Introduction

Water is one of the most important natural resources on earth. One of the fundamental human rights and a requirement for a healthy existence is having access to clean drinking water. Urbanization, population expansion, industrialization, and changes in consumption habits have all led to an increase in the demand for freshwater resources globally (Abu et al. 2024; Bhatt et al. 2024). Like other natural resources, proper management of water resources requires effective and efficient monitoring and assessment strategies which would ensure availability of clean water for present and future generations (Abba et al. 2024; Egbueri 2019). According to Ighalo et al. (2021), water quality monitoring and assessment help to determine its purity and safety. Since a lot of chemicals are utilized in our daily lives and could end up in water resources, monitoring water resources is becoming more and more difficult in the twenty-first century (Mhlongo et al. 2018). Prior to the adoption of data science, traditional methods were predominantly used for monitoring and assessment of natural and human resources. Traditional methods employed in the study of water resources generate sufficient data, which is now explored with the help of data science. Data science, popularly known as the “oil of the twenty-first century” (Shah et al. 2021), is the process of extracting clean information from raw data to create insights that can be put into practice (Yasmin 2019; Kumar 2015). Application of data science in the study of water resources has aided the determination of the quality of water resources (Karmakar et al. 2021; Chojnacki et al. 2017), discovery of complicated patterns and the causes of water pollution (Omeka and Egbueri 2022; Unigwe et al. 2022), and prediction of possible future status of water resources (Egbueri 2022). Data science has also offered solutions in a variety of other fields, such as engineering (Brunton and Kutz 2022), business (Gunjal 2022), marketing (Shi 2022), finance (Hasan and Alam 2021), and meteorology (Sarker 2021), among others. To build models and make predictions using algorithms and other methods, data scientists significantly rely on artificial intelligence (Usman et al. 2023; Yassin et al. 2023, 2022; Leonard et al. 2021), notably its subfields of machine learning (ML) (Egger 2022; El Mrabet et al. 2021) and deep learning (Franzen et al. 2021). Due to its strong nonlinear mapping and learning capabilities, high fault tolerance, and improved generalization capabilities, deep learning has recently become one of the most widely used methods for research on hydrological time series prediction (Jiang et al. 2021; Lu et al. 2019). The five stages of the typical data science life cycle are collecting, maintaining, processing, communicating, and predictive analysis (Bellatreche et al. 2022; Han and Trimi 2022; Sabharwal and Miah 2021). With the use of the data mining approach known as predictive modeling, different factors are examined for their potential impact on a future result (Aryadoust and Goh 2014). Regression, multivariate adaptive regression splines, classification and regression trees, neural networks, and their expansions known as ANFIS are a few important prediction models (Zhang et al. 2021, 2020; Chebrolu et al. 2005).

Artificial neural networks (ANNs), also referred to as neural networks (NNs), are mathematical nonparametric models that consist of a network of “neurons,” which are flexible and trainable processing units that store empirical information (Abba et al. 2023, 2020; Aryadoust and Goh 2014). Similar to the human brain, ANNs are made up of linked units or neurons that are capable of learning, pattern recognition, categorization, and prediction (Geetha et al. 2022; Karaca and Baleanu 2022; Aryadoust and Goh 2014). ANN has become one of the most popular ML tools, finding use in the domains of quality management of different water sources (Khan et al. 2020). The use of ANN as a tool does not necessitate prior understanding of the mathematical structures underlying interactions between inputs and associated outputs (Ewuzie et al. 2022; Nourani et al. 2011; Shahin et al. 2001). Due to their considerable adaptability, ANNs do not place any restrictions on the connections between dependent and independent variables, including those of normality, linearity, homogeneity of variance, and error independence (Aryadoust and Goh 2014). However, the main difficulty with using ANN is its complicated design problems, which might limit its capacity for data processing (Ewuzie et al. 2022). ANN can perform poorly when trained with less data (Ewuzie et al. 2022), since it needs a large number of parameters to get a good outcome (Khalil et al. 2005). Nevertheless, using a lot of weight might lead to overfitting (Farmaki et al. 2010; Zur et al. 2009). Despite these drawbacks, ANN has proven to be reliable in effective monitoring (Azrour et al. 2022; Nicklow et al. 2010; El-Shafie et al. 2009) and assessment (Kouadri et al. 2022; Than et al. 2021) of water resources.

Several studies have successfully predicted physical parameters (Saleh et al. 2022; Egbueri and Agbasi 2022a; Singh et al. 2009), chemical parameters (Egbueri and Agbasi 2022a; El-Safa et al. 2022), and metals (Alizamir and Sobhanardakani 2016; Egbueri 2021) in water with the aid of ANN. To predict using ANN, input/predictor variables and output/predicted variables are selected. Several factors impact the selection of input variables, including parameters that can be measured easily or cheaply, knowledge of the water source, knowledge from literature, theoretical understanding of the parameter(s), and available data from water authorities and monitoring stations (Ewuzie et al. 2022), among others. There are several kinds of neural networks, and they vary in terms of their structure, data flow, number of neurons employed, density, depth of activation filters, and others (Team 2020). They include multilayer perceptron, radial basis function, recurrent neural networks, convolutional neural networks, feedforward neural networks, and others (Praveena and Vivekanandan 2021; Team 2020; Burse et al. 2010). Additionally, a variety of optimizers are available for neural networks, such as gradient descent, stochastic gradient descent, mini-batch gradient descent, Nesterov accelerated gradient, and AdaGrad, among others (Haji and Abdulazeez 2021; Doshi 2019; Dogo et al. 2018; Ruder 2016). The learning rate and weights of NNs are two examples of the variables that these optimizers can adjust in order to change the NNs behavior and decrease losses (Doshi 2019). By selecting the appropriate optimization method, training time may be drastically decreased (Doshi 2019).

