# Improving Results of Existing Groundwater Numerical Models Using Machine Learning Techniques: A Review

## Abstract

**:**

## 1. Introduction

#### 1.1. Physically Based Models in Groundwater Management

#### 1.2. Uncertainty and Error Types of Physically-Based Models

#### 1.3. Machine Learning Models

#### 1.4. Machine Learning for Groundwater Level Forecasting: Current State of the Research

#### 1.5. Aim of This Work

## 2. Modelling Techniques Explored in This Review

#### 2.1. Physically Based Numerical Groundwater Flow Models

#### 2.2. Machine Learning Models

#### 2.2.1. Artificial Neural Networks (ANNs)

#### 2.2.2. Radial Basis Function Network (RBF)

#### 2.2.3. Adaptive Neuro-Fuzzy Inference System (ANFIS)

#### 2.2.4. Time Lagged Recurrent Neural Networks (TLRNs)

#### 2.2.5. Extreme Learning Machine (ELM)

#### 2.2.6. Bayesian Network (BN)

#### 2.2.7. Instance-Based Weighting (IBW)

#### 2.2.8. Support Vector Machine (SVM)

#### 2.2.9. Decision Trees (DT)

#### 2.2.10. Random Forest (RF)

#### 2.2.11. Gradient-Boosted Regression Trees (GBRT)

## 3. Bibliographic Review

#### 3.1. Comparing Results of a Physically-Based Model and a Surrogate Machine Learning Model

^{2}), mean squared error (MSE), and normalised mean squared error (NMSE). The water table was estimated with reasonable accuracy by all the models, but the ANN required lesser input data and took less time to run. However, the authors remarked two disadvantages of these networks: (i) the water table cannot be predicted in all observation wells by a single model with similar input parameters; and (ii) models are static and inputs and outputs from previous time steps are not considered (unless these are introduced explicitly). This results in a high difference between the observed and calculated GWL at some points. In order to overcome these difficulties, the authors tested TLRN to simulate the entire groundwater system with one model. The aim of TLRN is to predict a multivariate time series using past values and available covariates. Instead of using static feed-forward ANNs to model nonlinear relationships in water table level forecasting, the TLRNs approach takes into account the temporal nature of the data (i.e., the lagged inputs, see Section 2.2.4), and in this respect compares favourably with ANN multilayer perceptron networks. The model used in the TLRNs is the gamma model [71], which is characterised by a memory structure that is a cascade of leaky integrators. The neural network can control the depth of the memory by changing the value of the feedback parameter, instead of changing the number of inputs. Since the feedback parameter is recursive, a backpropagation through time algorithm was used to apply a more powerful learning rule. Considering the reduced computational costs and the lower data requirements, the authors concluded that a TRLN model can be effectively used in the field of GWL simulation.

- (1)
- The sensitivity of ANN performance to data availability was assessed by using different sizes of training sets. The results showed that, during validation, acceptable prediction accuracy was achieved with a relatively small number of training sets.
- (2)
- Input parameters groundwater withdrawals and rainfall were included in the sensitivity analysis. Results showed that, in the unconfined aquifer, short-term oscillations were correlated most strongly to rainfall, while in the underlying semiconfined aquifer the water level was mostly influenced by withdrawals. Since these results are in accordance with the hydrological conditions, the authors concluded that the physical dynamics of the system must be sufficiently understood by the modeller in order to identify the important predictor input variables.
- (3)
- The effect of measurement error and data noise (inherently present in most hydrologic data set) on ANN performance was assessed by introducing normally distributed random noise into the input variables of the training set. The results demonstrated that the ANN can filter out noise in the training data and effectively learn groundwater system behaviour.

