Genetic Algorithms Applied to Optimize Neural Network Training in Reference Evapotranspiration Estimation

Algoritmos Genéticos Aplicados para Otimizar o Treinamento de Redes Neurais na Estimativa da Evapotranspiração de Referência

Abstract

The increased consumption of natural resources, such as water, has become a global concern. Consequently, determining information that can minimize water consumption, such as evapotranspiration, is increasingly necessary. This research evaluates the capacity of Genetic Algorithms (GAs) in training and fine-tuning the parameters of Artificial Neural Networks (ANNs) (MLP-GA) to obtain daily values of reference evapotranspiration (ETo) in accordance with the Penman-Monteith FAO-56 method. The method is employed to estimate ETo at 14 weather stations in Brazil. The findings are assessed based on the coefficient of correlation (r), mean absolute error (MAE), root mean square error (RMSE), and mean percentage error (MPE), and are contrasted with the Hargreaves-Samani, Jensen-Haise, Linacre, Benavides & Lopez, and Hamon methods, along with the Multilayer Perceptron (MLP) neural network, which is conventionally trained and employs hyperparameter tuning techniques such as Grid Search (MLP-GRID) and Random Search (MLP-RD). The results show that the MLP-GA is, on average, 12 times faster than MLP-RD and 60 times faster than MLP-GRID, while achieving the highest precision indices in most regions, with an r of 0.99, MAE ranging from 0.11 mm to 0.20 mm, RMSE between 0.14 mm and 0.27 mm, and MPE between 2.49% and 7.09%. These findings suggest the results generated achieve an precision between 92.91% and 97.51% in comparison to the Penman-Monteith method. This confirms that employing Genetic Algorithms (GA) to automate the training and optimization of the model is effective and enhances the neural network's capacity to predict ETo.

Keywords
Penman-Monteith; hyperparameter tuning; Genetic Algorithms (GAs); grid search; random search

Resumo

O aumento do consumo de recursos naturais, como a água, tornou-se uma preocupação global. Consequentemente, determinar informações que possam minimizar o consumo de água, como a evapotranspiração, é cada vez mais necessário. Esta pesquisa avalia a capacidade dos Algoritmos Genéticos (AGs) em treinar e ajustar os parâmetros de Redes Neurais Artificiais (RNAs) (MLP-AG) para estimar os valores diários de evapotranspiração de referência (ETo) de cordo com o método Penman-Monteith FAO-56. O método é empregado para estimar a ETo em 14 estações meteorológicas no Brasil. Os resultados são avaliados com base no coeficiente de correlação (r), erro absoluto médio (MAE), raiz do erro quadrático médio (RMSE) e erro médio percentual (MPE), e são comparados com os métodos de Hargreaves-Samani, Jensen-Haise, Linacre, Benavides & Lopez e Hamon, juntamente com a rede neural Perceptron Multicamadas (MLP), que é treinada convencionalmente e através de técnicas de ajuste de hiperparâmetros, como Grid Search (MLP-GRID) e Random Search (MLP-RD). Os resultados mostram que o MLP-AG é, em média, 12 vezes mais rápido que o MLP-RD e 60 vezes mais rápido que o MLP-GRID, alcançando os maiores índices de acurácia na maioria das regiões, com r de 0,99, MAE variando de 0,11 mm a 0,20 mm, RMSE entre 0,14 mm e 0,27 mm, e MPE entre 2,49% e 7,09%. Essas descobertas sugerem que os resultados gerados atingem uma precisão entre 92,91% e 97,51% em comparação com o método Penman-Monteith. Isso confirma que o uso de Algoritmos Genéticos (AGs) para automatizar o treinamento e a otimização do modelo é eficaz e melhora a capacidade da rede neural de prever a ETo.

Palavras-chave
Penman-Monteith; ajuste de hiperparâmetros; Algoritmos Genéticos (AGs); grid search; random search

1. Introduction

Climate change, population growth, and the need to increase food production to ensure food security are among the biggest challenges facing countries in the coming decades. Research indicates that by the year 2050, the world population may exceed 9 billion inhabitants, making it necessary to expand food production by more than 70%. This improvement will only be possible through technological advancement combined with the irrigation technologies evolution (Cardoso and Justino, 2014; Feng et al., 2017; Mutiibwa et al., 2018; Saath and Fachinello, 2018; Cole et al., 2018; Pérez-Blanco et al., 2020; Top et al., 2022; Gyamfi et al., 2024).

On the other hand, agriculture is the activity that consumes the most water resources on the planet, accounting for approximately 70% of the available water, and most of it is used for irrigation (Amarasinghe and Smakhtin, 2014; Silva Junior et al., 2019; Abioye et al., 2021; Vásquez et al., 2022). Despite the significant use of water for irrigation in food production, it increases productivity, enables crop production during different times of the year, makes production viable in regions with unfavorable climatic characteristics, and guarantees greater profitability and profit for producers (Dantas et al., 2016). Nevertheless, the irresponsible use of this natural resource, combined with additional factors such as pollution, soil degradation, and deforestation, has directly impacted its replenishment and has become one of the primary causes of scarcity (Tao et al., 2018).

In Brazil, water crises have exposed the importance of water management in various sectors, particularly agriculture. Therefore, creating measures to minimize water consumption is highly relevant to society. One way to reduce the use of this resource is by understanding its consumption and the ideal amount for the production of each agricultural crop.

The water requirement can be obtained from the assessment of evapotranspiration (ET), a physical phenomenon that involves soil, plants, and meteorological conditions. This variable is responsible for 70% of the global water cycle, and in some regions, this percentage exceeds precipitation, making its understanding crucial for managing water in agriculture and modeling crop growth (Mello and Silva, 2013; Adnan et al., 2021a). ET is defined as the occurrence of water evaporation from the soil and plant transpiration (Silva et al., 2005; Fernandes et al., 2012). On the other hand, reference evapotranspiration (ETo) is the rate of ET for a hypothetical crop with an assumed height of 0.12 m, a surface resistance of 70 s/m, and an albedo of 0.23, similar to the evaporation from an extensive surface of uniformly tall green grass (Allen et al., 1998; Gharsallah et al., 2013).

ETo can be obtained directly through lysimeters or indirectly through empirical models (Mello and Silva, 2013). The implementation of lysimeters is a complex task that requires high financial cost and careful planning and experiments (Kumar et al., 2008). Due to these factors and because it presents values close to those recorded by lysimeters, the Food and Agriculture Organization (FAO) defined the Penman-Monteith FAO 56 (PM FAO-56) method as the standard model for quantifying ETo (Allen et al., 1998; Yassin et al., 2016). Nevertheless, calculating ETo through the PM FAO-56 method is complex and still requires a substantial amount of climatic data, which may not be available in all locations (Abdullah et al., 2015).

Consequently, throughout the years, methods like Jensen-Haise, Makkink, Priestley-Taylor, Hargreaves and Samani, Hamon, Linacre, Benavides & Lopez, Thornthwaite, and others have been evaluated as alternatives to PM FAO-56 (Hamon, 1961; Priestley and Taylor, 1972; Linacre, 1977; Hargreaves and Samani, 1985; Jensen et al., 1997; Borges and Mediondo, 2007; Sentelhas et al., 2010; Dantas et al., 2016; Mehdizadeh et al., 2017; Kaya et al., 2021). However, many of these methods require calibration and may not provide reliable estimates, often overestimating or underestimating ETo (Landeras et al., 2008; Dantas et al., 2016; Adnan et al., 2021a).

Over recent years, the utilization of machine learning methodologies to assess environmental, hydrological, and climatological parameters such as ETo has significantly increased (Patil and Deka, 2015; Mehdizadeh et al., 2017; Kaya et al., 2021). Saggi and Jain (2019) evaluated the aptitude of Multilayer Perceptrons (MLP), Generalized Linear Model (GLM), Random Forest (RF), and Gradient-Boosting Machine (GBM) frameworks to forecast daily ETo in the Hoshiarpur and Patiala districts in Punjab, India. Tikhamarine et al. (2020) evaluated the potential of a hybrid framework integrating support vector regression (SVR) with grey wolf optimizer (SVR-GWO) to compute monthly ETo in the Algiers, Tlemcen, and Annaba stations located in northern Algeria. Ferreira and Cunha (2020) assessed the potential of LSTM neural network frameworks, one-dimensional convolutional (CNN 1D), RF, and others to forecast daily ETo at 53 meteorological stations located in Minas Gerais, Brazil. Zhao et al. (2021) employed principal component examination together with SVM, gradient boosting decision tree (GBDT), particle swarm optimization (PSO)-SVM, and PSO-GBDT frameworks to develop a prediction system for ETo in southwest China. Adnan et al. (2021a) implemented a hybrid adaptive neuro-fuzzy inference system (ANFIS-WCAMFO) framework to compute monthly ETo at Dhaka and Mymensing stations in central-southern Bangladesh. Chia et al. (2021) utilized a hybrid approach with the Extreme Learning Machine (ELM) model and PSO, Moth-flame optimization algorithm (MFO), and whale optimization algorithm (WOA) to calculate ETo at three stations in Sabah and Sarawak in eastern Malaysia. Sattari et al. (2021) employed Gaussian process regression (GPR), SVR, and artificial neural artificial neural networks (ANNs) to compute monthly ETo in çorum (Turkey). Santos et al. (2023) evaluated the performance of various techniques, including ANN, RF, SVM, and multiple linear regression (MLR), for estimating the monthly mean reference evapotranspiration (ETo) in the state of Minas Gerais, Brazil. Khoja et al. (2023) investigated the use of various machine learning techniques to predict daily reference evapotranspiration (ETo) at three meteorological stations located in Tunisia: Beja, Mahdia, and Gafsa. Aly et al. (2023) applied an ensemble model to estimate ET₀ at 32 meteorological stations in Egypt, employing four machine learning techniques: Extra Tree Regressor, Support Vector Regression (SVR), K-Nearest Neighbors, and AdaBoost Regression. Bidabadi et al. (2024) applied the ANFIS and ANN with Grey Wolf Optimization (ANN-GWO) to estimate the reference evapotranspiration (ET0) at the Sirjan and Kerman stations in Iran.

Although research shows that machine learning techniques yield effective outcomes for estimating ETo, most models may have converged to local optima. This is because many approaches rely heavily on manual processes, where hyperparameter configurations and input variables are defined through tests and simulations. To address this issue and enhance the likelihood of achieving global optima, hyperparameter tuning methods have been proposed to fully or partially automate processes within machine learning models, including the training phase (Géron, 2019; Huyen, 2022; Raschka et al., 2022). Accordingly, this research employed an evolutionary approach using GA to automate the training and hyperparameter tuning of an MLP neural network, aiming to maximize the probability of obtaining the optimal configuration for estimating ETo in different locations across the southeastern region of Brazil. Furthermore, to assess the model's effectiveness, its outcomes were statistically compared with those from the empirical models of Hargreaves-Samani, Jensen-Haise, Linacre, Benavides & Lopez, Hamon, as well as a conventionally trained MLP model and an MLP model optimized through Grid Search and Random Search methods.

The remainder of this article is structured as follows. Section 2 outlines the methodology applied in this research. Section 3 presents the outcomes derived from the different models. Section 4 offers a discussion on the results, contrasting them with the findings reported in the literature. Finally, Section 5 highlights the key conclusions of this research.

2. Materials and Methods

2.1. Study area

The dataset used in this research was obtained from the Weather Forecasting and Climate Studies Center of the National Institute for Space Research (CPTEC - INPE). It consists of daily information derived from hourly measurements of average temperature (T_mean), maximum temperature (T_max), and minimum temperature (T_min) in degrees Celsius (°C), relative humidity (%) (R_H), wind speed (ms^-1) (u₂) and solar radiation (MJ m⁻²) (R_s) estimated using the Hargreaves and Samani method, recorded at various intervals from 14 stations located in the southeastern region of Brazil (Table 1).

Table 1
Meteorological and agrometeorological stations used for ETo calculation in the Southeast Region of Brazil. Legends: identification used for the city (ID), amount of daily information used (ADI), Köppen-Geiger Climate Classification (CKOP) Source: Lima et al., 2023; KoppenBrasil, 2024.