PTEs are elements that have the capacity to cause an upset when introduced into a medium (air, water, or soil). Studies have found arsenic (As), cadmium (Cd), chromium (Cr), copper (Cu), iron (Fe), manganese (Mn), nickel (Ni), nitrate (Ni), lead (Pb), zinc (Zn), and other PTEs in polluted levels within natural water bodies (Agbasi et al. 2024; Ayejoto et al. 2022; Fural et al. 2022). These elements can infiltrate water resources through natural processes (Abugu et al. 2024; Ayejoto et al. 2022; Egbueri and Agbasi 2022b) and human-induced processes (Abba et al. 2024; Fural et al. 2022; Jianfei et al. 2020). PTEs have been linked to a wide range of issues, which include anemia (Fural et al. 2022), anomalies in fetal development in pregnant women (Egbueri et al. 2022a), antisocial behaviors observed in children (Emenike et al. 2019), asthma (Rashid et al. 2022), cancer (Zhang et al. 2022), cardiovascular illness (Genchi et al. 2020), diarrhea (Ukah et al. 2019), mental retardation (Mgbenu and Egbueri 2019), respiratory disorders (Wan et al. 2016), sensory disorder (Egbueri 2020), and others. Despite the risks associated with these elements, research has also shown that some PTEs, when found within their allowable limits, have some health benefits (Alipour et al. 2021; Bini and Wahsha 2014). For instance, iron (Fe), when present in water within permissible limits, is advantageous to organisms as it helps circulate oxygen in the blood (Agbasi and Egbueri 2022; IDPH 2010). Globally, PTEs have impacted water resources significantly (Egbueri et al. 2022b; Wagh et al. 2017, 2018). In response, multiple studies have been conducted to understand the impact of PTEs on water resources (Ayejoto et al. 2022; Moghanm et al. 2020), identify their sources (Ricolfi et al. 2020), eradicate (Abbas 2021), and predict the chances of future occurrences (Egbueri and Agbasi 2022a). Several methods have been employed to forecast PTEs in water resources, which include ANN, ANFIS, M5P, multiple linear regression, and others (Egbueri 2021; Eid et al. 2021; Singha et al. 2021). However, most studies have employed ANN in predicting PTEs in water compared to other algorithms.

Studies have revealed that the choice of input variables influences the performance of ANN prediction (Egbueri and Agbasi 2022c; Lee et al. 2021; May et al. 2008). Several studies that used ANNs to predict various elements in water, including PTEs, have been reviewed (Ewuzie et al. 2022; Zounemat-Kermani et al. 2021; Rajaee et al. 2020; Sit et al. 2020). However, attention has not been given to analyzing the suitability of the input parameters utilized for the prediction of PTEs. It is important that the right input variables are selected for effective prediction of PTEs. There are speculative opinions that it is inappropriate to predict PTEs using physicochemical parameters as input variables. Moreover, trends in the applicability of MLP-NN, RBF-NN, and ANFIS algorithms for PTEs’ predictions and the performances of these three modeling techniques have not been analyzed in literature. Therefore, the present review examines the state of art and analyzes the input variables utilized for PTE predictions, the suitability of the input variables, the application trends of the three models, and the comparison of their performances. The specific objectives are to (1) identify the predominantly predicted PTEs in water resources; (2) identify the commonly used input variables for predicting PTEs in water; (3) analyze the suitability of input variables used for predicting PTEs in water; (4) identify the most commonly used ANNs; and (5) compare the performances of MLP-NN, RBF-NN, and ANFIS algorithms in forecasting PTEs in water. To the best of the authors’ knowledge, this is the first review conducted to analyze the suitability of input variables utilized in the prediction of PTEs in water, with a focus on the MLP-NN, RBF-NN, and ANFIS algorithms. This study is also novel, as it is the first to categorically analyze other aforementioned objectives. It is hoped that the findings of this study will aid researchers, water managers, and policymakers in selecting the right parameters for efficient prediction of PTEs in water resources. The current review paper is also expected to contribute economically, scholarly, and timely to the knowledge bank and understanding of PTEs’ predictions in water resources.

Brief history of ANNs

ANNs

ANNs have been used in numerous engineering and scientific applications since the 1940s (Jorjani et al. 2008). The neurophysiologist Warren McCulloch and logician Walter Pits developed the first artificial neuron model in 1943 (McCulloch and Pitts 1943). Simple electrical circuits were used to simulate a neural network by Warren McCulloch and Walter Pits (Zaqoot et al. 2017). The initial attempt to mimic a neural network was led by IBM researcher Nathanial Rochester (McCarthy et al. 1955). It failed at first, but further attempts were successful (Zaqoot et al. 2017). Different architectures, including at least three layers, are found in ANNs (input, hidden, and output layers). There may be one or more hidden layers, and they are situated between the input and output layers (Bayatzadeh Fard et al. 2017). Several neurons make up each layer depending on the layer’s location (Bayatzadeh Fard et al. 2017). Input layer neurons correspond to the number of input variables utilized for prediction, while output layer neurons correspond to the number of variables to be predicted (Bayatzadeh Fard et al. 2017). The neurons have learning, categorization, pattern recognition, and prediction abilities (Aryadoust and Goh 2014). By employing input or independent variables, mathematical functions like the multilayer perceptron (MLP) and radial basis function (RBF) are used to predict output or dependent variables in ANNs with the least amount of error (Aryadoust and Goh 2014). The designer chooses the training algorithm, learning rule, network topology, performance function, and criterion to end the training phase in artificial neural networks, but the system undoubtedly modifies the parameters (Adeoti and Osanaiye 2013). The feedforward topology and the recurrent topology are two significant designs that are frequently used to visualize an ANN (Zaqoot et al. 2017). Due to its link with the backpropagation learning algorithm, a dominant and very reliable learning technique, the feedforward topology is extremely well-liked (Zaqoot et al. 2017). Among the networks using the feedforward topology are the MLP network and the RBF network.

