1 Introduction

Because of a large number of complicated problems in most engineering applications, engineers depend on computational intelligence as well as soft computing analysis instead of following huge complicated calculations [1, 2]. In engineering problems, much sophisticated statistical analysis and mathematical modeling are introduced in order to solve engineering problems [3,4,5]. Challenges associated with the reliable engineering design solution and development of technology complicated the geotechnical engineering environment even more [6, 7]. Certainly, investigating the engineering properties of rock and soil masses show uncertain and varied behavior due to their imprecise and complex natures. On the other hand, many other materials in the field of civil engineering (e.g., steel, timber, and concrete) show far more homogeneity and isotropy. The artificial neural networks (ANNs), based on the sophisticated mathematical models and advanced software tools, can help to assess all the reliable choices available with respect to a predefined project outcome [8, 9]. The ANN, however, need to be used along with one optimization algorithm to reduce the rate of error especially in complex problems such as compressed sensing [10]. ANN offers the required tools for geotechnical engineers, working in large consultant companies, to make a fast and in most cases better decisions to improve the quality of their performance and to reduce risks [11]. Numerous researchers have discussed the operation and structure of ANNs (e.g., Wang [12], Choobbasti et al. [13], Gandomi and Alavi [14], Mukhlisin et al. [15], Lian et al. [16], Salsani et al. [17], Bahrami et al. [18], Mert [19], Moayedi and Rezaei [20]). As an alternative and effective approach, which has been proved to have a degree of success and reliability [21], is mainly based on the data alone to define the parameters and structure of the model [8]. The ANN was used in numerous academic subjects and projects, such as risk assessment [22, 23], health and medical [24], image processing [25, 26], mathematics [27,28,29,30], early warnings related to geotechnical problems [31], geosciences and remote sensing [32], business and management [33], civil engineering [14, 34,35,36] and particularly to the geotechnical engineering as the main concern for this study [32, 37].

In recent years some scholars have successfully attempted to generate, extend and present the new utility determining tools and approaches, as well as ANNs methods and techniques into the field of civil engineering. This interesting topic has been reviewed several times by researchers such as Flood and Kartam [38], Flood and Kartam [39], Adeli [3], Lu et al. [40], Lazarevska et al. [34] and Li and Hao [41]. Indeed, the use of ANNs method in the geotechnical engineering problems, as the first multicriteria assessment method, was presented in the early 1990s by Bolt [42]. Different subjects have been studied using the ANNs method such as faults modeling [42], underground openings [43], braced excavation [44], pile integrity testing [45], pile bearing capacity [20, 46,47,48,49,50,51,52,53,54,55,56], predicting geotechnical parameters [1, 57,58,59], modeling tunnel boring machine (TBM) performance [60], kinematic soil pile interaction response parameters [61], slope stability [62]. There are very few research studies that classified and reviewed the ANNs application for these approaches in various areas such as; principles and understanding of NNs in civil engineering [3], shallow foundations [63], pile foundations [64], corrosion monitoring [65]. Various approaches have been suggested regarding the previous findings on the ANN application in geotechnical engineering. However, the conducted surveys were limited to specific subjects such as pile foundation [66], shallow foundation [63] or general subject of geotechnical engineering [8] and did not keep up with the new challenges and changing situation in the field of geotechnical engineering. Thus, the authors think that there is an absence of a systematic review from the recently published studies performed in the highlighted area. Also, the authors believe that there is a great demand for a comprehensive review paper, combining the available methods as well as current studies.