Station	ID	State	Latitude	Longitude	Altitude (m)	Period	ADI	CKOP
Mendes	ME	RJ	-22,521 °S	-43,721 °W	475	2007-2015	3109	Cfa
Santa Maria Madalena	AS	RJ	-21,953 °S	-42,005 °W	615	2005-2015	3318	Aw
Teresópolis	TE	RJ	-22,407 °S	-42,793 °W	871	2005-2015	3572	Cfb
Jerônimo Monteiro	JE	ES	-20,750 °S	-41,288 °W	146	2005-2015	2087	Aw
Sooretama	SO	ES	-19,367 °S	-40,074 °W	75	2005-2015	3324	Aw
Venda Nova Imigrante	VI	ES	-20,384 °S	-41,187 °W	727	2005-2015	2909	Cfb
Guaratingueta	GT	SP	-22,801 °S	-45,189 °W	539	2005-2015	2640	Cwa
Itú	IT	SP	-23,285 °S	-47,288 °W	626	2005-2015	2280	Cfa
Jaboticabal	JA	SP	-21,242 °S	-48,301 °W	600	2005-2015	3733	Cwa
Miguelópolis	MI	SP	-20,188 °S	-48,010 °W	538	2005-2014	3409	Aw
Caratinga	CA	MG	-19,743 °S	-42,135 °W	899	2005-2015	3957	Cwa
Montes Claros	MC	MG	-16,751 °S	-43,888 °W	701	2005-2015	3871	As
Paracatu	PA	MG	-17,251 °S	-46,885 °W	625	2005-2015	3484	Aw
Santa Fé de Minas	SM	MG	-16,500 °S	-45,417 °W	493	2005-2015	3725	Aw

Several factors influence the use of these variables and locations in the research, including the recording period and the correlation indices of the data with ETo. The incorporation of these variables by different empirical models is noteworthy, and this region is among the largest consumers of irrigation in the country.

The Southeast region of Brazil is composed of the states of Rio de Janeiro (RJ), Espírito Santo (ES), São Paulo (SP), and Minas Gerais (MG). Covering an area of 924,511 km² and home to approximately 84,847,187 inhabitants, this region exhibits significant climatic diversity due to its topography, altitude, and proximity to the Atlantic Ocean (IBGE, 2014). The predominant climate is tropical, characterized by humid summers and dry winters. However, there are regional variations: in the southeastern part of the state of São Paulo, the humid subtropical climate prevails; in the Mantiqueira mountain range, in southern Minas Gerais, the humid subtropical climate with monsoon influence is observed; in the extreme south of Minas Gerais, the high-altitude subtropical climate predominates; and in the extreme north of the state, the semi-arid climate is dominant (Sparovek et al., 2015; Lima et al., 2023). Fig. 1 illustrates the distribution of the stations throughout this territory.

Figure 1
Brazil southeastern region map with the locations studied.

2.2. Model for estimating ETo

The model employs GA to maximize the probability of achieving the optimal configuration of an MLP neural network in an effort to estimate the ETo derived from the Penman-Monteith method (PM FAO-56). In this research, GA is utilized to automate the training process and optimize the neural network hyperparameters, and the obtained results are confronted with various empirical models and other commonly used hyperparameter tuning techniques. To accomplish this, it is essential to perform different tasks, from preprocessing meteorological data to assessing the resulting outcomes, as illustrated in Fig. 2.

Figure 2
Proposed model for estimating ETo.

During the preprocessing stage, inconsistencies in the dataset are corrected, and the necessary values for calculating ETo are estimated. Initially, faults and outliers are identified through boxplot analysis and the use of the interquartile range (IQR). These elements are subsequently removed to ensure a uniform and consistent dataset suitable for model training and testing. For instance, during the period of 10/10/2010 at 1:00 PM, if station x shows an inconsistency in the T_mean record, the measurements of T_max, T_min, R_H, and u₂ for the same time frame are removed, ensuring that only data corresponding to the same period are included. Subsequently, the daily averages for each parameter are calculated, and solar radiation is estimated using the Hargreaves and Samani method (Rivero et al., 2017; Comert et al., 2023). Fig. 3 shows the behavior of the variables for each location.

Figure 3
Boxplot of the variables T_mean, T_max, T_min, R_H, u₂, R_s, and FAO-56-PM, illustrating the distribution of these variables throughout the recording period.

After preprocessing, the variables are subjected to the empirical models of PM FAO-56, Hargreaves-Samani (HS), Jensen-Haise (JH), Linacre (LI), Benavides & Lopez (BL), and Hamon (HA) to estimate the daily ETo for each region. Utilizing the ETo derived from the PM FAO-56 method, linear correlations between ETo and the meteorological variables of T_mean, T_max, T_min, R_H, u₂, and R_s are examined to determine the input sets for the conventionally trained MLP. Evaluating the correlations identified by the variables across all stations, one can see that the highest indices are obtained by T_mean, R_s, and R_H, which exhibit a high inverse correlation with ETo, suggesting that the lower their values, the higher the ETo for the day (Table 2). Therefore, these variables are defined as the standard input set for this model.

Table 2
Correlation indexes obtained between the meteorological variables and the ETo of each region.

Variable	Stations														AVG
	ME	AS	TE	JE	SO	VI	GA	IT	JA	MI	CA	MC	PA	SM
Tmean	0.69	0.73	0.75	0.78	0.74	0.73	0.67	0.64	0.5	0.7	0.71	0.61	0.65	0.67	0.68
Tmax	0.57	0.63	0.62	0.57	0.57	0.62	0.55	0.42	0.25	0.47	0.59	0.39	0.49	0.47	0.52
Tmin	0.29	0.18	0.13	0.19	0.21	0.07	0.24	0.047	0.01	0.14	0.24	0.095	-0.12	-0.06	0.12
RH	-0.68	-0.7	-0.78	-0.8	-0.6	-0.83	-0.69	-0.77	-0.62	-0.57	-0.66	-0.59	-0.77	-0.68	-0.70
u2	0.46	0.38	0.18	0.43	0.49	0.6	0.38	0.21	0.48	0.51	0.32	0.58	0.47	0.6	0.44
Rs	0.89	0.76	0.84	0.9	0.7	0.91	0.68	0.72	0.66	0.64	0.92	0.66	0.76	0.57	0.76

After identifying the linear relationship, the data sets are normalized, changing the scaling of the values to a range between zero and one. This transformation aims to encode all attributes into similar intervals, making all data equally important. This enhances the calibration of training algorithms (Wu et al., 2020; Chia et al., 2021). The scaling is given by (Eq. (1)),

(1)$$x_j^{norm}=\frac{x_j-x^{min}}{x^{max}-x^{min}}$$

where $x_j^{norm}$ is the normalized variable, $x_j$ is the variable at position j, $x^{min}$ is the minimum observed value among the variables, and $x^{max}$ is the maximum observed value among the variables.

To confront the observed and estimated values, the dataset is divided into two parts: 70% for training and 30% for validation, and then subjected to the MLP, MLP-RD, MLP-GRID, and MLP-GA. Finally, the effectiveness of the empirical and machine learning models is assessed through statistical techniques used on the validation sets.

2.3. ETo estimation methods

2.3.1. Penman-Monteith method (PM FAO-56)

The PM FAO-56 method estimates the ETo in mmd^-1. t is defined by FAO as the standard method for estimating ETo due to its reliable estimates compared to values recorded in lysimeters (Allen et al., 1998). In this case, the ETo estimate is (Eq. (2)),

(2)$$ETo=\frac{0.408\delta \left(R_n-G\right)+\gamma \left(\frac{900}{T_{mean}+273}\right)u_2(e_s-e_a)}{\delta +\gamma (1+0.34u_2)}$$

In Eq. (2), ETo is the reference evapotranspiration (mmd^-1), δ is the slope of the vapor pressure curve with respect to temperature (kPa °C^-1), $R_n$ is the net solar radiation on the crop radiation (MJ m^-2 d^-1), G is the soil heat flux density (MJ m^-2 d^-1), γ is the psychrometric constant (kPa °C^-1), u₂ is the wind speed at (2 m, m s^-1), e_s is the saturation vapor pressure (kPa), e_a is the actual vapor pressure (kPa), and T_mean is the mean daily air temperature (°C).

Many meteorological stations do not have all the necessary variables for the application of PM FAO-56. Therefore, some of this information can be estimated using different equations (Allen et al., 1998; Gavilán et al., 2007; Bezerra et al., 2009; Delgado et al., 2015). In this research, the Hargreaves and Samani model is adopted to estimate R_s for stations that do not have records of this variable, but other models such as Bristow and Campbell, Donatelli and Campbell, Mahmood, and Hubbard, among others, can be used for the same purpose (Raziei and Pereira, 2013; Alencar et al., 2015; Quej et al., 2015; Alsamamra, 2019; Czekalski et al., 2020; Comert et al., 2023). The Hargreaves and Samani model for estimating solar radiation is given by (Eq. (3)),

(3)$$R_S=k_r\sqrt{(T_{max}-T_{min})}{R}_a$$

where R_s is the solar radiation, k_r is the adjustment coefficient, which varies from 0.16 for inland regions to 0.19 for coastal regions, T_max is the daily maximum temperature (°C), T_min is the daily minimum temperature (°C), and $R_a$ is the solar radiation at the top of the atmosphere (MJ m^-2 d^-1).

The variable u₂ recorded at 10 meters height is corrected according to (Allen et al., 1998; Bezerra et al., 2009; Feng et al., 2017), that is (Eq. (4)),

(4)$$u_2=u_z\frac{4.87}{\mathrm{l}\mathrm{n}(67.8z-5.42)}$$

where, u₂ is the wind speed at a 2 m height (ms^-1), u_z is the wind speed recorded at a 10 meters height (ms^-1) and z is the measurement height (10 m).

2.3.2. Hargreaves-Samani method (HS)

The HS method also estimates ETo in mmd^-1. It is applicable in the absence of solar radiation, relative humidity, and wind speed data, and can be expressed by (Eq. (5)) (Hargreaves and Samani, 1985; Silva et al., 2005),

(5)$$ETo=0.408\mathrm{*}0.0023\left(T_{mean}+17.8\right)(T_{max}-T_{min}{)}^{0.5}{R}_a$$

where R_a is the extraterrestrial solar radiation in (MJ m^-2 d^-1). T_mean, T_max and T_min have previously been defined in Eqs. (2) and (3).

2.3.3. Jensen-Haise method (JH)

The JH method is an alternative to PM FAO-56 for calculating ETo and is expressed by (Eq. (6)) (Jensen et al., 1997),

(6)$$ETo=R_s(0.025T_{mean}+0.08)$$

Where R_s is the global solar radiation converted into mm and T_mean is previously defined in Eq. (2).

2.3.4. Linacre method (LI)

The LI method also estimates evapotranspiration in mmd^-1. It can be obtained based on altitude, latitude, daily maximum and minimum temperatures, as well as dew point temperature, as shown below (Eq. (7)) (Linacre, 1977),

(7)$$ETo=\frac{700\frac{T_m}{100-\mathrm{\phi }}+15(T_{mean}-T_d) }{(80-T_{mean})}$$

where T_m is calculated as T_mean + 0.006z, being z is the altitude in meters, φ is the latitude in degrees, and T_d is the average dew point temperature. T_mean is previously defined in Eq. (2).

2.3.5. Benavides & Lopez method (BL)

According to Fanaya Júnior et al. (2012), this method is developed by Garcia Benavides and Lopez Dias (BL) in 1970. It is based solely on average air temperature and relative humidity data, and the ETo can be estimated as follows (Eq. (8)) (Matos and Silva, 2016),

(8)$$ETo=1.21\mathrm{*}10\left(\frac{7.45\mathrm{ }{T}_{mean}}{234.7+T_{mean}}\right)\left(1-0.01R_H\right)+0.21\mathrm{ }{T}_{mean}-2.30$$

where R_H is the average relative humidity (%) and T_mean is previously defined in Eq. (2).

2.3.6. Hamon method (HA)

The HA method uses the average air temperature and daily maximum sunshine duration to estimate evapotranspiration in mmd^-1. It can be expressed by (Eq. (9)) (Hamon, 1961),

(9)$$ETo=0.55\left(\frac{N}{12}\right)2\mathrm{ }\left(\frac{4.95{\mathrm{e}\mathrm{x}\mathrm{p}}^{0.062T_{mean}}}{100}\right)25.4$$

where N is the theoretical daily maximum sunshine duration based on latitude and time of year. The T_mean is previously defined in Eq. (2).