^{2}, MAE, RMSE), Nash–Sutcliffe efficiency (NSE), and mean percent deviation (Dv). Results revealed that the ANN model performed better for short-time predictions that require high accuracy, while numerical models were more appropriate for long-term predictions. Furthermore, the authors highlighted that physically based models provide the total water balance of the system, whereas the ANN models do not involve a description of the entire physics of the system. In the case of ANNs, a new model must be developed from the beginning to include any changes in the input or output parameters, differently from numerical models. Thus, the type of model should be selected in accordance with the type of problem.

#### 3.2. Comparing Results of a Physically-Based Model and Different Machine Learning Models

^{2}, RMSE, and NASH, and the best performance evaluated by a 2-year period groundwater hydrograph. BN models showed many advantages, such as the easier implementation, the higher forecasting accuracy, and the ability to deal with missing or incomplete data. Moreover, in the BN models, the variables were modelled by means of probability distributions; this allowed the authors to estimate uncertainty more accurately compared with other models other models [108,109,110].

^{2}). As for the multilayer perceptron, the hyperbolic tangent sigmoid transfer function was applied in the neurons of the hidden layer and the linear transfer function was applied in the output layer; the number of hidden neurons was identified by trial-and-error procedure. Trial-and-error was used also to identify the number of hidden neurons for the RBF network. In RBF, the Gaussian radial basis function was applied in the neurons of the hidden layer and linear transfer function was applied in the output layer, respectively. As for the SVM, Gaussian function (i.e., radial basis function) was used as a kernel function to compute the Gram matrix. Furthermore, for each of the machine learning models, the ratio between RMSE in the prediction stage times RMSE the in training stage was calculated as a measure of the models’ generalisation ability (GA). Machine learning models simulated historical data with higher performance with respect to numerical model, with the RBF model performing the best. In particular, SVM performed the best in the training stage, while RBF in the verification stage. Machine learning models showed much less computation cost in training and prediction stages than those of numerical model in calibration and verification stages. However, because of the physical based mechanism, the numerical model showed a better generalisation ability. Therefore, authors concluded that machine learning models are applicable to problems that require a high number of model runs without considering the physical mechanisms (e.g., optimisations, real-time models, sensitivity/uncertainty analysis).

#### 3.3. Testing Hybrid or Ensemble Models

^{2}groundwater model of Lake Michigan Basin [113] impedes the evaluation of local-scale impacts due to the long runtime and the too-coarse grid. The solution was to emulate the groundwater flow model using a dataset of collocated numerical model input and output to build a statistical learning model (“metamodel”, [114]), providing fast decision support to water managers which need to evaluate the permission to water abstraction. In practice, the numerical model was used to generate outputs reproducing several condition of the groundwater system; then, those outputs were used to train a statistical model, which could be subsequently used to make predictions without the need to run the regional model. The ability of the three techniques to extend MODFLOW predictions to areas with few samples was evaluated. K-fold cross validation (CV) was used to assess the models performance, as well as by hold-out data. The performance of the BN model (evaluated by means of R

^{2}and RMSE) was lower than the other two, and this could be due to the fact that the continuous input and output variables were both discretised into a small number of bins. All the three techniques can be implemented with commonly used commercial (in the case of BN) or open source (in the case of ANN or GBRT) software. The computational time is nearly instantaneous for all the three techniques while it takes longer to perform cross-validation. ANN or GBRT may be the best options for managers who need to achieve better predictive performance when a single response is considered. BN includes estimate of the uncertainty of predictions because all variables are treated as probability distributions. The authors concluded that the metamodelling approach is valid over a wide range of conditions and, as a screening approach, is helpful. A limitation of their approach is that it assumes that the response of the system to pumping rates is linear; thus, this assumption is violated at high pumping rates.

^{2}, the most accurate results were obtained with RF. The authors concluded that RF is able to reproduce time series trends in GWL as well as capture the variability in MODFLOW model predictions. In that way, the authors obtained a significant reduction of computational time: each MODFLOW run without the RF model would have taken approximately 36 years in a standard computing environment, instead of 24 h while simulating MODFLOW with a RF representation of the groundwater system. The procedure is integrated in a Robust Decision Making (RDM) process: the novel application of machine learning represents an improvement to the field of decision-making under deep uncertainty that allows reducing computational times and permits a greater exploration of the uncertainty space, such as future climate changes and drought conditions.