2 Literature review and distribution of the papers

Since the early 1900s, and up to the date of writing this paper, there are more than 4000 research scholar articles in the field of geotechnical engineering which indexed in the web of science (WOS). Distribution of papers published in the considered area, based on the source title, is presented in Table 1. In this regard, when the search narrows to the application of ANNs in the subject of geotechnical engineering only 152 articles, with a very limited number of source title, remained. Distribution of papers, based on the source title, in the use of ANN-based models in geotechnical engineering, is tabulated in Table 2. Figure 1 shows the subject of ANN application in the field of geotechnical engineering based on both total publications by year and sum of times cited by year. It can be seen that the number of publications has increased sharply from one publication in the year 2000 to twenty-two publications in the Year 2017. ANN-based methods have been applied increasingly as an effective methods in most geotechnical engineering subjects, including: tunneling [67, 68], mathematical constitutive modeling [69], underground openings [70, 71], geo-material properties [72, 73], bearing capacity of pile [20, 53, 64]; slope stability [47, 74,75,76,77,78,79]; liquefaction [80, 81], earth retaining structures [82, 83], soil swelling [84, 85], classification of soils [86, 87] and site characterization [88, 89]. Indeed, the fundamentals of modern ANN and hybrid ANN methods were developed (in some cases generated) in the 2000s and 1960s. The research and development of ANN methods increased between 2010 and 2017, but as presented in Fig. 1, it seems that the exponential (here means rapid increase) growth of this process continued. ANN application in pile bearing capacity prediction (both skin friction and end-bearing capacities), modeling soil behavior, earth retaining structures and excavations, site characterization, liquefaction, slope stability, and classification of soils are reviewed in this section. To make readers familiar with the geotechnical interest of research scholar authors listed the journals that mostly publish on the subject of geotechnical engineering (see Table 1). On the other hand, the distribution of papers in the subject of ANN application in geotechnical engineering according to the source title (see Table 2) is illustrated.

Table 1 Distribution of papers in the subject of geotechnical engineering based on the source title
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Table 2 Distribution of papers in the subject of ANN application in geotechnical engineering based on the source title
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Fig. 1
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The subject of ANN application in geotechnical engineering based on total publications by year and sum of times cited by year

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2.1 ANN application in modeling soil behavior

AttohOkine and Fekpe [90] employed adaptive NNs to model strength properties of local lateritic soils. They compared the results from generalized adaptive NNs (GANN) with some of the traditional back-propagation NN (BPNN) approaches for modeling the in situ soil strength properties based on raw California bearing ratio (CBR) data. Their results showed the GANN seems to be more effective. Zhu et al. [91] used recurrent NN (RNN) to model shearing characteristics of residual soil. The network was able to determine volumetric strains during shearing courses and abrupt changes in axial. Pal [92] modeled seismic liquefaction potential using a support vector machine (SVM). In this study, the data were collected from several field tests such as standard penetration test (SPT value) and cone penetration test (CPT) and utilized to assess the liquefaction potential using the SVM-based classification approach. The author concluded that the complex relationship between the liquefaction potential and different soil parameters can effectively be presented using the SVM. Pala et al. [93] employed the ANNs to analyses the dynamic soil-structure interaction of buildings. They used the back-propagation (BP) algorithm. The results showed that the solution time is quite fast and the analysis and modeling stages are minimized. The NNs has shown excellent performance for the solution of soil-structure interaction problems.

Nazzal and Tatari [94] and Park et al. [95] used genetic algorithms and ANN, respectively, to propose a practical model and predict the resilient modulus of subgrade soils. They concluded that the ANN-based predictive models work as a simple and reliable mathematical tool. Groholski and Hashash [96] developed of a framework for extracting dynamic soil performance and pore water pressure response from field data collected in downhole array test (after ASTM D7400). To represent pore water pressure generation during cyclic loading they introduced an ANN-based constitutive model. As a result, the successful presentation of the established model is well demonstrated. Distribution of the papers in the subject of ANN application in soil behavior according to publication years, research areas of their publication and source title is listed in Table 3.

Table 3 Distribution of the papers in the subject of ANN application in geotechnical engineering based on the publication years, research areas of their publication and source title
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2.2 ANN application in pile capacity prediction

Different techniques of ANN, along with experimental experiments, applied in other studies such as Moayedi and Rezaei [20] and Mosallanezhad and Moayedi [53], Nazir et al. [97] and Moayedi [98] to predict pile bearing capacity, pile settlement, pile skin friction and/or pile end-bearing capacity. One of the most basic researches on the pile is provided by Chan et al. [99]. They have released a training dataset using back-propagation neural network to develop a prediction model for the evaluation of the skin friction as well as end-bearing capacity in piles. After comparison between the generated networks, they generate more reliable outputs than a pile driving. In this regard, Ismail and Jeng [100] established a HON-PILE model (high-order neural network model) in order to model the load-settlement behavior of piles. Indeed, a total number of 121 research scholar articles on the applicability of ANNs on the pile bearing capacity (both lateral and axial) were indexed in the WOS (see Table 4). It can be seen that almost 86% of the publications are listed in the category of “engineering geology,” “engineering civil” and “geosciences multidisciplinary.”