2.4. Machine learning techniques

2.4.1. Genetic Algorithm (GA)

The GAs are part of the evolutionary algorithm field and are a global optimization heuristic based on natural selection, biological evolution, and the survival of the fittest individual, primarily drawing upon Darwin's theory of evolution (Xue et al., 2009; Xian et al., 2011). Their ability to generate good solutions for complex problems has allowed GAs to be applied in solving task allocation problems, route selection, combinatorial optimization, and machine learning (Wankhade et al., 2013).

GAs operate with a set of solutions originating from populations of individuals or chromosomes that are subjected to genetic operators. These operators use the characterization of the quality of each individual as a solution to the problem and generate a process of natural evolution of the individuals (Braga et al., 2012; Linden, 2012).

Thus, the main focus of a GA is to evolve a population until it reaches a possible optimal solution. GA begins by generating a population, and then, individuals are evaluated for their fitness to solve the problem, based on which the fittest individuals are selected, and a new population is created through recombination and mutation operators. This cycle, known as a generation, is repeated until the optimization criterion is achieved, and in the end, the fittest individual capable of solving the problem is taken as the solution (Cohoon and Paris, 1987; Linden, 2012). In the present research, this evolutionary process is used to automate the training of an MLP-type neural network and to attempt to find its optimal configuration by defining its characteristics.

2.4.2. Multilayer Perceptron (MLP)

The MLP is a feedforward artificial neural network (ANN) widely used for solving both classification and regression tasks, including applications such as weather forecasting, image analysis, tumor classification, text categorization, and seismic activity prediction (Saba et al., 2017; Asim et al., 2017; Waseem et al., 2023). It features a fully connected architecture, consisting of neurons organized into layers, including an input layer, one or more hidden layers, and an output layer (Asaduzzaman et al., 2010; Kumar et al., 2023). In this ANN, the dataset's features are extracted by adjusting the weights, which establish a mapping between the input and output data. The weight adjustments are performed using the Backpropagation algorithm or its variations, such as Quasi-Newton Backpropagation, Resilient Backpropagation, and Levenberg-Marquardt Backpropagation, among others (Haykin, 2001). In this research, a four-layer structure is employed, comprising an input layer, two hidden layers, and an output layer, as illustrated in Fig. 4.

Figure 4
Architecture of the ANN MLP applied to estimate ETo.

In the MLP, $x_1$, $x_2$, and $x_3$ represent T_mean, R_H, and R_s, respectively, $w_i$ denotes the weights corresponding to the layers. $H_j$ and $H_{k }$ refer to the neurons in the hidden layers (30 and 15, respectively) and the activation functions, chosen for these layers, are Hyperbolic Tangent (tansig) and Sigmoid (logsig) (Eqs. (10) and (11)) given by,

(10)$$\text{tansig}\left(u_i\right)= \frac{2}{\left(1+{\text{exp}}^{\left(-2 * ui\right)}\right)}-1$$

(11)$$\text{logsig}\left(u_i\right)=\frac{1}{\left(1+{\text{exp}}^{\left(-ui\right)}\right)}$$

These activation functions were determined through testing and simulations. Finally, $y_i$ represents the estimated ETo. The model training is also determined through simulations, where the Quasi-Newton Back-propagation algorithm (trainbfg) being selected. Regarding the model's training duration, it has been set to 4000 epochs.

2.4.3. MLP trained using the Grid Search method (MLP-GRID)

Unlike conventional approaches to configuring machine learning models, which require users to manually evaluate and modify their hyperparameters to find the optimal combination, the Grid Search method evaluates all possible combinations within a search space. In other words, based on a list of hyperparameters, such as input size, activation functions, training methods, and learning rates, it creates a grid with all possible configurations and performs an exhaustive search by evaluating the fitness of each one (Bergstra and Bengio, 2012; Gulli et al., 2019; Amr, 2020; Raschka et al., 2022). In this research, the Grid Search method is used to automate the training of the MLP neural network and assist in selecting the model configurations. This includes the input set (T_mean, T_max, T_min, R_H, u₂, and R_s) and activation functions for the first and second layers, which may be step-wise (satlin) or linear (purelin), that is (Eqs. (12) and (13)),

(12)$$\text{satlin}\left(u_i\right)=\left\{\begin{array}{l}1 \text{se} u_i \ge 0\\ 0 \text{se} u_i < 0 \end{array}\right.$$

(13)$$\text{purelin}\left(u_i\right)=u_i$$

sigmoid (logsig) (Eq. (11)) and hyperbolic tangent (tansig) (Eq. (10)), along with the type of training, which ranges from Back-propagation (traingd), Quasi-Newton Back-propagation (trainbfg), Resilient Back-propagation (trainrp), to Levenberg-Marquardt Back-propagation (trainlm). The number of neurons and the number of epochs follow the conventionally trained MLP model, with 30 neurons in the first hidden layer and 15 in the second. The training duration is set to 4000 epochs, and this initial set of parameters was defined based on various tests and simulations. Fig. 5 provides an overview of the configurations adopted by the MLP-GRID.

Figure 5
Characteristics defined by the MLP-GRID model. Legends: Activation function of the first hidden layer (F1), Activation function of the second hidden layer (F2), Training type adopted (TR).

2.4.4. MLP trained using the Random Search method (MLP-RD)

Techniques such as Grid Search, which evaluates all possible hyperparameter configurations of a model to find its optimal setup, can be computationally expensive and impractical when dealing with large feature sets (Raschka et al., 2022). As an alternative, the Random Search method performs a random exploration of configurations within a defined set of hyperparameters (Bergstra and Bengio, 2012).

Similarly to Grid Search, the Random Search method is employed to automate the training of MLP neural networks by generating and evaluating 1600 randomly created configurations that determine characteristics including the definition of inputs (T_mean, T_max, T_min, R_H, u₂, and R_s), activation functions for the first and second layers, such as satlin (Eq. (12)), purelin (Eq. (13)), logsig (Eq. (11)), and tansig (Eq. (10)), as well as training methods traingd, trainbfg, trainrp, and trainlm. The number of neurons and epochs are fixed, similar to the conventionally trained MLP model. Similarly to the Grid Search method, the parameter choices in this approach were also defined based on various tests and simulations. Fig. 6 illustrates the configurations adopted by the MLP-RD.

Figure 6
Characteristics defined by the MLP-RD model. Legends: Activation function of the first hidden layer (F1), Activation function of the second hidden layer (F2), Adopted training type (TR).

2.4.5. MLP trained using Genetic Algorithms (MLP-GA)

Establishing hyperparameters, including the quantity of inputs, activation functions, training algorithms, learning rates, among others, is a time consuming task for ANN users (Géron, 2019). This research proposes the use of GA to automate the training and adjustment process of the MLP, aiming to find its optimal configuration. Fig. 7 illustrates the functioning of the MLP-GA.

Figure 7
Functioning of the MLP-GA.

The GA is responsible for generating and evaluating various potential configurations for the MLP. It examines the input dataset, the activation functions of the first and second hidden layers, the training algorithm, the learning rate, and the momentum factor. The goal of the GA is to minimize the absolute global error, which is determined by comparing the ETo values calculated using the PM FAO-56 method on the validation set with those predicted by the MLP. This process involves assessing the MLPs generated from the configurations defined by each individual, represented by a 14-bit binary string. The binary representation streamlines the characterization of the problem and enhances the development of encoders and decoders, thereby automating the MLP training process. This method, due to its simplicity and efficiency, allows for the articulation of multiple features and has been extensively employed by various researchers (Lai and Wang, 2015; Feng et al., 2016; Ventura et al., 2019; Moayedi et al., 2020; Coutinho et al., 2023). Fig. 8 illustrates the encoding of chromosomes or individuals within a population.

Figure 8
Chromosome model used by GA to define the ANN configurations.

In the chromosome, the genes identified as (a) define the input variables (T_mean, T_max, T_min, R_H, u₂, and R_s). A value of 1 signifies that the variable should be included, while a value of 0 indicates that it should be excluded. The genes identified as (b) and (c) determine the activation functions used in the first and second hidden layers, which may include purelin (0 0), satlin (1 1), logsig (1 0), and tansig (0 1) (Eqs. 13, 12, 11, 10) (Braga et al., 2012; Asim et al., 2017). Identified genes as (d) are responsible for determining the training algorithm to be applied. Options include traingd (0 1), trainbfg (1 0), trainrp (0 0), and trainlm (1 1). Further details on these algorithms are available in Haykin, 2001; Ayoub and Demiral, 2011; Mathworks, 2017. The genes identified as (e) define the learning rate and momentum rate applied in some models. The learning rate can range from 0.01 to 0.6, and the momentum rate is obtained by multiplying the learning rate by specific constants. These multipliers are 0.10, 0.12, 0.15, and 0.8, corresponding to choices (0 1), (1 1), (0 0), and (1 0), respectively.

For the GA configuration, various tests and simulations were conducted, establishing a population size of 40 individuals and a maximum of 40 generations, enabling the evaluation of up to 1,600 neural networks. Additionally, it is specified that 50% of the population will consistently advance to the next generation. Elitism is applied to retain approximately 75% (15 individuals) with the highest fitness values, while the remaining 25% (5 individuals) with lower fitness values are selected through random sampling. New parents for generating offspring are chosen using a tournament selection method with a size of 5, both for the first and second parent, to maintain genetic diversity and prevent the loss of valuable traits from individuals with lower fitness. A mechanism is also in place to ensure the same individual is not selected twice as a parent. A two-point crossover operator is employed, limited to a maximum of 6 genes at each end, resulting in a crossover range of 2 to 12 genes in total. The number of neurons and epochs used in the MLP are kept consistent with those in the conventionally trained model. Some of the configurations adopted by the MLP-GA after reaching the termination criterion, i.e., upon completion of the process, are shown in Fig. 9.

Figure 9
Characteristics defined by the MLP-GA model. Legends: Function of the first hidden layer (F1), Function of the second hidden layer (F2), Type of training adopted (TR).

Finally, it is important to highlight that, for comparison purposes, all models were executed on a machine equipped with a 7th generation Intel Core i7 processor and 16 GB of DDR4 memory.

2.5. Performance evaluation

The effectiveness of the applied techniques can be assessed by comparing the ETo estimated by the techniques with that obtained through the PM FAO-56 method. For this purpose, the following statistical metrics are employed: Pearson correlation coefficient (r) (Eq. (14)), mean absolute error (MAE) (Eq. (15)), root mean square error (RMSE) (Eq. (16)), and mean percentage error (MPE) (Eq. (17)) (Manesh et al., 2014; Zhao et al., 2021). These metrics are defined as,

(14)$$r= \frac{\frac{\sum _{j=1}^n\left(x_j-\overline{x}\right)\left(O_j-\overline{O}\right)}{n} }{\sqrt{\frac{\sum _{j=1}^n{\left(x_j-\overline{x}\right)}^2}{n}} .\sqrt{\frac{\sum _{j=1}^n{\left(O_j-\overline{O}\right)}^2}{n}}}$$

(15)$$MAE= \frac{\sum _{j=1}^n\left|O_j-x_j\right|}{n}$$

(16)$$RMSE= \sqrt{\frac{\sum _{j=1}^n{\left(O_j-x_j\right)}^2}{n}}$$

(17)$$MPE=\frac{\sum _{j=1}^n\frac{\left|O_j-x_j\right|}{O_j}}{n}*100$$

where n is the number of data points used; $O_j$ is the ETo obtained with the PM FAO-56; $x_j$ is the ETo obtained by the other techniques; $\overline{O}$ is the mean of the ETo obtained with the PM FAO-56; $\overline{x}$ is the mean of the ETo obtained by the other techniques. Additionally, the metrics for mean (M), maximum (MAX), minimum (MIN), and standard deviation (SD) are utilized to compare the ETo estimated from the models with those derived from the PM FAO-56 method.