MLP

A perceptron model was first presented in 1958 by Cornell neurobiologist Frank Rosenblatt (Olazaran 1996). The hardware included the perceptron, which he discovered via his scientific efforts (Zaqoot et al. 2017). The MLP is a feedforward neural network ANN with input, hidden, and output layers. It always sends signals in the direction of the output layer (Bayatzadeh Fard et al. 2017). Each layer contains an activation function, which expresses the quantity of output based on the input data mathematically (Aryadoust and Goh 2014). Hyperbolic tangent and logistic functions are examples of mathematical activation functions in the neurons of MLP networks (Aryadoust and Goh 2014). The most commonly used mathematical function in ANN prediction is MLP (Maier and Dandy 2000).

RBF

According to Suen and Eheart (2003), radial basis function neural networks were created about the same time by Powell (1987) and Broomhead and Lowe (1988). Similar to the MLP, the RBF network also consists of an input layer, a hidden layer, and an output layer, with neurons present in each layer. In contrast to MLP, RBF only comprises weights between the hidden layer and the output layer. The hidden layer is where the most significant differences between RBF and MLP can be found. These differences may be divided into structural and functional differences (Ucun Ozel et al. 2020). The number of neurons and the training technique are responsible for the structural difference, but the hidden layer neurons’ inclusion of radial functions is responsible for the functional difference (Ucun Ozel et al. 2020). Because the RBF function depends on the distance from the origin to gather the input layer neurons, variations in it are depending on radial distance (Alizamir and Sobhanardakani 2017). The Gaussian function is primarily used as the activation function in the neurons of RBF networks (Asgharnia et al. 2019; Han et al. 2019).

ANFIS

An extension of artificial neural networks (ANNs) and adaptive neuro-fuzzy inference systems (ANFIS) merge ANNs with fuzzy (Landín et al. 2009). The fuzzy inference system (FIS) examines human thinking by incorporating uncertainty into if-then rules and human knowledge (Mousavi and Amiri 2012). The fuzzy rule base, membership functions, which specify the fuzzy sets of fuzzy rules, and a reasoning process are the three fundamental components of ANFIS (Ucun Ozel et al. 2020). In order to determine the membership function parameters, ANFIS employs gradient descent-based optimization techniques (Ucun Ozel et al. 2020). There are three different methodologies employed in developing the ANFIS model, namely the fuzzy C-means (FCM) technique, grid partitioning (GP), and the subtractive clustering method (SCM) (Bayatzadeh Fard et al. 2017). The SCM and FCM methods may be used for multi-output ANFIS, in contrast to the GP approach, which produces a single-output Sugeno-type ANFIS on the data (Bayatzadeh Fard et al. 2017). In general, an input-output data set is needed in order to use the ANFIS approach.

Selection of articles for the review study

The primary focus of this review study is on the adequacy of input variables utilized by previous authors for three ANN techniques (MLP, RBF, and ANFIS) to forecast PTEs in water resources. The relevant papers were found by conducting a keyword search of articles on Google, Goggle scholar, and Research Gate that had been published over the years on the subject, using terms like “ANN,” ‘predicting,” “forecasting,” “modelling,” “heavy metals,” “water,” and “potentially toxic elements,” alongside with the names of the modeling approaches, like “MLP,” “RBF,” and “ANFIS.” Then, among the search engine results displayed, the most pertinent articles were chosen after careful examinations of their contents.

Classification schemes of model performance metrics

Correlation coefficient (R) and coefficient of determination (R 2)

The correlation coefficient (R) is a statistical measure used to assess the degree of association between two quantitative variables recorded in each individual member of a group (Eq. 1; Nazar et al. 2023; Aggarwal and Ranganathan 2016). The coefficient of determination, also known as the square of the correlation coefficient (R2), is the percentage of variation in one variable that is accounted for by variation in the other variable (Eq. 2; Egbueri and Agbasi 2022c; Aggarwal and Ranganathan 2016). The values of R and R2 can range from −1.0 to +1.0. The qualitative description of R values according to Egbueri (2021) is as follows: strong correlation (0.75–1.00), moderate correlation (0.50–0.75), and weak correlation (r < 0.5).

$$R=\frac{\sum_{i=1}^n\kern0.1em \left({a}_i-{\overline{a}}_i\right)\left({p}_i-{\overline{p}}_i\right)}{\sqrt{\sum_{i=1}^n\kern0.1em {\left({a}_i-{\overline{a}}_i\right)}^2\sum_{i=1}^n\kern0.1em {\left({p}_i-{\overline{p}}_i\right)}^2}}$$
(1)
$${R}^2=1-\frac{\sum_{i=1}^n\kern0.1em {\left({X}_{\textrm{predicted}\ \textrm{value}}-{X}_{\textrm{measured}\ \textrm{value}}\right)}^2}{\sum_{i=1}^n\kern0.1em {\left({X}_{\textrm{predicted}\ \textrm{value}}-{X}_{\textrm{average}\ \textrm{value}}\right)}^2}$$
(2)

where pi and ai represent the ith predicted and observed scores, respectively. \({\overline{p}}_i\) and \({\overline{a}}_i\) denote the mean predicted and observed scores across n total observations.