#### 3.4. Reducing and Correcting Model Errors by Means of Machine Learning Approaches

^{2}model [115] developed to resolve water conflicts as growing water demand led to dramatically increased groundwater pumping. The second is the Spokane Valley–Rathdrum Prairie aquifer (SVRP) (USA), an 844 km

^{2}aquifer subjected to groundwater pumping stresses. The two models differ in various aspects, including parametrisation, calibration, grid resolution, data density, and calibration strategy, leading to different spatial patterns in model residuals.

## 4. General Results

#### 4.1. Key Area of Groundwater Model Use

#### 4.2. Input Variables Employed for Machine Learning Modelling

#### 4.3. Simulation Period of Physically-Based Models

^{2}and 20 layers [95] to 0.75 km

^{2}and 1 layer [55]. Results suggest that there are no machine learning techniques nor groundwater management problems specifically suitable for a given range of physically-based size.

#### 4.4. Time Step

#### 4.5. Data Set Size

^{2}) are covered by a relatively small number of data (e.g., ref. [3]). There is not any recommendation in the reviewed papers about the density of samples which optimises the model performance. However, denser distributed training data allow achieving the best performance in temporal prediction scenarios. For example, the ANNs’ ability to learn or generalise system behaviour is limited by the data with which it is trained. Machine learning models can fail to accurately predict GWL in areas where a scarce number of data for training is available, and results can be worse than those of numerical models.

#### 4.6. Subset for Machine Learning Model Training, Validation and Testing

#### 4.7. Used Software

## 5. Specific Results

#### 5.1. Properties of the Machine Learning Techniques Used in the Reviewed Papers

- -
- Feed-forward multilayer perceptron with a backpropagation learning algorithm was the most used ANN technique in the reviewed papers.
- -
- The training algorithms used in the reviewed papers were Levenberg Marquardt, Bayesian regularisation, scaled conjugate gradient, quick propagation algorithm, backpropagation algorithm, and resilient backpropagation. The most used were Levenberg Marquardt [60,117], which integrates the advantages of two training algorithms, namely the steepest descent, and Gaussian–Newton methods, and searches for the global minima function to optimise the solution [68]; some authors point out that this is the less time-consuming algorithm.
- -
- The transfer functions used for the hidden layer are: sigmoid, sine, hardlim, triangle basis, radial basis, hyperbolic tangent, linear, and logistic.
- -
- The most common structure of ANN in the reviewed papers is a feed-forward ANN with a single hidden layer, with sigmoid transfer function in the hidden layer and linear transfer function in output layer. The best structure and number of hidden neurons are chosen by trial-and-error or cross-validation.
- -
- The final structure of multilayer perceptron is usually chosen as the one resulting in minimum error and maximum efficiency during training.
- -
- ANNs are capable of achieving substantially higher predictive accuracy at observation wells than the physically-based numerical model, with fewer inputs and lower developmental effort and cost. The choice of the appropriate training data size is a key issue; it should be evaluated considering many aspects, such as the required model accuracy, the number of connection weights, the complexity, and the level of noise in the system [3]. Moreover, it is important to find the optimal ANN topology ensuring satisfactory generalisation capability for any given problem. This is generally achieved by testing different topologies and transfer functions.

^{2}and RMSE derived for all the observation points. In Fienen et al. [95], the BN was implemented with variables that were supposed to have the greatest influence on the source of water to wells: the distance to surface water, the surface water percent, the distance of 1st-order stream, and the percent of 1st-order stream. The continuous values of variables were discretised into bins; this permits performing predictions as discrete conditional probabilities without requiring a priori assumptions about distributions. Both the number of nodes and the number and ranges of bins were adjusted by 10-fold cross validation, and the set of parameters resulting in highest R

^{2}was selected as the optimal model.