3. Results

3.1. Estimation of daily ETo in the state of RJ regions

The indices highlighted in Table 3 reveal that the MLP, MLP-RD, MLP-GRID, and MLP-GA exhibit superior precision in relation to the results achieved from empirical methods. Evaluating the metrics r, RMSE, MAE, and MPE, it is verified that, across the three locations in the State of RJ, the MLP-GA shows greater precision. This is further confirmed by comparing the error metrics of the MLP-GA, which consistently show lower values in all locations, indicating a strong degree of precision in the estimates. Additionally, the values of M, MAX, and MIN obtained from the HS, HA, JH, LI, and BL clearly indicate that these models either underestimate or overestimate the daily ETo across all regions.

Table 3
Comparison between ETo estimated using the Penman-Monteith method and the values obtained from the models for the regions of RJ. Indexes: training time in minutes (TIME), generation of the individual chosen as the solution (G). Models: Hargreaves and Samani (HS), Hamon (HA), Jensen Haise (JH), Linacre (LI), Benavides Lopez (BL), MLP, MLP-RD, MLP-GRID, MLP-GA. The bolded values represent the most relevant metrics.

Penman-Monteith - ETo	M	MAX	MIN	SD
Mendes	3.46	8.97	0.47	1.78
Santa Maria Madalena	4.54	10.80	0.45	1.89
Teresópolis	3.76	8.98	0.45	1.68

Estimated ETo	Models	M	MAX	MIN	SD	(r)	RMSE	MAE	MPE (%)	TIME	G
Mendes	HS	4.41	8.64	1.33	1.49	0.64	1.69	1.32	68.11	X	X
	HA	3.89	7.69	1.44	1.24	0.66	1.39	1.11	55.49	X	X
	JH	4.22	13.11	0.02	2.55	0.94	1.30	0.98	26.87	X	X
	LI	5.25	11.61	1.83	1.50	0.80	2.06	1.83	85.90	X	X
	BL	5.85	12.89	0.93	1.99	0.82	2.65	2.39	97.75	X	X
	MLP	3.44	8.20	0.61	1.68	0.97	0.41	0.27	8.78	3.33	X
	MLP-RD	3.52	8.56	0.48	1.73	0.99	0.22	0.17	6.14	152.73	X
	MLP-GRID	3.59	7.89	-0.32	1.55	0.98	0.41	0.33	13.37	586.11	X
	MLP-GA	3.46	8.65	0.35	1.76	0.99	0.19	0.15	5.36	11.61	4

Estimated ETo	Models	M	MAX	MIN	SD	(r)	RMSE	MAE	MPE (%)	TIME	G
Santa Maria Madalena	HS	4.27	7.69	0.61	1.42	0.66	1.45	1.13	32.70	X	X
	HA	2.92	5.74	1.37	0.93	0.70	2.14	1.77	37.55	X	X
	JH	3.26	10.82	0.01	2.34	0.72	2.07	1.55	36.71	X	X
	LI	5.35	10.55	1.92	1.51	0.79	1.40	1.09	34.99	X	X
	BL	5.69	10.97	0.95	1.85	0.84	1.53	1.21	35.49	X	X
	MLP	4.50	8.74	0.43	1.72	0.93	0.67	0.45	11.59	3.41	X
	MLP-RD	4.36	9.55	0.17	1.82	0.98	0.34	0.24	5.95	134.93	X
	MLP-GRID	4.56	9.62	0.58	1.91	0.95	0.59	0.38	9.86	595.9	X
	MLP-GA	4.61	10.09	0.50	1.91	0.99	0.26	0.19	4.66	10.26	3

Estimated ETo	Models	M	MAX	MIN	SD	(r)	RMSE	MAE	MPE (%)	TIME	G
Teresópolis	HS	4.54	8.33	1.48	1.52	0.77	1.33	1.01	41.38	X	X
	HA	3.12	6.27	1.27	1.06	0.74	1.31	1.04	31.23	X	X
	JH	3.54	10.23	0.02	2.11	0.92	0.89	0.68	20.84	X	X
	LI	5.00	12.03	2.20	1.41	0.85	1.50	1.31	50.81	X	X
	BL	5.21	12.82	1.00	1.90	0.86	1.73	1.48	49.50	X	X
	MLP	3.91	8.03	-0.16	1.16	0.96	0.45	0.34	11.16	4.26	X
	MLP-RD	3.87	8.98	0.93	1.61	0.99	0.18	0.14	6.32	137.18	X
	MLP-GRID	3.76	8.45	0.49	1.59	0.98	0.27	0.20	6.61	631.55	X
	MLP-GA	3.84	9.00	0.93	1.62	0.99	0.19	0.13	5.46	2.56	1

For the Mendes station, besides the MLP-GA, the MLP, MLP-RD, MLP-GRID, and JH models also exhibit elevated r values. However, the RMSE, MAE, and MPE metrics reveal that the results produced by the MLP and MLP-RD are more reliable than those of the HS, HA, JH, LI, BL, and MLP-GRID. Nonetheless, the MLP-GA's performance stands out, with this model proving to be significantly more accurate than both the MLP and MLP-RD, achieving an MAE that is 80% below than that of the MLP and 13% smaller compared to the MLP-RD. The most notable advantage, however, lies in the execution time, as the MLP-GA completed the task in approximately 11 min and 61 s, more than 12 times faster than the MLP-RD.

The MLP-GA also achieves the highest r indexes for the Santa Maria Madalena station. When comparing its RMSE and MAE with those of other techniques, the RMSE was 150% below compared to that obtained by the MLP, 30% smaller than that achieved by the MLP-RD, 120% smaller than that yielded by the MLP-GRID, and more than seven times smaller than that of the HA method. In terms of MAE, it was 136% reduced compared to that of the MLP, 25% smaller than that of the MLP-RD, 100% smaller than that of the MLP-GRID, and eight times smaller than that of the HA method. Additionally, as observed in the scatter and violin plots (Fig. 10), the ETo obtained from the MLP-GA closely aligns with that obtained from the PM FAO-56, exhibiting a similar distribution. The execution time of the MLP-GA is completed in approximately 10 min, significantly faster than the MLP-RD and MLP-GRID, although nearly three times slower than the MLP.

Figure 10
Daily ETo scatter plot (A), MAE and RMSE indexes obtained by the models (B), and distribution of real and estimated ETo values (C) for the regions of the state of Rio de Janeiro (Brazil). Regions: Mendes (ME), Santa Maria Madalena (SM) and Teresópolis (TE).

For Teresópolis station, the MLP-GA also stands out, with the estimated ETo yielding low error values: 0.19 mm for RMSE, 0.13 mm for MAE, and 5.46% for MPE. These low errors suggest that the MLP-GA achieves an precision rate exceeding 94% for each estimated value. Moreover, the execution time of the MLP-GA is notably shorter than that of the other models, taking only 2 min and 56 s. Fig. 10 presents a dynamic perspective of the RMSE, MAE results, and data distribution for all regions.

3.2. Estimation of daily ETo in the state of ES regions

It is evident that the ANNs significantly outperform the HS, HA, JH, LI, and BL methods in estimating ETo. At the Jerônimo Monteiro station, the MLP-GA achieves high r values along with low RMSE and MAE indexes. Furthermore, the MPE generated by it is 2.96%, ensuring that its estimated values yield an average precision of 97%.

At the Sooretama station, the RMSE, MAE, and MPE indexes from the MLP-GA also indicate high precision. The obtained values show an MAE of 0.16 mm and an MPE of 3.89%, indicating an precision rate exceeding 96%. Another important observation for this location is that the M, MAX, MIN, and SD obtained by the MLP-GA closely align with the ETo measures estimated using the PM FAO-56, suggesting that the model exhibits minimal variation. In Table 4, the execution time of each model is presented, revealing that the MLP-GA required a training period four times longer than the conventional MLP, although this was still considerably shorter than the time needed for the MLP-RD and MLP-GRID.

Table 4
Comparison between ETo estimated using the Penman-Monteith method and the values obtained from the models for the regions of ES. Indexes: training time in minutes (TIME), generation of the individual chosen as the solution (G). Models: Hargreaves and Samani (HS), Hamon (HA), Jensen Haise (JH), Linacre (LI), Benavides Lopez (BL), MLP, MLP-RD, MLP-GRID, MLP-GA. The bolded values represent the most relevant metrics.

Penman-Monteith-ETo	M	MAX	MIN	SD
Jerônimo Monteiro	4.82	10.46	0.64	2.20
Sooretama	5.08	10.01	0.55	1.96
Venda Nova do Imigrante	3.61	8.97	0.53	1.73

Estimated ETo	Models	M	MAX	MIN	SD	(r)	RMSE	MAE	MPE (%)	TIME	G
Jerônimo Monteiro	HS	5.19	8.52	1.73	1.51	0.72	1.56	1.15	29.32	X	X
	HA	4.12	9.13	1.85	1.25	0.71	1.71	1.41	38.06	X	X
	JH	5.34	12.79	0.05	2.96	0.92	1.34	1.05	22.98	X	X
	LI	5.79	11.25	2.43	1.74	0.85	1.49	1.22	41.85	X	X
	BL	7.01	12.93	2.25	2.20	0.86	2.45	2.20	64.58	X	X
	MLP	4.77	9.49	0.71	2.14	0.96	0.56	0.42	9.97	2.85	X
	MLP-RD	4.83	10.17	0.79	2.11	0.99	0.17	0.12	3.88	102.55	X
	MLP-GRID	4.84	9.96	0.86	2.17	0.99	0.24	0.18	5.19	455.1	X
	MLP-GA	4.88	10.28	0.48	2.20	0.99	0.16	0.11	2.96	3.56	1

Estimated ETo	Models	M	MAX	MIN	SD	(r)	RMSE	MAE	MPE (%)	TIME	G
Sooretama	HS	4.65	8.35	1.45	1.24	0.68	1.48	1.17	29.97	X	X
	HA	3.77	9.02	2.03	0.95	0.67	1.98	1.65	34.12	X	X
	JH	4.09	10.41	0.05	2.22	0.66	1.98	1.49	29.04	X	X
	LI	5.17	12.02	2.48	1.15	0.83	1.19	0.98	26.89	X	X
	BL	6.44	13.69	2.50	1.53	0.85	1.71	1.41	41.04	X	X
	MLP	4.35	8.04	0.61	1.56	0.85	1.25	0.89	16.85	4.96	X
	MLP-RD	5.14	9.63	0.77	1.86	0.99	0.26	0.19	4.91	132.26	X
	MLP-GRID	5.29	9.36	0.88	1.96	0.96	0.52	0.33	8.96	607.1	X
	MLP-GA	5.08	10.06	0.51	1.96	0.99	0.21	0.16	3.89	24.61	23

Estimated ETo	Models	M	MAX	MIN	SD	(r)	RMSE	MAE	MPE (%)	TIME	G
Venda Nova do Imigrante	HS	4.61	8.54	1.34	1.32	0.74	1.52	1.21	55.58	X	X
	HA	3.12	7.18	1.44	0.91	0.68	1.38	1.10	36.93	X	X
	JH	3.83	10.68	0.01	2.20	0.91	0.94	0.70	19.94	X	X
	LI	4.66	10.37	2.07	1.20	0.85	1.40	1.21	52.64	X	X
	BL	5.02	11.00	1.17	1.68	0.86	1.67	1.46	54.76	X	X
	MLP	3.59	8.40	0.32	1.65	0.96	0.44	0.29	8.50	3.33	X
	MLP-RD	3.64	9.33	-0.22	1.74	0.99	0.15	0.11	3.84	121.6	X
	MLP-GRID	3.67	9.42	0.25	1.72	0.99	0.24	0.15	5.44	556.01	X
	MLP-GA	3.69	8.53	0.64	1.72	0.99	0.19	0.14	5.26	4.23	3

At the Venda Nova do Imigrante station, it is evident that, alongside the machine learning models, the JH method exhibited a strong correlation and minimal error metrics with the estimates produced by the PM FAO-56, achieving 0.91 for r, 0.94 mm for RMSE, 0.70 mm for MAE, and 19.94% for MPE. However, these results are less accurate than those attained by the ANNs, where the MLP achieved 0.29 mm for MAE and 8.50% for MPE, the MLP-RD recorded 0.11 mm for MAE and 3.84% for MPE, the MLP-GRID reached 0.15 mm for MAE and 5.44% for MPE, and the MLP-GA secured 0.14 mm for MAE and 5.26% for MPE, confirming that these models achieved an precision exceeding 90%. Moreover, the graphs shown in Fig. 11 indicate that the predictions made by the ANNs show a strong correlation, reduced errors, and a distribution closely matching those of the PM FAO-56.