Mean square error (MSE)

A statistical technique for calculating the difference between an estimator and an estimated outcome is the MSE, often known as the mean squared deviation (MSD) of an estimator (Sara et al. 2019). The MSD or MSE calculates the average of the square of the errors between an estimator and an estimated outcome (Eq. 3; Pande et al. 2024; Sara et al. 2019). The key benefit of utilizing MSE is that it squares the error, which penalizes or prominently highlights large errors (Allwright 2022). Therefore, it is helpful when working on models where it is necessary to minimize sporadic significant errors (Allwright 2022). Models with MSE values nearer to zero are more accurate. Nevertheless, a “good” value for MSE does not exist (Allwright 2022). This is because MSE is an absolute metric specific to each use case; results may only be compared to other MSE values computed for the same dataset (Allwright 2022).

$$\textrm{MSE}=\frac{1}{n}\sum_{i=1}^n\kern0.1em {\left({y}_{-}i-{\hat{y}}_{-}i\right)}^2$$
(3)

where n is the total count of data points. For the ith sample, y−i refers to the true, observed value of the target variable, while \({\hat{y}}_{-}i\) shows the value forecasted by the model for that same sample.

Root mean square error (RMSE)

The square root of the MSE is what generates the RMSE (Eq. 4; Pande et al. 2024; Sara et al. 2019). According to Draper et al. (2013), the RMSE is also referred to as the root mean square deviation (RMSD). The difference between an estimator’s forecasted value and the actual value is often measured using the RMSE (Sara et al. 2019). It calculates the differences in predicting errors from the various estimators for a certain variable and analyzes the error size, making it the ideal accuracy measure (Sara et al. 2019). Like the MSE, models with RMSE values closer to zero are more accurate.

$$\textrm{RMSE}=\sqrt{MSE}$$
(4)

Predominantly predicted PTEs in water resources

MLP

The predicted PTEs in water resources using the MLP-NN algorithm by the studies reviewed (24 articles) include Al, As, Cd, Cr, Cu, Fe, Hg, Mn, Ni, NO3, Pb, Sb, Sr, Ti, and Zn. The degree to which they were predicted by the studies is as follows: Pb > Zn > As > Fe > Mn > Cu > NO3 > Ni > Cd > Al = Cr = Hg = Sb = Sr = Ti. A graphical representation of their distribution can be found in Fig. 1a.

Fig. 1
figure 1

Bar charts showing PTEs and number of studies that predicted them using a MLP-NN, b RBF-NN, c ANFIS

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RBF

The predicted PTEs in water resources using the RBF-NN algorithm by the studies reviewed (8 articles) include As, Cd, Cu, Fe, Hg, Mn, Ni, NO3, Pb, and Zn. The degree to which they were predicted by the studies is as follows: NO3 > Zn = Pb > As = Cd, Cu = Fe = Hg = Mn = Ni. A graphical representation of their distribution can be found in Fig. 1b.

ANFIS

The predicted PTEs in water resources using the ANFIS algorithm by the studies reviewed (10 articles) include As, Cd, Cu, Fe, Mn, Ni, NO3, Pb, and Zn. The degree to which they were predicted by the studies is as follows: NO3 = Fe = Mn = Zn > As = Cd = Cu = Pb > Ni. A graphical representation of their distribution can be found in Fig. 1c.

Commonly used input parameters for predicting PTEs in water resources

From the articles reviewed (42 articles), seven groups of input variables were found to be commonly used for predicting PTEs in water resources. These groups of input variables include Group 1, a combination of physical parameters (P), chemical parameters (C), and metals (M); Group 2, a combination of P and C only; Group 3, a combination of P and M only; Group 4, a combination of C and M only; Group 5, P only; Group 6, C only; and Group 7, M only. Physical parameters include pH, temperature, TDS, SS, BOD, and COD; chemical parameters include HCO3, SO4, NO3, and Cl; and metals include As, Cd, Cr, Pb, and Zn. The full details of the input parameters utilized by 42 articles for forecasting PTEs in water resources can be found in Tables 1, 2, and 3.

Table 1 Synopsis of MLP-NN algorithm used for forecasting PTEs in water resources
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Table 2 Synopsis of RBF-NN algorithm used for forecasting PTEs in water resources
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Table 3 Synopsis of ANFIS algorithm used for forecasting PTEs in water resources
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MLP

The pictorial representation of the distribution of input variables used to predict PTEs in water resources with the aid of the MLP-NN algorithm can be visualized in Fig. 2a. It was observed that most of the studies used Group 1 input variables for their prediction. The degree to which the reviewed studies utilized input variables when the MLP-NN algorithm was employed is as follows: Group 1 > Group 5 > Group 7 > Group 2 = Group 4 > Group 6 > Group 3.

Fig. 2
figure 2

Bar charts showing input variables utilized for prediction of PTEs employing a MLP-NN, b RBF-NN, c ANFIS

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RBF

The pictorial representation of the distribution of input variables used to predict PTEs in water resources with the aid of the RBF-NN algorithm can be visualized in Fig. 2b. It was observed that most of the studies used Groups 5 and 7 input variables for their prediction. The degree to which the reviewed studies utilized input variables when the RBF-NN algorithm was employed is as follows: Group 5 = Group 7 > Group 1 = Group 2 = Group 3 > Group 4 = Group 6.

ANFIS

The pictorial representation of the distribution of input variables used to predict PTEs in water resources with the aid of the ANFIS algorithm can be visualized in Fig. 2c. It was observed that most of the studies used Group 1 input variables for their prediction. The degree to which the reviewed studies utilized input variables is as follows: Group 1 > Group 2 = Group 5 > Group 7 > Group 3 = Group 4 = Group 6.