^{2}score.

#### 5.2. Comparison between Machine Learning Techniques

- -
- The performance of ANN with RBF as the activation function performed the best in simulating groundwater dynamics in arid basins, compared with ANN multilayer perceptron and SVM [63]. In detail, SVM performed the best in the training stage, while RBF in the verification stage; ANN’s performance was lower than these two.
- -
- Regarding ANFIS, no improvements are remarked with respect to ANNs, although greater performance with respect to the MODFLOW numerical model is documented [68].
- -
- With respect to multilayer perceptron ANN, TLRNs can provide an appropriate tool for processing time-varying information. The main advantage is that TLRNs require a lower memory compared to multilayer perceptron, due to their lower network size. Furthermore, TLRNs have a low sensitivity to noise.
- -
- Compared to simple ANN, ELM showed better performance, much less modelling time, less modelling error, and less weights norm [100].
- -
- With respect to ANN, BN models provided easier implementation, higher prediction accuracy, and a greater ability to deal with missing or incomplete data [46]. It allows an uncertainty estimation more accurate than other machine learning models because the variables are modelled by means of probability distributions. When used as a metamodel, replacing a regional groundwater model to simulate the source of water-to-well [95], BN showed lower cross validation predictive skill compared with ANN and GBRT. However, the BN includes estimates of the uncertainty of predictions as part of the technique. GBRT required the least time with respect to BN and ANN. Thus, in this case, the choice between a statistical learning approach such as ANN or GBRT and the BN approach depends upon the preference of the modeller and the aims of the problem.
- -
- When used to predict the annual change in GWL as effect of managed recharge, RF produced the most accurate average basin GWL representation respect to observations, compared with SVM and ANN [99].

#### 5.3. Results of Testing Hybrid or Ensemble Models

- -
- ELM and WA-ELM were both used to simulate GWL in an arid basin [100]. However, the ELM model with the db2 mother wavelet for data pre-processing showed a better performance with a significant accuracy improvement compared with the physically-based models.
- -
- The hybrid approach of Nikolos et al. [101] provides a fast way to integrate the physically-based models within an evolution-based optimisation procedure (DE algorithm) by replacing the calls of the PTC model with an ANN. The ANN provides a tool to perform an optimisation run with the DE algorithm with very short time, serving as a fast and accurate surrogate model.
- -
- The hybrid modelling approach HANN [102] showed a high model structure strength since it integrated a robust data pre-processing and input variable selection techniques.
- -
- Using machine learning models in hierarchical approach can significantly improve the results of physics-based models [82]; moreover, by that way, advantages and disadvantages of different machine learning models are identified and insights are provided into which data are most valuable to long-term monitoring objectives and which are not. In particular, Michael et al. [82] found that DT consistently provided the most accurate predictions of hydraulic head compared with IDW and ANN. However, when using all of the data across time, IDW showed substantial improvements. Given that IDW is simple to use and is widely accepted among practitioners, it could be considered as an optimum choice.
- -
- The computational time of regional physically-based models can be substantially reduced by introducing an empirical (or statistical) representation of numerical models; this consists of machine learning models trained using numerical models inputs and outputs, which can be used to make predictions of variable of interest [95,99].

#### 5.4. Results of Machine Learning Models Used to Reduce or Correct Errors in Physically-Based Models

## 6. Discussion

- -
- The numerical models are comparatively more reliable. While showing a lower prediction error than the physical models, machine learning models cannot return many of the outputs of a physical model, such as flux estimates or total water balance.
- -
- Xu et al. [79] found that data-driven models are difficult to interpret physically. The updated head no longer conserved mass for the given model inputs, which can confound the physical interpretation of the results and prevent understanding errors in the conceptualisation of the groundwater system.
- -
- Numerical models exhibit a higher generalisation ability than machine learning methods because they are based on the physics of the system [63]. Conversely, machine learning models are applicable to problems that require a high number of model runs without considering the physical system (e.g., optimisations, real-time models, sensitivity/uncertainty analysis).
- -
- Usually, while the machine learning models may be more efficacious for predicting short-term GWL and reproducing highly localised flow impacts, numerical modelling is more appropriate for long-term projections, or in areas where field data are insufficient for the given problem. However, it should be remarked that Almuhaylan et al. [68] were able to use machine learning models to perform long-term prediction (up to 50 years), by training the ANN/ANFIS model for the prediction of changes in groundwater levels instead of the direct simulation of water levels.