Figure 11
Scatter plot of daily ETo (A), MAE and RMSE indexes obtained by the models (B), and distribution of real and estimated ETo values (C) for the regions of Espírito Santo state (Brazil). Regions: Jerônimo Monteiro (JE), Sooretama (SO), and Venda Nova do Imigrante (VI).

3.3. Estimation of daily ETo in the state of SP regions

The graphs presented in Fig. 12 reveal that the ETo estimated by the MLP-GA for the four regions of SP closely aligns with the values derived from the PM FAO-56 method. Moreover, the metrics r, RMSE, MAE, and MPE (Table 5) suggest that the MLP-GA surpasses the performance of competing models in most cases.

Figure 12
Dispersion of daily ETo (A), MAE and RMSE indexes obtained by the models (B), and distribution of real and estimated ETo values (C) for the regions of São Paulo state (Brazil). Regions: Guaratinguetá (GT), Itú (IT), Jaboticabal (JA) and Miguelópolis (MI).

Table 5
Comparison between ETo estimated using the Penman-Monteith method and the values obtained from the models for the regions of SP. Indexes: training time in minutes (TIME), generation of the selected solution (G). Models: Hargreaves Samani (HS), Hamon (HA), Jensen Haise (JH), Linacre (LI), Benavides Lopez (BL), MLP, MLP-RD, MLP-GRID, MLP-GA. The bolded values represent the most relevant metrics.

Penman-Monteith - ETo	M	MAX	MIN	SD
Guaratingueta	5.38	11.92	0.79	2.04
Itú	4.97	10.50	0.37	2.48
Jaboticabal	4.48	10.35	0.64	1.71
Miguelópolis	5.39	11.65	0.66	1.91

Estimated ETo	Models	M	MAX	MIN	SD	(r)	RMSE	MAE	MPE (%)	TIME	G
Guaratinguetá	HS	6.30	9.62	2.29	1.65	0.56	1.99	1.59	42.90	X	X
	HA	5.06	9.37	1.29	1.46	0.58	1.71	1.37	31.79	X	X
	JH	3.46	14.16	0.03	2.32	0.68	2.59	2.16	42.18	X	X
	LI	8.45	14.52	1.76	2.22	0.69	3.49	3.08	73.02	X	X
	BL	9.57	14.97	0.74	2.36	0.70	4.52	4.18	95.49	X	X
	MLP	5.38	10.19	0.92	1.75	0.84	1.08	0.84	17.73	1.06	X
	MLP-RD	4.45	10.46	0.16	1.92	0.99	0.96	0.94	19.70	119.90	X
	MLP-GRID	5.53	11.05	0.85	1.90	0.99	0.30	0.23	5.73	514.06	X
	MLP-GA	5.36	11.03	0.83	2.00	0.99	0.19	0.13	2.94	7.3	2

Estimated ETo	Models	M	MAX	MIN	SD	(r)	RMSE	MAE	MPE (%)	TIME	G
Itú	HS	4.56	11.00	1.19	1.45	0.39	2.36	1.89	78.77	X	X
	HA	3.18	6.83	1.29	1.06	0.44	2.85	2.39	63.25	X	X
	JH	3.78	12.19	0.02	2.11	0.72	2.10	1.53	32.55	X	X
	LI	4.99	8.87	1.85	1.46	0.93	1.24	1.03	46.99	X	X
	BL	5.32	10.11	0.75	1.97	0.93	1.03	0.78	36.91	X	X
	MLP	4.00	7.87	0.33	1.66	0.94	1.43	1.06	19.81	7.33	X
	MLP-RD	5.04	9.93	0.46	2.35	0.98	0.48	0.35	14.05	103.53	X
	MLP-GRID	5.26	9.77	0.29	2.22	0.96	0.70	0.56	27.05	515.91	X
	MLP-GA	4.94	10.22	-0.20	2.50	0.99	0.27	0.20	7.09	10.15	11

Estimated ETo	Models	M	MAX	MIN	SD	(r)	RMSE	MAE	MPE (%)	TIME	G
Jaboticabal	HS	5.28	8.31	2.29	1.37	0.54	1.69	1.27	42.24	X	X
	HA	4.00	6.98	1.50	1.27	0.48	1.62	1.32	35.62	X	X
	JH	3.92	11.59	0.02	2.15	0.73	1.58	1.12	25.54	X	X
	LI	6.37	11.84	2.46	1.47	0.76	2.20	1.92	57.56	X	X
	BL	7.26	13.00	1.65	1.84	0.77	3.03	2.79	76.44	X	X
	MLP	3.98	9.03	0.55	2.15	0.67	1.68	1.02	24.74	2.9	X
	MLP-RD	4.43	9.86	0.03	1.67	0.99	0.24	0.17	5.52	151.35	X
	MLP-GRID	4.66	9.51	0.74	1.68	0.99	0.28	0.21	5.71	695.28	X
	MLP-GA	4.45	9.59	0.26	1.66	0.99	0.17	0.12	3.25	7.43	4

Estimated ETo	Models	M	MAX	MIN	SD	(r)	RMSE	MAE	MPE (%)	TIME	G
Miguelópolis	HS	5.48	8.60	1.76	1.22	0.64	1.46	1.13	29.18	X	X
	HA	3.81	6.45	1.47	0.99	0.62	2.18	1.83	34.34	X	X
	JH	4.51	14.59	0.21	2.42	0.71	1.91	1.32	25.59	X	X
	LI	6.34	11.07	2.40	1.43	0.73	1.60	1.22	33.32	X	X
	BL	7.38	12.41	1.79	1.75	0.76	2.35	2.00	49.27	X	X
	MLP	5.35	9.56	0.95	1.65	0.89	0.85	0.59	12.93	5.63	X
	MLP-RD	5.35	11.41	0.64	1.85	0.99	0.16	0.11	2.53	126.91	X
	MLP-GRID	5.35	10.36	-1.48	1.82	0.97	0.43	0.30	8.07	598.00	X
	MLP-GA	5.40	10.66	0.70	1.85	0.99	0.17	0.12	2.90	13.23	13

At the Guaratinguetá station, the M, MAX, MIN, and SD values generated by the machine learning models MLP, MLP-GRID, and MLP-GA are closely aligned with those derived from the PM FAO-56 method. However, it is important to highlight that the MPE value produced by the MLP-GA is five times smaller than the MPE reported by the MLP, 5.7 times smaller than that of the MLP-RD, and 95% below that of the MLP-GRID. Once again, the most significant advantage is the execution time, as detailed in Table 5, where the MLP-GA completed its run in approximately 7 min, while the MLP-RD and MLP-GRID took 119 and 514 min, respectively.

The MAE of the MLP-GA at the Itu station shows that the predicted ETo had an average error of 0.20 mm. Additionally, the (r) remains at 0.99, confirming a strong relationship between the estimated values and those obtained from the PM FAO-56 method. In terms of training times, Table 5 illustrates that while the MLP-GA required 38% more time to train compared to the MLP, its execution time was significantly shorter than that of the MLP-RD and MLP-GRID models.

At the Jaboticabal station, the MLP-GA, as in other locations, produced the most accurate results for RMSE, MAE, and MPE 0.17 mm, 0.12 mm, and 3.25%, respectively representing the lowest error values. As for the MLP-GA execution time, Table 5 reveals it took approximately 7 min to train, compared to the 151 and 695 min required by the MLP-RD and MLP-GRID models, respectively.

For the Miguelópolis station, Table 5 highlights the superior performance of neural networks over other models. However, although the MLP-RD model delivered slightly more accurate metrics than the MLP-GA, it required a significantly longer execution time. Fig. 12 visualizes the error distribution, showing differences between RMSE and MAE, and illustrates the predicted ETo distribution across the models.

3.4. Estimation of daily ETo in the state of MG regions

The data in Table 6 reveal that the LI and BL methods underestimated the M, MAX, and MIN values produced by PM FAO-56 in the regions of Caratinga, Paracatu, Montes Claros, and Santa Fé de Minas. Similarly, the HA method also showed underestimation in all four regions. In contrast, the neural network models MLP, MLP-RD, MLP-GRID, and MLP-GA exhibit greater precision, with the RMSE, MAE, and MPE metrics significant the effectiveness of the MLP-GA in estimating ETo for nearly all regions. Fig. 13 illustrates the strong alignment between the ETo values predicted by MLP-GA and those from PM FAO-56. At the Caratinga station, the high correlation coefficient (r) indicates a strong relationship between the values predicted by MLP-RD and those from PM FAO-56. Evaluating the RMSE, MAE, and MPE, which are 0.13 mm, 0.09 mm, and 3.24%, respectively, reveals that the predictions have an precision of over 96%. However, when considering training times, MLP-RD required nearly 150 min, which is substantially longer than the training times of the MLP and MLP-GA.

Table 6
Comparison between ETo estimated using the Penman-Monteith method and the values obtained from the models for the regions of MG. Indexes: training time in minutes (TIME) and generation of the selected individual (G). Models: Hargreaves Samani (HS), Hamon (HA), Jensen Haise (JH), Linacre (LI), Benavides Lopez (BL), MLP, MLP-RD, MLP-GRID and MLP-GA. The bolded values represent the most relevant metrics.

Penman-Monteith - ETo	M	MAX	MIN	SD
Caratinga	3.82	9.13	0.48	1.60
Paracatu	5.11	10.86	0.59	2.06
Montes Claros	5.50	9.75	0.71	1.75
Santa Fé de Minas	5.73	11.59	0.45	2.10

Estimated ETo	Models	M	MAX	MIN	SD	(r)	RMSE	MAE	MPE (%)	TIME	G
Caratinga	HS	4.82	8.55	1.94	1.22	0.75	1.45	1.17	46.84	X	X
	HA	3.33	6.10	1.70	0.89	0.75	1.20	0.94	27.91	X	X
	JH	4.40	14.03	0.02	2.37	0.93	1.19	0.93	24.16	X	X
	LI	5.45	11.88	2.36	1.37	0.77	1.92	1.68	61.64	X	X
	BL	5.95	12.73	1.50	1.77	0.79	2.39	2.15	71.37	X	X
	MLP	3.75	8.12	0.59	1.52	0.96	0.43	0.29	8.45	5.26	X
	MLP-RD	3.84	8.82	0.59	1.57	0.99	0.13	0.09	3.24	149.81	X
	MLP-GRID	3.29	10.31	0.01	1.58	0.98	0.58	0.53	16.66	662.91	X
	MLP-GA	3.87	8.34	0.44	1.58	0.99	0.18	0.13	4.29	8.96	3

Estimated ETo	Models	M	MAX	MIN	SD	(r)	RMSE	MAE	MPE (%)	TIME	G
Paracatu	HS	6.03	9.58	2.85	1.36	0.70	1.72	1.33	42.57	X	X
	HA	3.74	6.44	1.89	0.98	0.51	2.15	1.74	36.22	X	X
	JH	5.28	13.49	0.12	2.46	0.83	1.35	1.02	20.67	X	X
	LI	6.03	11.85	2.51	1.72	0.83	1.44	1.12	32.63	X	X
	BL	7.01	13.18	2.09	2.11	0.85	2.21	1.91	48.69	X	X
	MLP	5.16	10.30	0.39	1.99	0.95	0.58	0.40	8.73	4.48	X
	MLP-RD	5.12	10.53	0.84	2.01	0.99	0.19	0.14	4.11	138.93	X
	MLP-GRID	5.09	10.47	0.97	2.04	0.99	0.23	0.17	4.50	624.65	X
	MLP-GA	5.10	11.33	0.55	2.02	0.99	0.22	0.15	3.86	14.98	6