Table 4 Distribution of the papers on the applicability of the ANNs in pile capacity based on the research areas
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2.3 ANN application in earth retaining structures

Research on the earth retaining structures has been always one of the main interests between the geotechnical engineers. Countries such as Peoples R China (20.743%), USA (12.384%), England (6.502%), Japan (5.882%), France (5.623%), Italy (4.664%), Canada (4.025%) provided the most published articles on the subject of retiming structures in the WOS. However, when it comes to the ANN applicability, there are very few studies on the applicability of the neural network on the estimation of retaining structures behaviors. Studies such as Li et al. [101] and Chen and Wang [102] worked on the deformation prediction of the pile-anchor retaining structure. Li et al. [101] investigated on the application of neural network to predict displacement of deep foundation pit retaining structure. Their research indicated that the soft computing method is a useful and valid method for prediction of deformation in the foundation pit. Similarly, Chen and Wang [102] used ANN to predict the deformation characteristics of pile-anchor structure (one of the established retaining system) in deep foundation pit. Distribution of the papers on the applicability of the ANNs in earth retaining structures as well as excavation according to the source title is tabulated in Table 5.

Table 5 Distribution of the papers on the applicability of the ANNs in earth retaining structures as well as excavation based on the source title
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The ANN was also used to formalize and synthesize data derived from FE modeling studies. Up to the date of writing this article, there are 48 research scholars indexed in the WOS on the use of the neural network on excavations. In general, the input parameters used in the provided models were the wall stiffness, the soil layer thickness/excavation width ratio, excavation width, soil unit weight, soil undrained shear strength, the height of excavation, and undrained soil modulus/shear strength ratio [103, 104]. The maximum wall deflection was selected as the only output. For instance in braced excavation and in soft clay Goh et al. [44] established an ANN model to estimate maximum wall deflections (normally in the top of the wall). The results produced high accuracy with coefficients of correlation equivalent to 0.984 and 0.967 for the training and testing datasets, respectively.

Kwon and Wilson [105] used NNs to explore the impact of a deep excavation on other adjacent excavations. They applied NNs to investigate the influence of each parameter and the deformation increase on the deformation variation derived from extensometer measurements. Jan et al. [106] also investigated the use of ANN prediction model in the deep excavation. To collect the required data for training and verification, eighteen different case histories of deep excavations, with a minimum of four and maximum of seven excavation (construction) stages, were selected. The results of simulation show that not only the ANN can determine the maximum deflection of the diaphragm wall but also it can predict the location and the magnitude which the maximum deformations occur. Chua and Goh [107] used Bayesian NNs to determine wall deformation behavior in a deep excavation. It is found that the trained model could be used as a reliable and simple prediction tool. They could calculate the maximum wall deformation. Huang et al. [108] investigated the ANN-based reliability analysis for deep excavation. Chern et al. [109] applied a neural network to predict successfully lateral wall deflection in the top–down excavation. Yu et al. [110] used ANNs in artificial intelligent prediction model in order to calculate shallow settlement adjacent to the excavation of a foundation pit.

