN−T and DNN. The convergence epoch of learning by DNN training was larger than that by DNN-T training, indicating that DNN−T has a more reasonable time and technical training. That is, the learning rate of DNN−T was faster. Moreover, it solves the overfitting problem that occurs as DNN layers deepen. Therefore, DNN−T can solve overfitting without degrading the performance of the DNN.

4. Discussions

Analysis methods for permeability predictions showed different error results. Among them, multiple regression analysis and ANFIS show large errors in prediction results. Pradeep et al. (2021) found no significant difference in the performance of the ANFIS and DNN in a study that predicted rock deformation based on experimental data [51]. However, the DNN and ANFIS in this study exhibit different results. This is because ANFIS is not suitable for high−dimensional data, and its performance decreases as the input size increases [48]. Moreover, Sadrmomtazi et al. (2013) emphasized that ANNs can be used more effectively for predictions than ANFIS [52].

There is no clear standard for the ratio of training data to validation data in artificial neural network analysis. First, 7:3 and 8:2 showed good results compared to 6:4 regarding indicators, such as r, NSE, RMSE, and MAPE. The performance of the model varies according to the data distribution in this study, which can be due to the considerably limited number of data and the diverse composition of the test data. That is, the test data may consist only of data that the model trained on can make good predictions. Depending on the model’s hyperparameters, the model may perform better or worse. However, in this study, the change in performance based on the structure of the model was shown, and the problem of data distribution is not within the scope of this study. Therefore, the problem of data distribution is left as a limitation of this study and will be addressed in future studies. Regarding the efficiency of the model based on the distribution of data, it is difficult to make a clear judgment as each previous study presents different results.

Therefore, the DNN model can supplement and solve the disadvantages (accuracy, time, and reliability) of existing methods in predicting ground problems. In particular, it is necessary to re-evaluate the analysis model through additional research on DNN−T.

5. Conclusions

In this study, MR, ANFIS, DNN, and DNN−T were used to predict and evaluate the permeability coefficient. The analysis results showed that the DNN was better than the MR and ANFIS. The DNN and DNN-T showed good performance, with a strong correlation of r = 0.75 or higher. Evidently, the DNN−T performance in terms of r, NSE, and RMSE improved more than that of the DNN. However, the difference between the MAPE of DNN−T and DNN was that the error of DNN was small (11%). The analysis results of the DNN and DNN−T models were similar. However, in DNN−T, MAPE, r, and NSE are more reliable, and the training time of the input data is faster. Moreover, overfitting was solved without degrading the DNN performance. This is because DNN−T considers the input variables according to the change in weight immediately before the input data are injected into the DNN by modifying the function and structure of the intermediate neurons. The overfitting problem can be rapidly resolved because the intermediate neuron stabilizes the circuit through feedback timing. We believed that DNN−T is an alternative for estimating the soil permeability coefficient in safety inspections in the construction field and is an excellent methodology for saving time and budget.

That is, both the DNN and DNN−T can be alternatives for estimating soil permeability at construction sites. Predicting the permeability coefficient using particle size analysis data as a factor can contribute to the maintenance of agricultural reservoirs in the future. However, the predictive performance of neural network analysis may be underestimated due to the lack of analytical data. Thus, building a DB through data collection and standardization can improve the accuracy of predicting ground property information on mechanical properties, such as the permeability coefficient, which can be used to evaluate the stability of fill dams, such as reservoirs.

However, the influence of the input variables and the additional structure of the model constitute the scope of future studies to further improve the accuracy of the DNN model.