Suitability of input variables in the prediction of PTEs in water resources

To ascertain the suitability of input variables in the prediction of PTEs in water resources using MLP-NN, RBF-NN, and ANFIS algorithms, data from 148 models (MLP-NN algorithm, 86 models, RBF-NN algorithm, 29 models, and ANFIS algorithm, 33 models) was extracted from the 42 articles reviewed. However, 9 models (MLP-NN algorithm-1 model, RBF-NN algorithm-2 models, and ANFIS algorithm-6 models) did not report R2 or R values and thus were not considered (Tables 4, 5, and 6). Thus, a total of 139 models (MLP-NN algorithm, 85 models; RBF-NN algorithm, 27 models; and ANFIS algorithm, 27 models) were used to ascertain the suitability of input variables in forecasting PTEs in water resources using MLP-NN, RBF-NN, and ANFIS algorithms. The models were evaluated based on the seven groups of input variables mentioned in the previous section. The seven groups of input variables utilized by the three ML algorithms were compared with the model’s R2 or R. In scenarios where R2 and R were given, the outcomes of R2 were taken. This is because R2 has a more straightforward interpretation and is more widely used than R for ANN modeling. Moreover, it aligns more with the study objectives, as it represents the proportion of variance in the dependent variable that is explained by the independent variable(s) in the model. Based on their R2 or R values, the performances of the models were classified into weak, moderate, and strong correlation.

Table 4 Models of PTEs in water resources produced using MLP-NN algorithm
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Table 5 Models of PTEs in water resources produced using RBF-NN algorithm
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Table 6 Models of PTEs in water resources produced using ANFIS algorithm
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MLP

Physical parameters, chemical parameters, and metals combined as input variables

A total of 22 models combined Group 1 parameters (P, C, and M) as input variables for prediction of PTEs in water resources using the MLP-NN algorithm. Based on the performance metrics used (R2 or R values), 1/22, 2/22, and 19/22 of the models can be classified as having weak, moderate, and strong correlations, respectively (Table 7). The distribution of the model performance can be visualized in Fig. 3a. The distribution shows that 4.50%, 9.10%, and 86.40% of models that used Group 1 parameters as input variables for predicting PTEs in water resources had weak, moderate, and strong correlations, respectively. In summary, 95.5% of the models evaluated had an acceptable model performance (moderate-strong correlation). Thus, it can be concluded that a combination of only physical parameters, chemical parameters, and metals as input variables is suitable for predicting PTEs in water resources employing the MLP-NN algorithm.

Table 7 Groups of input variables and model performances in forecasting PTEs in water using the MLP-NN algorithm
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Fig. 3
figure 3

Bar charts showing performance of MLP-NN models that predicted PTEs in water using the following as input variables: a Group 1, b Group 2, c Group 3, d Group 4, e Group 5, f Group 6, and g Group 7

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Physical and chemical parameters only combined as input variables

A total of 22 models combined Group 2 parameters (P and C only) as input variables for prediction of PTEs in water resources using the MLP-NN algorithm. Based on the performance metrics used, 0/22, 6/22, and 16/22 of the models can be classified as having weak, moderate, and strong correlations, respectively (Table 7). The distribution of the model performance can be visualized in Fig. 3b. The distribution shows that 0%, 27.27%, and 72.73% of models that used Group 2 parameters as input variables for predicting PTEs in water resources had weak, moderate, and strong correlations, respectively. In summary, 100% of the models evaluated had acceptable model performance. Thus, it can be concluded that a combination of only physical and chemical parameters as input variables is suitable for predicting PTEs in water resources employing the MLP-NN algorithm.

Physical parameters and metals only combined as input variables

A total of 4 models combined Group 3 parameters (P and M only) as input variables for prediction of PTEs in water resources using the MLP-NN algorithm. Based on the performance metrics used, 1/4, 2/4, and 1/4 of the models can be classified as having weak, moderate, and strong correlations, respectively (Table 7). The distribution of the model performance can be visualized in Fig. 3c. The distribution shows that 25%, 50%, and 25% of models that used Group 3 parameters as input variables for predicting PTEs in water resources had weak, moderate, and strong correlations, respectively. In summary, 75% of models that used Group 3 parameters as input variables had an acceptable model performance. Thus, it can be concluded that a combination of physical parameters and metals as input variables is suitable for predicting PTEs in water resources employing the MLP-NN algorithm.

Chemical parameters and metals only combined as input variables

A total of 6 models combined Group 4 parameters (C and M only) as input variables for prediction of PTEs in water resources using the MLP-NN algorithm. Based on the performance metrics used, 2/6, 2/6, and 2/6 of the models can be classified as having weak, moderate, and strong correlations, respectively (Table 7). The distribution of the model performance can be visualized in Fig. 3d. The distribution shows that 33.33%, 33.33%, and 33.33% of models that used Group 4 parameters as input variables for predicting PTEs in water resources had weak, moderate, and strong correlations, respectively. In summary, 66.66% of the models evaluated had acceptable model performance. Thus, it can be concluded that a combination of chemical parameters and metals as input variables is suitable for predicting PTEs in water resources employing the MLP-NN algorithm.

Physical parameters only as input variables

A total of 13 models combined Group 5 parameters (P only) as input variables for prediction of PTEs in water resources using the MLP-NN algorithm. Based on the performance metrics used, 0/13, 2/13, and 11/13 of the models can be classified as having weak, moderate, and strong correlations, respectively (Table 7). The distribution of the model performance can be visualized in Fig. 3e. The distribution shows that 0%, 15.38%, and 84.62% of models that used Group 5 parameters as input variables for predicting PTEs in water resources had weak, moderate, and strong correlations, respectively. In summary, 100% of the models evaluated had acceptable model performance. Thus, it can be concluded that utilizing only physical parameters as input variables is suitable for predicting PTEs in water resources employing the MLP-NN algorithm.