2.4 ANN application in site characterization

In all geotechnical engineering problems, site characterization is known as an important step that needs to be considered. It is essential to explore the subsurface before doing any project analysis. Many researchers such as Huang and Siller [111], Yilmaz et al. [112], Garcia-Fernandez and Jimenez [113], Orhan et al. [114], Kim et al. [115], Cao et al. [116], Wang [117], Aladejare and Wang [118] and Roy and Jakka [119] worked on this subject. Several researchers also applied ANNs to improve the estimation of the site characterization. For example, in order to represent the data obtained from borehole Huang and Siller [111] developed a fuzzy set-based model which uses to infer the subsurface profile. Bagtzoglou and Hossain [88] used RBFN for hydrologic inversion. The RBFN was used as a reliable method in the context of site characterization. In this regard, Samui and Sitharam [120] employed a relevance vector machine and least-square (LS) SVM based on corrected SPT data in order to estimate site characterization. Samui and Sitharam [121] modeled site characterization using ANN and Kriging. An extensive number of data (2700 field SPT values) were collected from SPTs in 3D subsurface of Bangalore, India.

2.5 ANN application in liquefaction

Soil liquefaction defined as a phenomenon whereby a partially or fully saturated soil (in most cases sands) substantially loses stiffness and strength in response to a specifically applied stress. In this regard, the applied stresses usually are induced by earthquake shaking causing the soil to behave like a liquid (with no shear strength). The liquefaction often leads to extensive damage and very high defamation to most infrastructures. The reason behind such large deformation is that the soil will lose its basic shear strength due to an increase in the pore pressure. Indeed, the soil liquefaction is introduced as one of the multicriteria tasks to assess in geotechnical earthquake engineering. Many experts stated that the assessment of soil liquefaction, due to a lot of variables, is the most complicated phenomena in geotechnical engineering [122,123,124,125,126]. Table 6 presents the distribution of the papers on the applicability of the ANNs in liquefaction evaluation. As one of the earliest researches on the use of ANN in liquefaction assessment, Goh [127] used ANN to solve the complex relationship between different soil parameters and seismic loading applied in order to explore liquefaction potential. The network model was trained using 13 case recorded real-world earthquakes. The study included eight input variables (SPT value, the mean grain size, the fines content, the earthquake magnitude, the equivalent dynamic shear stress, the total and effective stress, and the maximum horizontal acceleration at ground surface) and only one output variable. The output was assigned a no liquefaction (binary value of 0) and, for sites with extensive or moderate liquefaction potential (value of 1). The results gained by the proposed neural network model were compared with the conventional mathematical method that further developed by Seed et al. [128]. In comparison with the success rate of 84% from the method presented by Seed et al. [128], the study revealed that the ANN model gave reliable predictions in 95% of cases. Goh [129], Juang et al. [130], Liu et al. [131] and Chern and Lee [81] used ANN to evaluate liquefaction resistance based on raw CPT data. The results from the neural network showed a minimum success rate of 94%, which is acceptable in comparison with previous evaluation method presented by Shibata and Teparaksa [132] with a success rate of 84%. Wang and Rahman [133] presented a neural network-based model for liquefaction phenomena caused by horizontal ground displacement. Young-Su and Byung-Tak [134] used ANNs to predict liquefaction resistance of sands. Hsu et al. [135] applied ANN to model liquefaction resistance. In their study, a total of 217 sets of shear wave velocity data, 31 from Taiwan after 1999 and 186 from the previous reports and studies, were collected and synthesized. Zhang et al. [136] evaluated soil liquefaction based on multivariate adaptive regression splines and capacity energy concept.

Table 6 Distribution of the papers on the applicability of the ANNs in liquefaction assessment based on the source title
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On the other hand, many others also investigated the use of SVM method in landslide assessment. For instance, Pal [92] employed SVM-based modeling to assess liquefaction potential induced by seismic loading. The data were collected from actual SPT and CPT field data. In fact, SVMs can provide better performance and required few user-defined parameters in comparison with the ANN approach. Similarly, Goh and Goh [137] explored the use of SVM in geotechnical engineering with the main focus on seismic liquefaction data. They trained and tested the SVM model based on a relatively large data set comprising 226 field records of CPT measurements and liquefaction performance. The results of classification showed that the overall success rate for the entire data set is 98%.