Estimated ETo	Models	M	MAX	MIN	SD	(r)	RMSE	MAE	MPE (%)	TIME	G
Montes Claros	HS	5.84	9.22	2.82	1.25	0.56	1.49	1.16	29.73	X	X
	HA	3.79	6.39	1.93	0.88	0.57	2.23	1.92	34.99	X	X
	JH	4.77	12.07	0.02	2.32	0.74	1.70	1.24	24.31	X	X
	LI	7.00	11.73	2.67	1.55	0.23	2.54	2.04	52.05	X	X
	BL	8.12	12.81	2.30	1.81	0.76	2.89	2.62	56.87	X	X
	MLP	4.40	7.61	0.68	1.31	0.85	1.43	1.15	19.73	3.23	X
	MLP-RD	5.84	11.11	-1.5	1.85	0.99	0.42	0.36	7.25	153.3	X
	MLP-GRID	5.05	8.45	0.83	1.28	0.82	1.09	0.83	15.88	662.61	X
	MLP-GA	5.54	9.94	0.45	1.71	0.99	0.19	0.14	2.99	7.16	4

Estimated ETo	Models	M	MAX	MIN	SD	(r)	RMSE	MAE	MPE (%)	TIME	G
Santa Fé de Minas	HS	6.25	9.58	3.33	1.27	0.61	1.73	1.32	40.64	X	X
	HA	4.42	7.74	2.41	1.03	0.53	2.20	1.86	38.45	X	X
	JH	4.24	12.08	0.02	2.32	0.63	2.41	1.75	31.42	X	X
	LI	7.66	13.28	2.86	1.88	0.76	2.37	1.98	50.06	X	X
	BL	9.08	14.43	2.71	2.11	0.78	3.63	3.35	76.59	X	X
	MLP	5.61	9.16	0.46	1.84	0.86	1.05	0.78	15.12	3.45	X
	MLP-RD	5.62	12.33	-0.03	2.10	0.99	0.23	0.18	4.36	152.11	X
	MLP-GRID	5.71	12.72	-0.24	2.18	0.93	0.77	0.58	13.18	641.43	X
	MLP-GA	5.78	11.66	0.53	2.09	0.99	0.14	0.11	2.49	9.9	3

Figure 13
Dispersion of daily ETo (A), MAE and RMSE indexes obtained by the models (B), and distribution of real and estimated ETo values (C) for the regions of Minas Gerais state (Brazil). Regions: Caratinga (CA), Paracatu (PA), Montes Claros (MC) and Santa Fé de Minas Gerais (SM).

At the Paracatu station, Table 6 reveals that the MLP-RD, MLP-GRID, and MLP-GA exhibited very minimal error rates, showing MLP-GA achieving the lowest MPE at 3.86%, indicating superior precision relative to the other models. Table 6 also outlines the training durations assigned to each model, where it is clear that while MLP-GA required three times more training time than MLP, it still completed training faster than both the MLP-RD and MLP-GRID.

At the Montes Claros station, it becomes apparent that the MLP and MLP-GRID produced higher RMSE, MAE, and MPE compared to the MLP-RD and MLP-GA, suggesting that the former models were less suited for ETo estimation at this site (Table 6). On the other hand, MLP-GA achieved a RMSE of 0.19 mm, a MAE of 0.14 mm, and a MPE of 2.99%, delivering highly accurate predictions.

At the Santa Fé de Minas station, the estimates produced by MLP-GA show less variation when compared to PM FAO-56, setting it apart from the alternative models. This is further illustrated by the graphs in Fig.13, which highlight how the error metrics for MLP-GA are smaller than those produced by the other models, and how its data distribution closely resembles that of PM FAO-5.

4. Discussion

Most approaches involve manually evaluating different models and input combinations to determine those that reveal the best fit for estimating ETo (Landeras et al., 2008; Saggi and Jain, 2019; Fan et al., 2019; Adnan et al., 2021a). This task requires a significant amount of time from users, who may still fail to achieve the optimal configurations. This limitation can cause models to become stagnant in local optima, resulting in unsatisfactory performance and preventing them from reaching the global optimum, thus hindering the full exploration of their potential.

This becomes evident when comparing the results of the MLP neural network trained using a conventional approach. Although this approach produced more accurate results than the HS, HA, JH, LI, and BL models, its performance was still inferior to that of the MLP-GRID, MLP-RD, and MLP-GA. Upon evaluating each model trained with hyperparameter tuning techniques, it is noted that MLP-GRID exhibits higher error rates and requires a longer execution time. Bergstra and Bengio (2012) have previously highlighted that Grid Search can be a time-consuming and less efficient approach compared to other techniques. Conversely, MLP-RD shows RMSE, MAE, and MPE values similar to those of MLP-GA in certain regions; however, in most cases, its performance is inferior, and its execution requires significantly more time.

Another relevant point pertains to the differences in characteristics defined by each training approach. The conventional training and Grid Search establish distinct parameters compared to Random Search techniques and GA. These approaches reveal both similarities and differences in defining certain aspects of the MLP, such as the training and activation functions. The GA selects as input variables those that exhibit the highest linear correlation with ETo, as revealed in Tables 2 and 5. Although MLP-GA has access to a broader set of parameters, it is observed that the GA determines different activation functions and training methods compared to the conventional approach and other hyperparameter tuning methods. This underscores that, even when applied to configurations with parameter constraints, the GA can identify combinations that yield satisfactory ETo estimates. This feature may facilitate the implementation of machine learning techniques, allowing for the determination of parameters as efficiently as a human supervisor.

One potential reason for the differences in results and characteristics defined by each type of training in the fact that the conventional approach employed in the MLP, even when conducted through tests, simulations, and evaluations, is directly influenced by the human supervisor and their prior expertise regarding the problem. In contrast, the other hyperparameter tuning approaches evaluate a significantly greater number of combinations, particularly the evolutionary approach adopted by the GA, which is solely guided by its fundamental principle of enhancing the individual to ensure a search for the globally optimal combination. This allows the MLP-GA to consistently exhibit low error rates, with minimal variations across each region, as illustrated in Table 7.

Table 7
Variation of the results obtained by the MLP, MLP-RD, MLP-GRID, and MLP-GA in estimating ETo for all locations. Indexes: average training time in minutes (TIME (M)), generation of the individual chosen as the solution (G).

Variation of machine learning model results
Models	Stations	(r)	RMSE	MAE	MPE (%)	TIME (M)	(G)
MLP	Rio de Janeiro	0.93-0.97	0.41-0.67	0.27-0.45	8.78-11.59	3.66	X
	Espírito Santo	0.85-0.96	0.44-1.25	0.29-0.89	8.50-16.85	3.71	X
	São Paulo	0.67-0.94	0.85-1.68	0.59-1.06	12.93-24.74	4.23	X
	Minas Gerais	0.85-0.96	0.43-1.43	0.29-1.15	8.45-19.73	4.10	X
Models	Stations	(r)	RMSE	MAE	MPE (%)	TIME (M)	(G)
MLP-RD	Rio de Janeiro	0.98-0.99	0.18-0.34	0.14-0.24	5.95-6.32	141.61	X
	Espírito Santo	0.99	0.15-0.26	0.11-0.19	3.84-4.91	118.80	X
	São Paulo	0.98-0.99	0.16-0.96	0.11-0.94	2.53-19.70	125.42	X
	Minas Gerais	0.99	0.13-0.42	0.09-0.36	3.24-7.25	148.53	X
Models	Stations	(r)	RMSE	MAE	MPE (%)	TIME (M)	(G)
MLP-GRID	Rio de Janeiro	0.95-0.98	0.27-0.59	0.20-0.38	6.61-13.37	604.52	X
	Espírito Santo	0.96-0.99	0.24-0.52	0.15-0.33	5.19-8.96	539.40	X
	São Paulo	0.96-0.99	0.28-0.70	0.21-0.56	5.71-27.05	580.81	X
	Minas Gerais	0.82-0.99	0.23-1.09	0.17-0.83	4.50-16.66	647.9	X
Models	Stations	(r)	RMSE	MAE	MPE (%)	TIME (M)	(G)
MLP-GA	Rio de Janeiro	0.99	0.19-0.26	0.13-0.19	4.66-5.46	8.14	1 - 4
	Espírito Santo	0.99	0.16-0.21	0.11-0.16	2.96-5.26	10.80	1 - 23
	São Paulo	0.99	0.17-0.27	0.12-0.20	2.90-7.09	9.52	2 - 13
	Minas Gerais	0.99	0.14-0.22	0.11-0.15	2.49-4.29	10.25	3 - 6

However, it is considerable that achieving these results requires the MLP-GA, on average, two and a half times more training time than the conventional training used by the MLP. This is attributed to the automation process driven by the integration of the GA and MLP, which involves generating populations of individuals defining the characteristics of the neural network, testing, evaluating, and recombining new individuals. This process can result in up to 1,600 executions if the maximum number of generations is reached. Nevertheless, the model executed 60 times faster than the MLP-GRID and 12 times faster than the MLP-RD, with the longest time for a viable solution being approximately 25 min. Still, it can be acknowledged that the complexity of the problem and the numerous parameters can directly influence its execution time. When comparing the MLP-GA results to those derived from different methodologies reported in the literature in Table 8, it is evident that the achieved indexes are comparable to those identified by other researchers.

Table 8
Indexes obtained by different models in estimating the ETo found in the literature.

Models	(r)	RMSE	MAE	MPE (%)	References
MLP	X	0.23	0.17	X	Landeras et al. (2008)
MVRVM	X	0.65	X	X	Torres et al. (2011)
WNN	0.89	1.03	X	X	Falamarzi et al. (2014)
Wavelet-ANN	0.96	0.63	X	X	Patil and Deka, (2015)
GFF	0.75	0.87	0.99	X	Luo et al. (2015)
ELM	X	0.33	X	X	Patil and Deka, (2016)
MLP	0.92	X	0.70	X	Traore et al. (2016)
MLP	X	0.19	0.14	X	Yassin et al. (2016)
MARS e SVM-RBF	X	0.36	0.28	X	Mehdizadeh et al. (2017)
GEP	0.86	0.99	0.73	X	Traore et al. (2017)
ANFIS-FA	X	0.24	0.18	3.57	Tao et al. (2018)
MLP	X	0.19	X	X	Saggi and Jain, (2019)
LSTM	X	0.38	X	X	Afzaal et al. (2020)
RF	0.99	0.33	0.25	X	Wu, M., et al. (2020)
CAT	X	0.56	X	X	Wu, T., et al. (2020)
PSO-GBDT	X	0.21	0.28	X	Zhao et al. (2021)
MLP	X	0.37	0.30	X	Kaya et al. (2021)
GMDHNN	X	0.42	0.34	X	Adnan et al. (2021b)
ANN	X	0.43	0.18	X	Spontoni et al. (2023)
ANN	X	0.15	X	X	Treder et al. (2023)
XGBoost	X	0.34	X	X	Huang et al. (2025)
RF	0.98	0.22	0.15	X	Koç and Paksoy, (2025)
XGB-MPA	X	0.17	X	X	Achite et al. (2025)
RF	X	0.31	0.23	5.21	Tausif et al. (2025)

The RMSE, MAE, and MPE indices of the MLP-GA obtained in the present study ranged from 0.14 mm to 0.27 mm, 0.11 mm to 0.20 mm, and 2.49% to 7.09%, indicating a high precision of the estimated ETo. Comparable outcomes are reported by Landeras et al. (2008); Yassin et al. (2016); Mehdizadeh et al. (2017); Tao et al. (2018); Saggi and Jain (2019); Wu et al. (2020); Zhao et al. (2021); Kaya et al. (2021); Adnan et al. (2021b); Treder et al. (2023); and Koç and Paksoy (2025), who assess various models and configurations to identify an optimal model for estimating or predicting ETo.