Chemical parameters only as input variables

A total of 3 models combined Group 6 parameters (C only) as input variables for prediction of PTEs in water resources using the MLP-NN algorithm. Based on the performance metrics used, 0/3, 3/3, and 0/3 of the models can be classified as having weak, moderate, and strong correlations, respectively (Table 7). The distribution of the model performance can be visualized in Fig. 3f. The distribution shows that 0%, 100%, and 0% of models that used Group 6 parameters as input variables for predicting PTEs in water resources had weak, moderate, and strong correlations, respectively. In summary, 100% of the models evaluated had acceptable model performance. Thus, it can be concluded that utilizing only chemical parameters as input variables is suitable for predicting PTEs in water resources employing the MLP-NN algorithm.

Metals only as input variables

A total of 15 models combined Group 7 parameters (M only) as input variables for prediction of PTEs in water resources using the MLP-NN algorithm. Based on the performance metrics used, 1/15, 2/15, and 12/15 of the models can be classified as having weak, moderate, and strong correlations, respectively (Table 7). The distribution of the model performance can be visualized in Fig. 3g. The distribution shows that 6.67%, 13.33%, and 80% of models that used Group 7 parameters as input variables for predicting PTEs in water resources had weak, moderate, and strong correlations, respectively. In summary, 93.33% of the models evaluated had acceptable model performance. Thus, it can be concluded that utilizing only metals as input variables is suitable for predicting PTEs in water resources employing the MLP-NN algorithm.

RBF

Physical parameters, chemical parameters, and metals combined as input variables

A total of 2 models combined only physical parameters (P), chemical parameters (C), and metals as input variables for prediction of PTEs in water resources using the RBF-NN algorithm. Based on the performance metrics used, 0/2, 0/2, and 2/2 of the models can be classified as having weak, moderate, and strong correlations, respectively (Table 8). The distribution of the model performance can be visualized in Fig. 4a. The distribution shows that 0%, 0%, and 100% of models that combined P, C, and M parameters as input variables for predicting PTEs in water resources had weak, moderate, and strong correlations, respectively. In summary, 100% of the models evaluated had acceptable model performance. Thus, it can be concluded that a combination of only physical parameters, chemical parameters, and metals as input variables is suitable for predicting PTEs in water resources employing the RBF-NN algorithm.

Table 8 Groups of input variables and model performances in forecasting PTEs in water using the RBF-NN
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Fig. 4
figure 4

Bar charts showing performance of RBF-NN models that predicted PTEs in water using the following as input variables: a Group 1, b Group 2, c Group 3, d Group 5, and e Group 7

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Physical and chemical parameters only combined as input variables

A total of 3 models combined only physical and chemical parameters as input variables for the prediction of PTEs in water resources using the RBF-NN algorithm. Based on the performance metrics used, 0/3, 1/3, and 2/3 of the models can be classified as having weak, moderate, and strong correlations, respectively (Table 8). The distribution of the model performance can be visualized in Fig. 4b. The distribution shows that 0%, 33.33%, and 66.67% of models that combined only P and C parameters as input variables for predicting PTEs in water resources had weak, moderate, and strong correlations, respectively. In summary, 100% of the models evaluated had acceptable model performance. Thus, it can be concluded that a combination of only physical and chemical parameters as input variables is suitable for predicting PTEs in water resources employing the RBF-NN algorithm.

Physical parameters and metals only combined as input variables

A total of 3 models combined only physical parameters and metals as input variables for prediction of PTEs in water resources using the RBF-NN algorithm. Based on the performance metrics used, 0/3, 1/3, and 2/3 of the models can be classified as having weak, moderate, and strong correlations, respectively (Table 8). The distribution of the model performance can be visualized in Fig. 4c. The distribution shows that 0%, 33.33%, and 66.67% of models that combined only P and M parameters as input variables for predicting PTEs in water resources had weak, moderate, and strong correlations, respectively. In summary, 100% of the models evaluated had acceptable model performance. Thus, it can be concluded that a combination of physical parameters and metals as input variables is suitable for predicting PTEs in water resources employing the RBF-NN algorithm.

Chemical parameters and metals only combined as input variables

Out of the 139 models considered in the review, none combined only chemical parameters and metals as input variables for prediction of PTEs in water resources using the RBF-NN algorithm (Table 8). Therefore, there is no conclusion with regard to the suitability of combining only chemical parameters and metals as input variables for predicting PTEs in water resources employing the RBF-NN algorithm.

Physical parameters only as input variables

A total of 13 models combined only physical parameters as input variables for the prediction of PTEs in water resources using the MLP-NN algorithm. Based on the performance metrics used, 0/13, 6/13, and 7/13 of the models can be classified as having weak, moderate, and strong correlations, respectively (Table 8). The distribution of the model performance can be visualized in Fig. 4d. The distribution shows that 0%, 46.15%, and 53.85% of models that combined only P parameters as input variables for predicting PTEs in water resources had weak, moderate, and strong correlations, respectively. In summary, 100% of the models evaluated had acceptable model performance. Thus, it can be concluded that utilizing only physical parameters as input variables is suitable for predicting PTEs in water resources employing the RBF-NN algorithm.

Chemical parameters only as input variables

Out of the 139 models considered in the review, none combined only chemical parameters as input variables for prediction of PTEs in water resources using the RBF-NN algorithm (Table 8). Therefore, there is no conclusion with regard to the suitability of combining only chemical parameters as input variables for predicting PTEs in water resources employing the RBF-NN algorithm.