2.6 ANN application in slope stability

The slope stability analysis has been always a big challenge for geotechnical engineers. This is of course because of a wide variety of variables affecting the slope stability. Indeed, for most civil engineers working with software (i.e., include many details and variables) is not usually acceptable. Researchers such as Lu and Rosenbaum [138], Li and Liu [139], Liu et al. [140], Zhang et al. [36], Aghajani et al. [141], Rahul et al. [142], Gordan et al. [143], Kostic et al. [11] and Li et al. [144] studied on the subject of ANN application in slope stability. In their approaches, the input parameters were horizontal profile, gradient, location, height, vertical profile, soil texture, geological origin, the direction of slopes, depth of weathering, vegetation, maximum precipitation hour, and maximum daily precipitation. The slope failure potential was taken as the main output. Table 7 presents the distribution of the papers on the applicability of the ANNs in slope stability based on the source title.

Table 7 Distribution of the papers on the applicability of the ANNs in the subject of slope stability based on the source title
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Most practical applications prove that the estimation of slope stability analysis using ANN is achievable. In this regard, a well-trained neural network learning system reveals an extremely fast convergence, a better generalization and a high degree of accuracy for the slope stability problems. Lu and Rosenbaum [138] employed Grey and ANNs systems for the prediction of slope stability. Li and Liu [139] used AI forecast procedures for the slope stability. Liu et al. [140] applied a fast robust NN model called Extreme Learning Machine (ELM) to find a solution for the prediction of slope stability problems. After comparing several ANN techniques, the results prove that, in most of the common slope stability analysis, the ELM act as a helpful way to the genetic algorithm and the GRNN-based models. Gordan et al. [143] combined Particle Swarm Optimization (PSO) and neural network to predict slope stability induced by seismic loading. Kostic et al. [11] developed a model for prediction slope stability based on the ANNs. In this regard, they employed multilayer feed-forward network. The obtained results indicated a high level of statistical reliability.

2.7 ANN application in landslides assessment

The use of ANN in landslide susceptibility mapping is indeed well established. Perhaps the most well-known applicability of the ANN is in the subject of landslides [16, 35]. Indeed, the use of ANNs method in the landslide hazard mapping problems, as one of the multicriteria evaluation method, was introduced in the late 1990s by Yamagami et al. [145], Cai et al. [146] and Kobayashi et al. [147]. Different subjects have been studied using the ANNs method in landslide such as risk assessment [148], susceptibility analysis [149, 150], prediction [151, 152], earthquake-induced/triggering [153, 154], susceptibility mapping by geographical information system [155, 156]. Figure 2 shows the applicability of the NN in the subject of landslide based on both total publications by year and sum of times cited by year (Fig. 2). It can be seen that the number of publications has increased sharply from 10 publications in the year 2005 to 85 publications in the year 2016. Similarly, the number of citations per year raised to 3200 citations per year. This is showing that the subject is still one of the main interests of the researchers. Table 8 presents the distribution of the papers on the applicability of the ANNs in landslide assessment.

Fig. 2
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The subject of ANN application in landslide assessment based on total publications by year and sum of times cited by year

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Table 8 Distribution of the papers on the applicability of the ANNs in landslide assessment based on the journal of their publications, authors, research areas
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3 Research methodology

For the presented review study a new technique called PRISMA method, proposed by Moher et al. [157] was used. It is important to note that the PRISMA statement has two different parts; (1) systematic reviews and (2) meta-analysis. This method is well described in Shamseer et al. [158] and Mardani et al. [159]. Systematic reviews found out about research topics in order to provide summaries from the objectives and what has been written in the literature [157]. Generally, the systematic review part tries to present a full overview of research scholars performed on a specific subject (here it was applicability of ANNs in geotechnical engineering) until the present date. However, when it comes to the meta-analysis part, it offers main findings of statistical approach from previously published works. The main objective of the PRISMA method is to assist practitioners and researchers in finding a complete, simple, and clear literature review [158, 160]. There are many good examples of previous studies which used the PRISMA method in various fields. In overall they intend to present a comprehensive review of the most recent published articles as a literature review [158, 160, 161]. In our review study, we considered three main steps (1) search in articles indexed in WOS, (2) selection of the eligible published articles, and (3) extraction of datasets and summarizing the data [159].