Table 8 further confirms that high precision indices for estimating ETo are similarly identified by Falamarzi et al. (2014), who utilized Wavelet neural networks to estimate ETo in Redesdale, Australia (r = 0.89 and RMSE = 1.03 mm); Patil and Deka (2015), who evaluated hybrid Wavelet-ANN and Wavelet-ANFIS models to estimate ETo in arid regions in India (r = 0.96 and RMSE = 0.63 mm); Luo et al. (2015), who examined the capability of four ANNs to predict ETo at the Gaoyou station located in Jiangsu Province, China (r = 0.75, RMSE = 0.87, and MAE = 0.99); Traore et al. (2016), who assessed the ability of four machine learning models to predict ETo in Dallas, Texas, USA (r = 0.92 and MAE = 0.70); and Traore et al. (2017), who applied the gene expression programming algorithm to estimate daily ETo in the city of Gaoyou, located in Jiangsu Province, China (r = 0.86, RMSE = 0.99 mm, and MAE = 0.73 mm). However, a slight discrepancy can be noted between the reported results and those obtained with MLP-GA. Nevertheless, it is acknowledged that several factors, including the quality of meteorological data, data quantity, forecasting period, adopted methodologies, and normalization, among others, can directly impact the outcomes produced by machine learning models. Nevertheless, the evaluations conducted and the similarities between the indices reported in the literature and the results obtained by the MLP-GA suggest that the evolutionary process employed by the GA is as effective as conventional methods. This approach enables the exploration of a broad range of potential combinations, significantly increasing the likelihood of identifying an optimal configuration. Consequently, the MLP-GA demonstrated the ability to capture critical features for ETo estimation, surpassing other models in terms of performance and precision.

5. Conclusion

The results of the models in estimating ETo indicate that empirical methods perform inferiorly compared to the neural network models MLP, MLP-GRID, MLP-RD, and MLP-GA. The error indices also reveal that, while the MLP neural network can estimate ETo efficiently, its performance is smaller than that of the MLP-GRID, MLP-RD, and MLP-GA. This discrepancy may arise from the conventional approach, which limits the evaluation of hyperparameter combinations, in contrast to other methodologies that have extensively explored a broader range of combinations. When comparing the MLP-GRID, MLP-RD, and MLP-GA, differences in configurations, execution times, and results are evident. Despite yielding promising outcomes, MLP-GRID and MLP-RD require significantly longer execution times than MLP-GA, complicating their application to more complex problems. Another notable aspect is that the evolutionary process implemented by GA to automate the training of the neural network enables the model to adjust to the variations present in each region, demonstrating efficiency comparable to that of a human supervisor. Therefore, given its superior performance, it can be concluded that MLP-GA is a viable tool for estimating ETo in the evaluated regions.

Acknowledgments

This research was funded by the Research Support Foundation of the State of Rio de Janeiro (FAPERJ). The authors would also like to thank the Center for Weather Forecasting and Climate Studies of the National Institute for Space Research (CPTEC-INPE) for providing the meteorological data. The Federal Center of Technological Education of Rio de Janeiro (CEFET-ANGRA) for research support. The Federal Rural University of Rio de Janeiro (UFRRJ-SEROPéDICA) and the financial support provided by the Postgraduate Program in Civil Engineering at the Federal University of Rio de Janeiro (UFRJ - COPPE) and Fluminense Federal University (UFF).