- -
- when few field data exist, the results of numerical models can be improved by training machine learning models, which allow to obtain accurate groundwater level forecasting at specific observation wells;
- -
- machine learning models cannot substitute a numerical model as one single model, but can be used to simulate water table fluctuation at every individual observation well with reduced computational time;
- -
- accurate results of machine learning models in specific test sites can be used to obtain the best GWL data required by the numerical model as input;
- -
- the physical dynamics of the system must be sufficiently understood by the modeller in order to identify the important predictor input variables of machine learning models. Results of numerical models help to understand the physical system; this can help, in turn, choosing the input parameters for machine learning models. Coppola et al. [3] suggested using ANNs to perform a sensitivity analysis on the interrelationships between input and output variables;
- -
- Numerical models can simulate different scenarios, allowing for detection areas requiring particular management strategies, thereby supporting the design of an effective monitoring network, which, in turn, may improve both machine learning predictive capability and performance.

- -
- The aim of the work, for example: improvement of prediction at some well location, numerical model error correction, numerical model updating;
- -
- the need to produce a probability distribution of the results and obtain uncertainty estimation within the model, (i.e., in areas with few data);
- -
- the availability of data for training and testing (number and spatial-temporal distribution);
- -
- the need to speed up decision making processes and reduce the computational time;
- -
- the degree of expertise of the modeller, which should drive the searching for a good compromise between model complexity and prediction performance.

## 7. Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 5.**Percentage of the training and testing datasets used in machine learning modelling. Data from: Mohammadi, 2009 [59], Coppola et al., [3], Banerjee et al., 2011 [96], Mohanty et al., 2013 [97], Parkin et al., 2007 [98], Moghaddam et al., 2019 [76], Almuhaylan at al., 2020 [68], Chen et al., 2020 [63], Fienen et al., 2016 [95], Miro et al., 2021 [99], Malekzadeh et al., 2019 [100], Nikolos et al., 2008 [101], Sahoo et al., 2013 [102], Michael et al., 2005 [82], Xu et al., 2014 [79], Demissie et al., 2009 [85].

**Table 1.**List of the reviewed studies. P: precipitation. T: temperature. E: evapotranspiration. SWL: surface water level. EP: effective precipitation.

n | Reference | Region of Study (Country) | Key Area of Model Use | Used ML Models | Input Variables to ML Model | ML Model Time Step | Range of Total Data (Number of Data or Observation Wells) | PB Simulation Time (Time Step) | Size of the PB Model Domain | Field of Application of the ML Technique | Journal (202+C:L0 IF) | Aquifer or Basin Hydrostratigraphy |
---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | Mohammadi, 2009 [59] | Chamchamal plain (Iran) | Farming and agricoltural systems | ANN | MODFLOW output | monthly | 1986–1998 (144 sets) | 1 year (monthly) | 145.7 Km^{2} | using the results of PB models to train a single ML model | Pratical hydroinformatics (book) | Alluvial (karst bedrock) |

2 | Coppola et al., 2003 [3] | Northwest Hillsborough Wellfield (USA) | Planning and supply | ANN | GWL, pumping rates, P, T, dew point, wind speed conditions, stress period lenghts | weekly | January 1995–August 2000, (212 sets) | 20 years (monthly) | 10,359.9 Km^{2} | using the results of PB models to train a single ML model | Journal of hydrologic engineering (2.064) | Highly permeable limestone overlain by low permeability clay and, above, sand with interbedded clay |