4 Literature search

The database of WOS was used to provide a systematic review of applications and methodologies of ANN-based models in the subject of geotechnical engineering. The most recent published papers were found based on the searching several keywords such as bearing capacity, pile, retaining structures, excavation, site characterization, liquefaction, liquefaction susceptibility, slope stability, landslides and different ANN-based model approaches use in the field of geotechnical engineering. We have chosen those articles from the literature which were published between 2010 and 2018. In this regard, and according to our strategy search, a total of 734 scholarly papers were extracted. In the next step, and after a double check, we duplicated articles with repeated information. As a result, 108 papers were selected (see Fig. 3). Then, to remove the duplicated articles from the selected list, we eliminated 22 records due to duplicates. In the end, screened papers were selected based on different structures of the papers such as titles, keywords, topics, abstracts, and studies that were unrelated to the topic of this review were removed.

Fig. 3
figure 3

Flowchart of the study in regard to the (i) identification, (ii) screening, (iii) eligibility, and (iv) included of articles

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4.1 ANN

The ANN is known as a tool to model the multicriteria and complex systems involved in approximation problems. The theoretical background of the ANN is comprehensively discussed by Hill et al. [162], and Wang [12]. Tables 9 and 10 are listed the distribution of papers in the subject of ANN applicability in the field of geotechnical engineering based on the research areas and source title, respectively. A typical structure of most neural network-based models consists of a number of nodes (or processing elements, PEs), that are typically arranged in form of several layers: one or more hidden layers, an output layer and an input layer [8]. A general flowchart for the ANN models is shown in Fig. 4.

Table 9 Distribution of papers in the subject of ANN applicability in the field of geotechnical engineering based on the research areas
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Table 10 Distribution of papers in the subject of ANN applicability in the field of geotechnical engineering based on the source title
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Fig. 4
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A general flowchart for the ANN models [163]

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4.2 FFNN

FFNN is a simple NN model used for modeling numerous nonlinear phenomena [164, 165]. As established by Hornik [164] it is a standard multilayer FFNN which are able to give a prediction value, in a very specific and satisfying sense, to any measurable function. One interesting capability of FFNN method is to pre-mapping the input data before sending it to the hidden layer for further processing. The selected data set for the training the network is firstly multiplied by Wh (a specific weight matrix). The results will later be added to a bias vector (bh). As in the final step, a transfer function will be applied (e.g., during the data processing in the hidden layer). In fact, the process of network training is a modification of the introduced weight matrixes and bias vectors. This is because the outcome of the trained network needs to be minimized (the distance between both training data and network results). The increase in the number of neurons and layers, for instance, to get a better result, is not desirable. This is because such an increase can lead to a more complex network which later produces problems in both convergence and training of the networks. In this regard, the FFNN method is well described in the literature. For instance, several good examples of using the FFNN technique in the subject of geotechnical engineering are in Han et al. [166], Uncuoglu et al. [167], Lian et al. [168] and Protopapadakis et al. [169]. The FFNN method has been very popular among the researchers. Figure 5 shows the recent use of FFNN method in the scholar papers between 1999 and 2017. In this regard, the total publications by year increased to about 550 publications (indexed in the web of science only) and the summation of times cited by year.

Fig. 5
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FFNN method used in the scholar papers between 1999 and 2017

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4.3 RBFN

RBFN is another widely used MLP network. The RBFN is also well recommended because of its less time consuming during the training of the networks. The structure of an RBFN is like a single layer feed-forward network as shown in Fig. 6. The structure of the RBFN and its application in the subject of geotechnical engineering is well described in Mustafa et al. [170], Shu and Gong [171], Kang et al. [172] and Moayedi and Hayati [74]. The only difference is that in this function (e.g., in all the hidden layers) a radial basis (also called radbas) function is used in comparison with the FFNN function which it was a sigmoid function. The RBFN can simply be defined as Eq. (1).