Metals only as input variables

A total of 6 models combined only metals as input variables for prediction of PTEs in water resources using the MLP-NN algorithm. Based on the performance metrics used, 0/6, 0/6, and 6/6 of the models can be classified as having weak, moderate, and strong correlation, respectively (Table 8). The distribution of the model performance can be visualized in Fig. 4e. The distribution shows that 0%, 0%, and 100% of models that combined only M parameters as input variables for predicting PTEs in water resources had weak, moderate, and strong correlations, respectively. In summary, 100% of the models evaluated had acceptable model performance. Thus, it can be concluded that utilizing only metals as input variables is suitable for predicting PTEs in water resources employing the RBF-NN algorithm.

ANFIS

Physical parameters, chemical parameters, and metals combined as input variables

A total of 3 models combined only physical parameters (P), chemical parameters (C), and metals as input variables for prediction of PTEs in water resources using the ANFIS algorithm. Based on the performance metrics used, 0/3, 0/3, and 3/3 of the models can be classified as having weak, moderate, and strong correlations, respectively (Table 9). The distribution of the model performance can be visualized in Fig. 5a. The distribution shows that 0%, 0%, and 100% of models that combined P, C, and M parameters as input variables for predicting PTEs in water resources had weak, moderate, and strong correlations, respectively. In summary, 100% of the models evaluated had acceptable model performance. Thus, it can be concluded that a combination of only physical parameters, chemical parameters, and metals as input variables is suitable for predicting PTEs in water resources employing the ANFIS algorithm.

Table 9 Groups of input variables and model performances in forecasting PTEs in water using the ANFIS
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Bar charts showing performance of ANFIS models that predicted PTEs in water using the following as input variables: a Group 1, b Group 2, c Group 3, d Group 5, and e Group 7

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Physical and chemical parameters only combined as input variables

A total of 11 models combined only physical and chemical parameters as input variables for the prediction of PTEs in water resources using the ANFIS algorithm. Based on the performance metrics used, 0/11, 1/11, and 10/11 of the models can be classified as having weak, moderate, and strong correlations, respectively (Table 9). The distribution of the model performance can be visualized in Fig. 5b. The distribution shows that 0%, 9.09%, and 90.91% of models that combined only P and C parameters as input variables for predicting PTEs in water resources had weak, moderate, and strong correlations, respectively. In summary, 100% of the models evaluated had acceptable model performance. Thus, it can be concluded that a combination of only physical and chemical parameters as input variables is suitable for predicting PTEs in water resources employing the ANFIS algorithm.

Physical parameters and metals only combined as input variables

A total of 4 models combined only physical parameters and metals as input variables for prediction of PTEs in water resources using the ANFIS algorithm. Based on the performance metrics used, 0/4, 3/4, and 1/4 of the models can be classified as having weak, moderate, and strong correlations, respectively (Table 9). The distribution of the model performance can be visualized in Fig. 5c. The distribution shows that 0%, 75%, and 25% of models that combined only P and M parameters as input variables for predicting PTEs in water resources had weak, moderate, and strong correlations, respectively. In summary, 100% of the models evaluated had acceptable model performance. Thus, it can be concluded that a combination of physical parameters and metals as input variables is suitable for predicting PTEs in water resources employing the ANFIS algorithm.

Chemical parameters and metals only combined as input variables

Out of the 139 models considered in the review, none combined only chemical parameters and metals as input variables for prediction of PTEs in water resources using the ANFIS algorithm (Table 9). Therefore, there is no conclusion with regard to the suitability of combining only chemical parameters and metals as input variables for predicting PTEs in water resources employing the ANFIS algorithm.

Physical parameters only as input variables

A total of 8 models combined only physical parameters as input variables for the prediction of PTEs in water resources using the ANFIS algorithm. Based on the performance metrics used, 0/8, 2/8, and 6/8 of the models can be classified as having weak, moderate, and strong correlations, respectively (Table 9). The distribution of the model performance can be visualized in Fig. 5d. The distribution shows that 0%, 25%, and 75% of models that combined only P parameters as input variables for predicting PTEs in water resources had weak, moderate, and strong correlations, respectively. In summary, 100% of the models evaluated had acceptable model performance. Thus, it can be concluded that utilizing only physical parameters as input variables is suitable for predicting PTEs in water resources employing the ANFIS algorithm.

Chemical parameters only as input variables

Out of the 139 models considered in the review, none combined only chemical parameters as input variables for prediction of PTEs in water resources using the ANFIS algorithm (Table 9). Therefore, there is no conclusion with regard to the suitability of combining only chemical parameters as input variables for predicting PTEs in water resources employing the ANFIS algorithm.

Metals only as input variables

One model combined only metals as input variables for prediction of PTEs in water resources using the ANFIS algorithm. Based on the performance metrics used, the model can be classified as having a strong correlation (Table 9). The distribution of the model performance can be visualized in Fig. 5e. The distribution shows that 100% of models that combined only M parameters as input variables for predicting PTEs in water resources had a strong correlation. Therefore, 100% of the models evaluated had acceptable model performance. Thus, it can be concluded that utilizing only metals as input variables is suitable for predicting PTEs in water resources employing the ANFIS algorithm.