3 | Banajeree et al., 2011 [96] | Kavaratti, island of the Lakshadweep archipelago (India) | Coastal water management | ANN | not mentioned in the paper | monthly | 2005–2007 (23 sets) | 5 years (monthly) | 2D model, with section lenght = 2650 m and depth 1000 m | using the results of PB models to train a single ML model | Journal of hydrology (5.722) | Coastal |

4 | Mohanty et al., 2013 [97] | Kathajodi-Surua Inter-basin of Odisha (India) | Coastal water management | ANN, TLRNs | GWL, P, E, river stage, SWL, pumping rates | weekly | February 2004–May 2007 (174 sets) | 3 years, (weekly) | 114.5 m^{2} | using the results of PB models to train a single ML model | Journal of hydrology (5.722) | Alluvial |

5 | Parkin et al., 2007 [98] | Winterbourne stream, Thames Basin, Berkshire (UK) | aquifer-river interaction | ANN | GWL, river flow depletion | daily | Not specified (1 well) | 25 years (daily) | Regional aquifer: 200 Km^{2}. Valley aquifer: 2 Km^{2} | using the results of PB models to train a single ML model | Journal of hydrology (5.722) | Alluvial |

6 | Moghaddam et al., 2019 [76] | BirjandAquifer, South Khorasan (Iran) | Drought-prone regions | ANN, BN | GWL, E, T, EP, Discharge | monthly | 2002–2014 (1872 sets) | 12 years | 277.8 Km^{2} ([99]) | using the results of PB models to train and compare different ML models | Groundwater for sustainable development (no IF) | Alluvial |

7 | Almuhaylan at al., 2020 [68] | Saq Aquifer in Quassim (Saudi Arabia) | Drought-prone regions | ANN, ANFIS | GWL, pumping rates | not specified | 1980–2018 (55 wells) | not specified | 600 Km^{2} | using the results of PB models to train and compare different ML models | Water (3.103) | Sandstone |

8 | Chen et al., 2020 [63] | Heihe River Basin (China) | Drought-prone regions | ANN, RBF, SVM | pumping rates, recharge, streamflow rates | monthly | 1986–2008 (11,088 sets) | 22 years (monthly) | 21,120 Km^{2} | using the results of PB models to train and compare different ML models | Scientific reports (4.380 | Alluvial |

9 | Fienen et al., 2016 [95] | Lake Michigan Basin (USA) | Planning and supply | ANN, GBRT, BN | parameters expected to have predictive power to the source of water to wells | not specified | 1864–2005, (4911 sets) | 141 years (variable, [100]) | 204,764.4 Km^{2} | using the results of PB models to train and compare different ML models | Environmental modelling and software (5.288) | Glacial deposits |

10 | Miro et al., 2021 [101] | San Bernardino and Rialto-Colton basins, San Bernardino Valley Municipal Water District - Valley District (USA) | Drought-prone regions | RF, SVM, ANN | Recharge, pumping rates | not specified | 2015–2050 (not specified) | 35 years (monthly) | 3000 Km^{2} | using the results of PB models to train and compare different ML models | Climate risk management (4.090) | Basin comprising ancient metamorphic bedrock, eolic sands, ancient fans, recent alluvium |

11 | Malekzadeh et al., 2019 [102] | Kabodarahang Plain, Hamadan (Iran) | Farming and agricoltural systems | ELM, WA-ELM | decomposed sub-series of observed GWL | monthly | August 1990–September 2015 (301 sets) | 10 years (monthly) | not specified | using the results of PB models to train a single ML model | Groundwater for sustainable development (no IF) | Alluvial (limestone bedrock) |

12 | Nikolos et al., 2008 [103] | Northern Rhodes Island (Greece) | Coastal water management | ANN combined with DE algorithm | GWL, pumping rates | daily | 1997–1998 (3125 sets) | 1 year (2 seasonal stress periods) | 217 Km^{2} | using the results of PB models to test hybrid or ensamble modelling approaches | Hydrological processes (3.565) | Coastal |