Fig. 6
figure 6

The structure of a FFNN, b GRNN and c RBFN models

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$${\text{radbas}}\,\left( x \right) = \exp \,\left( { - x^{2} } \right)$$
(1)

The RBFN method has been employed successfully in various research areas. Distribution of papers in the subject of RBFN applicability based on the research areas is tabulated in Table 11. However, the use of RBFN in the field of geotechnical engineering is still remained unknown. In recent years, there are very few studies use RBFN in the field of geotechnical engineering; modeling free-surface seepage flow [173], reliability analysis [174], predicting rock mass deformation modulus [175], three-dimensional simulations of tensile cracks [176], reliability analysis of geotechnical engineering [177], groutability prediction of permeation grouting [178, 179].

Table 11 Distribution of papers in the subject of RBFN applicability based on the research areas
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4.4 GRNN

GRNN includes four separate layers: the first layer is the input layer where the data will be introduced to the network and prepared for the training. The second layer is the pattern layer which follows a specific pattern function. Thirdly, it is the summation layer and finally, the results are generated from the output layer. The use of GRNN has been widely used in the subject of civil engineering and more particularly in the field of geotechnical engineering. Many researchers such as Ibric et al. [180], Pal and Deswal [181], Jiang et al. [182], Goorani and Hamidi [183] and Rajesh and Choudhury [184] used the GRNN through their studies. In this regard, the structure and application of GRNN are also well discussed in Cigizoglu and Alp [185] and Li et al. [186]. In training the neural network, linear activation and RBFN are used in both output and hidden layers. Each pattern layer unit is associated with the two different neurons in the summation layer (called D and S summation neurons). D summation neuron is used to calculate un-weighted outputs of pattern neurons, while S summation neuron computes the sum of weighted responses of the pattern layer [187].

$$Y_{i}^{'} = \frac{{\mathop \sum \nolimits_{i = 1}^{n} y_{i} \cdot \exp - D\left( {x,x_{i} } \right)}}{{\mathop \sum \nolimits_{i = 1}^{n} \exp - D\left( {x,x_{i} } \right)}}$$
(2)

The term D is Gaussian function and it is defined in Eq. (3):

$$D\left( {x,x_{i} } \right) = \mathop \sum \limits_{k = 1}^{m} \left( {\frac{{x_{i} - x_{ik} }}{\sigma }} \right)^{2}$$
(3)

where n the training pattern’s number, m the number of elements applied in the input vector, yi the weight connection (connection between the ith neuron in the pattern layer and the neuron in the S summation), D the defined Gaussian function, xk and xik are the jth elements of x and xi, respectively.

A search in the topic of GRNN shows a total number of 1771 articles in different research areas. Distribution of research scholar papers in the subject of GRNN applicability based on the research areas is listed in Table 12. However, similar to the RBFN, the applicability of the GRNN method in the field of civil engineering is still considered a new topic. There are only a few studies that use the GRNN in the field of geotechnical engineering; compressive strength analysis of reinforced soil [188], slope stability inference [36, 140], lateral load bearing capacity modeling of piles [189], determination of ultimate bearing capacity of concrete driven piles in sand [190], expansive soil characterization [191] and three-dimensional site characterization [89].

Table 12 Distribution of papers in the subject of GRNN applicability based on the research areas
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4.5 Adaptive neuro-fuzzy inference system (ANFIS)

ANFIS is one of the strong learning systems for prediction of complex functions. ANFIS was first proposed by Jang and Sun [192]. Among fuzzy inference systems, it is one of the most commonly used training systems. In fact, ANFIS uses a Takagi–Sugeno fuzzy inference system (FIS). The structure and procedure of the ANFIS are presented and discussed by Jang [193]. General use of ANFIS in geotechnical engineering is well described in other studies such as Cabalar et al. [194]. However, the most common use of ANFIS is in the subject of landslide susceptibility mapping [195, 196], through landslide risk management [197, 198], rock-cutting trencher [199], constitutive modeling [200], prediction of uniaxial strength of rocks [201], liquefaction prediction [202], swelling potential [203], and permeability estimation [204].