Performances of MLP-NN, RBF-NN, and ANFIS in predicting PTEs in water

To understand the overall performance of the three algorithms (MLP-NN, RBF-NN, and ANFIS) in forecasting PTEs in water resources, we aggregated their performances across the seven groups of input variables. For the MLP-NN algorithm, 85 models were evaluated and 5.88%, 22.35%, and 71.77% of the models were classified into weak, moderate, and strong correlation, respectively (Table 10, Fig. 6a). Thus, 94.12% of the models that used the MLP-NN in forecasting PTEs in water resources had acceptable model performance (moderate-strong correlation). For the RBF-NN algorithm, 27 models were evaluated and 0%, 29.63%, and 70.37% of the models were classified into weak, moderate, and strong correlation, respectively (Table 11, Fig. 6b). Thus, 100% of the models that used the RBF-NN in forecasting PTEs in water resources had acceptable model performance. For the ANFIS algorithm, 27 models were evaluated, and 0%, 22.22%, and 77.78% of the models were classified into weak, moderate, and strong correlation, respectively (Table 12, Fig. 6c). Thus, 100% of the models that used ANFIS in forecasting PTEs in water resources had acceptable model performance. Based on the percentage of acceptable models of PTEs produced by the three algorithms, the RBF-NN and ANFIS clearly outperformed the MLP-NN. Nevertheless, the ANFIS algorithm had a higher percentage of strongly correlated models (77.78%), compared to 70.37% ascertained by the RBF-NN algorithm (Tables 10 and 12). Therefore, the overall performance of the three algorithms in forecasting PTEs in water resources can be rated as follows: ANFIS > RBF-NN > MLP-NN.

Table 10 Overall performance of the MLP-NN models in forecasting PTEs in water
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Fig. 6
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Bar charts showing overall performance of models that predicted PTEs in water using a MLP-NN, b RBF-NN, and c ANFIS

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Table 11 Overall performance of the RBF-NN models in forecasting PTEs in water
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Table 12 Overall performance of the ANFIS models in forecasting PTEs in water
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Conclusions

It is anticipated that the findings of the review study would ultimately help in protecting the lives of a wide range of human populations, especially those who are the most vulnerable to water pollution by PTEs. In addition, insights drawn from this study will aid cost-effective and efficient water quality monitoring, assessment, prediction, management, and sustainability. The current review study identified the most commonly used ANN algorithm, the most commonly predicted PTEs, the most commonly used input variables, the most suitable input variables, and the most efficient ANN algorithm for predicting PTEs in water. These were achieved after a careful evaluation of 139 models from 42 articles that employed the three ML algorithms (MLP-NN, RBF-NN, and ANFIS) to predict PTEs in water resources. Based on the findings, the following conclusions were made:

  • The MLP-NN was the most commonly employed algorithm among the three analyzed ML algorithms.

  • The most commonly predicted PTEs using the MLP-NN algorithm were Fe, Zn, and As. For the RBF-NN algorithm, they were NO3, Zn, and Pb, and for the ANFIS, they were NO3, Fe, and Mn.

  • The input variables utilized by the three ML algorithms to predict PTEs in water resources can be grouped into seven: Group 1 consists of physical parameters (P), chemical parameters (C), and metals (M). Group 2 contains only P and C; Group 3 contains only P and M; Group 4 contains only C and M; Group 5 contains only P; Group 6 contains only C; and Group 7 contains only M.

  • The MLP-NN algorithm used parameters in Group 1 most and Group 3 least as input variables for prediction of PTEs in water resources. The RBF-NN algorithm used parameters in Groups 5 and 7 most and least used those in Groups 4 and 6 as input variables for prediction of PTEs in water resources. The ANFIS algorithm used parameters in Group 1 most, while using Groups 3, 4, and 6 least as input variables for prediction of PTEs in water resources.

  • For MLP-NN, RBF-NN, and ANFIS algorithms, the use of Groups 1, 2, 3, 5, and 7 parameters proved to be suitable input variables for forecasting PTEs in water resources. Thus, we encourage future research to make use of mentioned input variables to predict PTEs. However, the suitability of Groups 4 and 6 parameters using the RBF-NN and ANFIS algorithms could not be ascertained due to non-selection of the mentioned groups as input variables. Nevertheless, models that utilized Groups 4 and 6 parameters as input variables using the MLP-NN algorithm showed that they were suitable for the predictions.

  • The overall order of performance of the three algorithms in predicting PTEs in water resources is ANFIS > RBF-NN > MLP-NN.

  • The findings of the review agree with previous studies which suggest that MLP-NN, RBF-NN, and ANFIS are reliable algorithms for predicting PTES in water resources.

While key input variables have been identified for predicting PTEs in water bodies, it is crucial to recognize that regional peculiarities can significantly influence model performance. Therefore, in addition to the identified variables, dominant regional water quality influencers should be accounted for to enhance the robustness and applicability of the predictive models across diverse geographical contexts.

Recommendations and perspectives for future research

Based on the findings of the literature review and analysis, the following concerns and recommendations are provided to be addressed in future studies:

  • The key challenge faced during the course of this study was extracting data from literature. Future studies should ensure model performances are summarized in a table for easier access.

  • Future studies could consider the suitability of Group 4 (chemical and physical parameters only) and Group 6 (chemical parameters only) as input variables in modeling of PTEs in water resources using RBF-NN and ANFIS algorithms.

  • Future studies could analyze the suitability and sensitivity of the specific (individual) input variables in the prediction of PTEs in water resources.

  • Future studies could analyze the suitability of groups of input variables in the prediction of specific (individual) PTEs (i.e., the outputs) in water resources.

  • Future studies could investigate new groups of input variables that can be used individually or combined to forecast PTEs in water resources using the ML algorithms.

  • For regions where existing data is available, principal component analysis and correlation techniques can facilitate the selection of the most relevant input variables from the identified high-performing groups of predictors.

  • Future studies could employ more advanced ML algorithms in forecasting PTEs in water resources and compare them with these neural network algorithms.