13 | Sahoo et al., 2017 [104] | High Plains aquifer and Mississippi River Valley aquifer (USA) | Farming and agricoltural systems | Automated hybrid artificial neural network (HANN) | GWL, P, T, streamflow, climate indexes, irrigation demand, NAO index | monthly | 1980–2012, (HPA: 263,808 sets. MRVA: 115,368 sets) | 33 years (monthly) | MRVA: 405,720 Km^{2} ([105]). HPA: 3.34 Km^{2} ([106]) | using the results of PB models to test hybrid or ensamble modelling approaches | Water resource research (5.240) | High Plain Aquifer: ancient alluvial fans and quaternary deposits. Mississippi River Valley Alluvial Aquifer: Tertiary and Quaternary clay, silt, sand and gravel deposits. |

14 | Michael et al., 2005 [82] | Argonne National Laboratory, Illinois (USA) | Contaminant/phytoremediation | DT, IDW, ANN | GWL, P | quartely | November 1999–March 2001 (22 wells with quarterly data); May 2001 (7 wells with hourly data) | 6 years, (monthly) | 4.8 Km^{2} ([107]) | using the results of PB models to test hybrid or ensamble modelling approaches | Water resource research (5.240) | Glacial deposits |

15 | Xu et al., 2014 [79] | Republican River Basin and Spokane Valley-Rathdrum Prairie aquifer (USA) | Planning and supply | Cluster analysis, IBW, SVM | GWL, well location, observation time | monthly | RRCA: 1918–2007 (300,000 sets). SVRP: 1990–2005 (2191 sets) | RRCA: 89 years. SVRP: 15 years, (monthly) | RRCA: 79,396 Km^{2}. SVRP: 844.3 Km^{2} | using ML techniques for PB models errors reduction/correction | Groundwater (2.671) | Alluvial |

16 | Demissie et al., 2009 [85] | Argonne National Laboratory, Illinois (USA) | Contaminant/phytoremediation | ANN, DT, SVM, IBW | GWL, EVP, stress periods | monthly | 2000–2005, (3600 sets) [107] | 6 year (monthly) | 0.75 Km^{2} | using ML techniques for PB models errors reduction/correction | Journal of hydrology (5.722) | Glacial deposits |

Machine Learning Model | Software | Commercial/Free | n of Times |
---|---|---|---|

ANN | Matlab | c | 3 |

R-neuralnet package | f | 1 | |

LINGO | c | 1 | |

not specified | 7 | ||

RBF | Matlab | c | 1 |

ANFIS | Matlab | c | 1 |

TLRN | NeuroSolution | c | 1 |

ELM, WA-ELM | Matlab, Matlab wavelet toolbox | c | 1 |

BN | Hugin Lite 8.3; | c | 1 |

netica Software, CVNetica (for cv) | c | 1 | |

IBW | Matlab Statistic Toolbox TM | c | 1 |

not specified | 1 | ||

SVM | Matlab Statistic Toolbox TM | c | 2 |

not specified | 2 | ||

DT | not specified | 1 | |

RF | R (randomForest package) | f | 1 |

not specified | 1 | ||

GBRT | Phyton (scikit-learn library) | f | 1 |

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Di Salvo, C.
Improving Results of Existing Groundwater Numerical Models Using Machine Learning Techniques: A Review. *Water* **2022**, *14*, 2307.
https://doi.org/10.3390/w14152307

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Di Salvo C.
Improving Results of Existing Groundwater Numerical Models Using Machine Learning Techniques: A Review. *Water*. 2022; 14(15):2307.
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2022. "Improving Results of Existing Groundwater Numerical Models Using Machine Learning Techniques: A Review" *Water* 14, no. 15: 2307.
https://doi.org/10.3390/w14152307