4.6 Imperialist competitive algorithm (ICA)

The imperialist competitive algorithm function (also called ICA) is a global method of search population-based that was firstly proposed by Atashpaz-Gargari and Lucas [205] an followed by many other researchers such as Ahmadi et al. [206], Marto et al. [207], Mosallanezhad and Moayedi [53] and Moayedi and Armaghani [50]. The ICA has been used in many optimization problems. This is because it involves a procedure similar to many other evolutionary algorithms such as those used by Thangavelautham et al. [208], Manouchehrian et al. [209], Lian et al. [210] and Gandomi and Kashani [211]. The imperialist competitive algorithm begins with a candidate solution (or initial population), which, along with the imperialist competitive algorithm itself, consists of many countries [205]. In this step, all countries are separated into two main categories (shown in Fig. 7): (1) some of the best countries which called imperialists) and (2) the remaining countries which called colonies. In order to make an empire, first, the colonies required to be distributed through the best countries (called here imperialists or stronger countries). The distribution of the colonies is according to the relative strength of the countries in which the stronger countries could get a higher number of colonies. This competition will continue as the empires intend to expand their territories and control over more colonies. At the end of the competition algorithm (as mentioned by title imperialist competitive algorithm), the stronger empires expanded their power by taking control of weaker colonies. The process is like variables with higher relevancy could impact more on the output layer. Once a predefined stopping criterion is satisfied, the process will stop. A more detailed description of the designed steps in the imperialist competitive algorithm alone is discussed the literature by researchers such as Ghorbani and Jokar [212], and Al Dossary and Nasrabadi [213]. An overview of the imperialist competitive algorithm is depicted in Fig. 8.

Fig. 7
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The schematic procedure of imperialist competitive algorithm to take control of the weaker colony [205]

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Fig. 8
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Overview of the imperialist competitive algorithm [50]

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5 Articles eligibility

For the purpose of eligibility, in this step of review, we independently reviewed each of the full text. However, the selected articles were shortlisted in the last step of the review. The shortlisted articles were selected according to the main objectives of this review paper on the subject of applicability of soft computing and ANN-based techniques in the field of geotechnical engineering. In this regard manuscript that applied ANN (e.g., any source of neural network modeling approach) in the subject of geotechnical engineering were chosen. Noteworthy, editorial notes, book chapters, master and doctoral dissertations, unpublished working papers, a non-English language written papers and textbooks were not involved. In addition, several previous studies have employed other techniques such as geostatistical, limit equilibrium, FE etc.; therefore, in this step also we did not include those studies. At the end of article eligibility decision, we selected 196 articles which used directly the ANN models in the geotechnical engineering. These articles could meet the considered selection criteria.

6 Summarizing and data extraction

In this step of review, some required data was collected and finally 196 articles were summarized and reviewed. In flowing, all selected papers were categorized into different application areas including; engineering, geology, computer science, water resources, environmental sciences, ecology, mechanics, materials science, geochemistry geophysics, mathematics, construction building technology, meteorology atmospheric sciences, mining mineral processing, instruments instrumentation, oceanography, science technology other topics, transportation and operations research management science (see Table 13). In addition, articles were reviewed and summarized according to various criteria such as journals and conferences names, the year of publication, authors, the method and technique used, research objectives, solution and modeling, and results and findings.

Table 13 Distribution of papers in the subject of ANN application in geotechnical engineering based on the application areas
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7 Conclusion

This review paper discussed the applications and theory with several ANN recent developments in the subject of geotechnical engineering. The ANN is introduced as a reliable tool for complex problems. The authors believe the ANN-based model, as a user-friendly and time-saving, is a good alternative to the FEM and conventional mathematical modeling. This is because, normally, problems in the field of geotechnical engineering evolve with many variables which make it hard to be modeled using conventional mathematical methods. Since the ANN-based methods are able to (1) rank the variables and alternatives, (2) evaluate stronger and weaker criteria, and (3) performing the comparative analysis, recently, ANN research interest increased largely in the geotechnical problems. In contrast, it is important to note that the neural networks have begun to replace by deep structured learning. Throughout this review paper, the use of ANN in the subject of geotechnical engineering were categorized into twenty-five research areas. Apart from the research areas, articles were classified according to the authors, source titles (either journal or conferences names), the year of publication, research areas, the used technique, solution and modeling, and outcomes.