References

• ABIOYE, E.A.; ABIDIN, M.S.Z.; AMAN, M.N.; MAHMUD, M.S.A.; BUYAMIN, S. A model predictive controller for precision irrigation using discrete lagurre networks. Computers and Electronics in Agriculture, v. 181, 105953, 2021. doi
  » https://doi.org/10.1016/j.compag.2020.105953
• ABDULLAH, S.S.; MALEK, M.A.; ABDULLAH, N.S.; KISI, O.; YAP, K.S. Extreme learning machines: A new approach for prediction of reference evapotranspiration. Journal of Hydrology, v. 527, p. 184-195, 2015. doi
  » https://doi.org/10.1016/j.jhydrol.2015.04.073
• ACHITE, M.; NASIRI, H.; KATIPOğLU, O.M.; ABDALLAH, M.; MOAZENZADEH, R.; et al. A coupled extreme gradient boosting-MPA approach for estimating daily reference evapotranspiration. Theoretical and Applied Climatology, v. 156, n. 113, p. 1-23, 2025. doi
  » https://doi.org/10.1007/s00704-024-05313-x
• ADNAN, R.M.; MOSTAFA, R.R.; ISLAM, A.R.M.T.; KISI, O.; KURIQI, A.; et al. Estimating reference evapotranspiration using hybrid adaptive fuzzy inferencing coupled with heuristic algorithms. Computers and Electronics in Agriculture, v. 191, 106541, 2021a. doi
  » https://doi.org/10.1016/j.compag.2021.106541
• ADNAN, R.M.; HEDDAM, S.; YASEEN, Z.M.; SHAHID, S.; KISI, O; et al. Prediction of potential evapotranspiration using temperature-based heuristic approaches. Sustainability, v. 13, n. 1, p. 297, 2021b. doi
  » https://doi.org/10.3390/su13010297
• AFZAAL, H.; FAROOQUE, A.A.; ABBAS, F.; ACHARYA, B.; ESAU, T. Computation of evapotranspiration with artificial intelligence for precision water resource management. Applied Sciences, v. 10, n. 5, p. 1621, 2020. doi
  » https://doi.org/10.3390/app10051621
• ALENCAR, L.P.; SEDIYAMA, G.C.; MANTOVANI, E.C. Estimativa da evapotranspiração de referência (ETo padrão FAO), para Minas Gerais, na ausência de alguns dados climáticos. Engenharia Agrícola, v. 35, n. 1, p. 39-50, 2015. doi
  » https://doi.org/10.1590/1809-4430-Eng.Agric.v35n1p39-50/2015
• ALLEN, R.G.; PEREIRA, L.; RAES, D.; SMITH, M. Crop Evapotranspiration - Guidelines for Computing Crop Water Requirements. FAO Irrigation and Drainage Paper No. 56 Rome: Food and Agriculture Organization of the United Nations, 1998.
• ALSAMAMRA, H. Estimation of global solar radiation from temperature extremes: A case study of Hebron City, Palestine. Journal of Environmental and Natural Resources, 2019. v. 8, n. 1, p. 1-5, doi
  » https://doi.org/10.11648/j.jenr.20190801.11
• ALY, M.S.; DARWISH, S.M.; ALY, A.A. High performance machine learning approach for reference evapotranspiration estimation. Stochastic Environmental Research and Risk Assessment, v. 38, n. 2, p. 689-713, 2024. doi
  » https://doi.org/10.1007/s00477-023-02594-y
• AMARASINGHE, U.A.; SMAKHTIN, V. Global Water Demand Projections: Past, Present and Future. IWMI Researche Report n. 156 Colombo: International Water Management Institute (IWMI), p. 34, 2014. doi
  » https://doi.org/10.5337/2014.212
• AMR, T. Hands-On Machine Learning with Scikit-Learn and Scientific Python Toolkits: A Practical Guide to Implementing Supervised and Unsupervised Machine Learning Algorithms in Python Birmingham: Packt Publishing Ltd, 2020.
• ASADUZZAMAN, M.; AHMED, S.U.; KHAN, F.E.; SHAHJAHAN, M.; MURASE, K. Making use of damped noisy gradient in training neural network. In: The 2010 International Joint Conference on Neural Networks (IJCNN), p. 1-5, 2010. doi
  » https://doi.org/10.1109/IJCNN.2010.5596284
• ASIM, K.M.; AWAIS, M.; MARTíNEZ-áLVAREZ, F.; IQBAL, T. Seismic activity prediction using computational intelligence techniques in northern Pakistan. Acta Geophysica, v. 65, p. 919-930, 2017. doi
  » https://doi.org/10.1007/s11600-017-0082-1
• AYOUB, M.A.; DEMIRAL, B.M. Application of resilient back-propagation neural networks for generating a universal pressure drop model in pipelines. University of Khartoum Engineering Journal (UOKEJ), v. 1, n. 2, p. 9-21, 2011.
• BEZERRA, J.R.C.; DIAS, J.M.; PEREIRA, J.R.; GUEDES, F.X. Consumo Hídrico do Algodoeiro BRS 200 Marrom. Circular Técnica 129-EMBRAPA Campina Grande: Embrapa Algodão, p. 11, 2009.
• BERGSTRA, J.; BENGIO, Y. Random search for hyper-parameter optimization. Journal of Machine Learning Research, v. 13, p. 281-305, 2012.
• BIDABADI, M.; BABAZADEH, H.; SHIRI, J.; SAREMI, A. Estimation reference crop evapotranspiration (ET0) using artificial intelligence model in an arid climate with external data. Applied Water Science, v. 14, n. 3, 2024. doi
  » https://doi.org/10.1007/s13201-023-02058-2
• BORGES, A.C.; MENDIONDO, E.M. Comparação entre equações empíricas para estimativa da evapotranspiração de referência na bacia do Rio Jacupiranga. Revista Brasileira de Engenharia Agrícola e Ambiental, v. 11, n. 3, p. 293-300, 2007. doi
  » https://doi.org/10.1590/S1415-43662007000300008
• BRAGA, A.P.; CARVALHO, A.C.P.L.F.; LUDEMIR, T.B. Redes Neurais Artificiais - Teoria e Aplicações Rio de Janeiro: LTC, 2012.
• CARDOSO, G.M.; JUSTINO, F. Simulação dos componentes da evapotranspiração sob condições climáticas atuais e de cenários climáticos futuros de aquecimento global com o uso de modelos de clima-vegetação. Revista Brasileira de Meteorologia, v. 29, n. 1, p. 85-95, 2014. doi
  » https://doi.org/10.1590/S0102-77862014000100009
• CHIA, M.Y.; HUANG, Y.F.; KOO, C.H. Swarm-based optimization as stochastic training strategy for estimation of reference evapotranspiration using extreme learning machine. Agricultural Water Management, v. 243, 106447, 2021. doi
  » https://doi.org/10.1016/j.agwat.2020.106447
• COHOON, J.P.; PARIS, W.D. Genetic placement. IEEE - Transactions on Computer Aided Design, v. 6, n. 6, p. 956-964, 1987. doi
  » https://doi.org/10.1109/TCAD.1987.1270337
• COLE, M.B.; AUGUSTIN, M.A.; ROBERTSON, M.J.; MANNERS, J.M. The science of food security. NPJ Science of Food, v. 2, n. 1, p. 14, 2018. doi
  » https://doi.org/10.1038/s41538-018-0021-9
• COMERT, M.M.; ADEM, K.; ERDOGAN, M. Comparative analysis of estimated solar radiation with different learning methods and empirical models. Atmósfera, v. 37, p. 273-284, 2023. doi
  » https://doi.org/10.20937/atm.53131
• COUTINHO, E.R.; MADEIRA, J.G.F.; SILVA, R.M.; OLIVEIRA, E.M.D.; DELGADO, A.R.S. Application of a computational hybrid model to estimate and fill gaps for meteorological time series. Revista Brasileira de Meteorologia, v. 38, e38220030, 2024. doi
  » https://doi.org/10.1590/0102-778638220030
• CZEKALSKI, D.; OBSTAWSKI, P.; BAKOń, T. Possibilities to estimate daily solar radiation on 2-axis tracking plane using a model based on temperature amplitude. Sustainability, v. 12, n. 23, p. 9909, 2020. doi
  » https://doi.org/10.3390/su12239909
• DANTAS, G.F.; OLIVEIRA, V.M.R.; DALRI, A.B.; PALARETTI, L.F.; SANTOS, M.G.; et al. Desempenho de métodos na estimativa de evapotranspiração de referência para o estado da Paraíba, Brasil. Irriga, v. 21, n. 3, p. 481-490, 2016. doi
  » https://doi.org/10.15809/irriga.2016v21n3p481-490
• DELGADO, B.; PAREDES, M.; MARTíNEZ, M. Software application for calculating solar radiation and equivalent evaporation in mobile devices. Agricultural Water Management, v. 151, p. 30-36, 2015. doi
  » https://doi.org/10.1016/j.agwat.2014.09.012
• FALAMARZI, Y.; PALIZDAN, N.; HUANG, Y.F.; LEE, T.S. Estimating evapotranspiration from temperature and wind speed data using artificial and wavelet neural networks (WNNs). Agricultural Water Management, v. 140, p. 26-36, 2014. doi
  » https://doi.org/10.1016/j.agwat.2014.03.014
• FAN, J.; MA, X.; WU, L.; ZHANG, F.; YU, X.; et al. Light gradient boosting machine: An efficient soft computing model for estimating daily reference evapotranspiration with local and external meteorological data. Agricultural Water Management, v. 225, 105758, 2019. doi
  » https://doi.org/10.1016/j.agwat.2019.105758
• FANAYA JúNIOR, E.D.; LOPES, A.S.; OLIVEIRA, G.Q.; JUNG, L.H. Métodos empíricos para estimativa da evapotranspiração de referência para a Aquidauana, MS. Irriga, v. 17, n. 4, p. 418-438, 2012. doi
  » https://doi.org/10.15809/irriga.2012v17n4p418
• FENG, Y.; JIA, Y.; CUI, N.; ZHAO, L.; LI, C.; et al. Calibration of Hargreaves Model for reference evapotranspiration estimation in Sichuan basin of southwest China. Agricultural Water Management, v. 181, p. 1-9, 2017. doi
  » https://doi.org/10.1016/j.agwat.2016.11.010
• FENG, Y.; CUI, N.; ZHAO, L.; HU, X.; GONG, D. Comparison of ELM, GANN, WNN and empirical models for estimating reference evapotranspiration in humid region of Southwest China. Journal of Hydrology, v. 536, p. 376-383, 2016. doi
  » https://doi.org/10.1016/j.jhydrol.2016.02.053
• FERNANDES, L.C.; PAIVA, C.M.; ROTUNNO FILHO, O.C. Evaluation of six empirical evapotranspiration equations - case study: Campos dos Goytacazes/RJ. Revista Brasileira de Meteorologia, v. 27, n. 3, p. 272-280, 2012. doi
  » https://doi.org/10.1590/S0102-77862012000300002
• FERREIRA, L.B.; CUNHA, F.F. Multi-step ahead forecasting of daily reference evapotranspiration using deep learning. Computers and Electronics in Agriculture, v. 178, 105728, 2020. doi
  » https://doi.org/10.1016/j.compag.2020.105728
• GAVILáN, P.; BERENGENA, J.; ALLEN, R.G. Measuring versus estimating net radiation and soil heat flux: Impact on Penman-Monteith reference ET estimates in semiarid regions. Agricultural Water Management, v. 89, n. 3, p. 275-286, 2007. doi
  » https://doi.org/10.1016/j.agwat.2007.01.014
• GéRON, A. Hands-On Machine Learning with Scikit-Learn, Keras, and Tensorflow: Concepts, Tools, and Techniques to Build Intelligent Systems Newton: O'Reilly Media, Inc., 2019.
• GHARSALLAH, O.; FACCHI, A.; GANDOLFI, C. Comparison of six evapotranspiration models for a surface irrigated maize agro-ecosystem in Northern Italy. Agricultural Water Management, v. 130, p. 119-130, 2013. doi
  » https://doi.org/10.1016/j.agwat.2013.08.009
• GULLI, A.; KAPOOR, A.; PAL, S. Deep Learning with TensorFlow 2 and Keras Regression, ConvNets, GANs, RNNs, NLP, and More with TensorFlow 2 and the Keras API Birmingham: Packt Publishing Ltd, 2019.
• GYAMFI, E.K.; ELSAYED, Z.; KROPCZYNSKI, J.; YAKUBU, M.A.; ELSAYED, N. Agricultural 4.0 leveraging on technological solutions: Study for smart farming sector. In: 2024 IEEE 3rd International Conference on Computing and Machine Intelligence (ICMI), p. 1-9, 2024. doi
  » https://doi.org/10.1109/ICMI60790.2024.10585910
• HAMON, W.R. Estimating potential evapotranspiration. Journal of Hydraulics Division, v. 87, n. 3, p. 107-120, 1961. doi
  » https://doi.org/10.1061/JYCEAJ.0000599
• HARGREAVES, G.H.; SAMANI, Z. Reference crop evapotranspiration from temperature. Journal of Applied Engineering in Agriculture, v. 1, n. 2, p. 96-99, 1985. doi
  » https://doi.org/10.13031/2013.26773
• HAYKIN, S. Redes Neurais Princípios e Pratica PortoAlegre: Bookmam, 2001.
• HUANG, G.; DONG, J.; WU, L.; LUO, J.; QIU, R.; et al. A new regional reference evapotranspiration model based on quantile approximation of meteorological variables. Agricultural Water Management, v. 308, 109299, 2025. doi
  » https://doi.org/10.1016/j.agwat.2025.109299
• HUYEN, C. Designing Machine Learning Systems: An Iterative Process for Production-Ready Applications Newton: O'Reilly Media, Inc., 2022.
• INSTITUTO BRASILEIRO DE GEOGRAFIA E ESTATíSTICA. Estimativas da População dos Municípios Brasileiros com Data de Referência em 1° de Julho de 2014 Available in https://www.ibge.gov.br/arquivo/noticias/pdf/analise_estimativas_2014.pdf, accessed in October 2024.
  » https://www.ibge.gov.br/arquivo/noticias/pdf/analise_estimativas_2014.pdf
• JENSEN, D.T.; HARGREAVES, G.H.; TEMESGEN, B.; ALLEN, R.G. Computation of ETo under nonideal conditions. Journal of Irrigation and Drainage Engineering, v. 123, n. 5, p. 394-399, 1997.
• KAYA, Y.Z.; ZELENAKOVA, M.; üNEş, F.; DEMIRCI, M.; HLAVATA, H.; et al. Estimation of daily evapotranspiration in Košice City (Slovakia) using several soft computing techniques. Theoretical and Applied Climatology, v. 144, p. 287-298, 2021. doi
  » https://doi.org/10.1007/s00704-021-03525-z
• KHOJA, I.; LATRECH, B.; LASRAM, A.; LADHARI, T.; M'SAHLI, F.; et al. Application of machine learning in agriculture the reference evapotranspiration model prediction. In: 2023 IEEE International Conference on Artificial Intelligence & Green Energy (ICAIGE), p. 1-6, 2023. doi
  » https://doi.org/10.1109/ICAIGE58321.2023.10346360
• KOPPENBRASIL. Classificação Climática de Köppen para os Municípios Brasileiros Available in https://koppenbrasil.github.io/, accessed in December 2024.
  » https://koppenbrasil.github.io/
• KOç, D.L.; PAKSOY, S.T. Daily reference evapotranspiration prediction using empirical and data-driven approaches: A case study of Adana plain. Journal of Agricultural Sciences, v. 31, n. 1, p. 207-229, 2025. doi
  » https://doi.org/10.15832/ankutbd.1481207
• KUMAR, M.; BANDYOPADHYAY, A.; RAGHUWANSHI, N.S.; SINGH, R. Comparative study of conventional and artificial neural network-based ETo estimation models. Irrigation Science, v. 26, p. 531-545, 2008. doi
  » https://doi.org/10.1007/s00271-008-0114-3
• KUMAR, V.; AZAMATHULLA, H.M.; SHARMA, K.V.; MEHTA, D.J.; MAHARAJ, K.T. The state of the art in deep learning applications, challenges, and future prospects: A comprehensive review of flood forecasting and management. Sustainability, v. 15, n. 13, 10543, 2023.
• LAI, C.L.; WANG, H.F. Application of soft computing techniques to forecast monthly electricity demand. In: 2015 International Conference on Industrial Engineering and Operations Management (IEOM), v. 1, p. 1-7, 2015. doi
  » https://doi.org/10.1109/IEOM.2015.7093922
• LANDERAS, G.; BARREDO, A.O.; LóPEZ, J.J. Comparison of artificial neural network models and empirical and semi-empirical equations for daily reference evapotranspiration estimation in the Basque Country (Northern Spain). Agricultural Water Management, v. 95, n. 5, p. 553-565, 2008. doi
  » https://doi.org/10.1016/j.agwat.2007.12.011
• LIMA, R.F.D.; APARECIDO, L.E.D.O.; TORSONI, G.B.; ROLIM, G.D.S. Climate change assessment in Brazil: Utilizing the Köppen-Geiger (1936) climate classification. Revista Brasileira de Meteorologia, v. 38, e38230001, 2024. doi
  » https://doi.org/10.1590/0102-77863810001
• LINACRE, E.T. A simple formula for estimating evaporation rates in various climates, using temperature data alone. Agricultural Meteorology, v. 18, n. 6, p. 409-424, 1977.
• LINDEN, R. Algoritmos Genéticos Rio de Janeiro: Ciência Moderna Ltda., 2012.
• LUO, Y.; TRAORE, S.; LYU, X.; WANG, W.; WANG, Y.; XIE, Y.; JIAO, X.; FIPPS, G. Medium range daily reference evapotranspiration forecasting by using ANN and public weather forecasts. Water Resources Management, v. 29, p. 3863-3876, 2015. doi
  » https://doi.org/10.1007/s11269-015-1033-8
• MANESH, S.S.; AHANI, H.; ZADEH, M.R. ANN-based mapping of monthly reference crop evapotranspiration by using altitude, latitude and longitude data in Fars province, Iran. Environment, Development and Sustainability, v. 16, n. 1, p. 103-122, 2014. doi
  » https://doi.org/10.1007/s10668-013-9465-x
• MATHWORKS. Gradient Descent Backpropagation Available in https://la.mathworks.com/help/deeplearning/ref/traingd.html?searchHighlight=traingd&s_tid=doc_srchtitle, accessed in February, 2017.
  » https://la.mathworks.com/help/deeplearning/ref/traingd.html?searchHighlight=traingd&s_tid=doc_srchtitle
• MATOS, R.M.; SILVA, P.F. Análise de métodos de estimativa da evapotranspiração de referência mensal para o município de Barbalha-CE. Agropecuária Científica no Semiárido, v. 12, n. 1, p. 10-21, 2016.
• MEHDIZADEH, S.; BEHMANESH, J.; KHALILI, K. Using MARS, SVM, GEP and empirical equations for estimation of monthly mean reference evapotranspiration. Computers and Electronics in Agriculture, v. 139, p. 103-114, 2017. doi
  » https://doi.org/10.1016/j.compag.2017.05.002
• MELLO, C.R.; SILVA, A.M. Hidrologia: Princípios e Aplicações em Sistemas Agrícolas Lavras: UFLA, 2013.
• MOAYEDI, H.; KHARI, M.; BAHIRAEI, M.; FOONG, L.K.; BUI, D.T. Spatial assessment of landslide risk using two novel integrations of neuro-fuzzy system and metaheuristic approaches; Ardabil Province, Iran. Geomatics, Natural Hazards and Risk, v. 11, n. 1, p. 230-258, 2020. doi
  » https://doi.org/10.1080/19475705.2020.1713234
• MUTIIBWA, D.; FLEISHER, D.H.; RESOP, J.P.; TIMLIN, D.; REDDY, V.R. Regional food production and land redistribution as adaptation to climate change in the U.S. Northeast Seaboard. Agricultural Water Management, v. 154, p. 54-70, 2018. doi
  » https://doi.org/10.1016/j.compag.2018.08.026
• PATIL, A.P.; DEKA, P.C. Performance evaluation of hybrid wavelet-ANN and wavelet-ANFIS models for estimating evapotranspiration in arid regions of India. Neural Computing and Applications, v. 28, p. 275-285, 2015. doi
  » https://doi.org/10.1007/s00521-015-2055-0
• PATIL, A.P.; DEKA, P.C. An extreme learning machine approach for modeling evapotranspiration using extrinsic inputs. Computers and Electronics in Agriculture, v. 121, p. 385-392, 2016. doi
  » https://doi.org/10.1016/j.compag.2016.01.016
• PéREZ-BLANCO, C.D.; HRAST-ESSENFELDER, A.; PERRY, C. Irrigation technology and water conservation: A review of the theory and evidence. Review of Environmental Economics and Policy, v. 14, n. 2, p. 216-239, 2020. doi
  » https://doi.org/10.1093/reep/reaa004
• PRIESTLEY, C.H.B.; TAYLOR, R.J. On the assessment of surface heat-flux and evaporation using large-scale parameters. Monthly Weather Review, v. 100, n. 2, p. 81-92, 1972. doi
  » https://doi.org/10.1175/1520-0493(1972)100<0081:OTAOSH>2.3.CO;2
• QUEJ, V.H.; ALMOROX, J.; IBRAKHIMOV, M.; SAITO, L. Empirical models for estimating daily global solar radiation in Yucatán Peninsula, Mexico. Energy Conversion and Management, v. 110, p. 448-456, 2016. doi
  » https://doi.org/10.1016/j.enconman.2015.12.050
• RASCHKA, S.; LIU, Y.H.; MIRJALILI, V. Machine Learning with PyTorch and Scikit-Learn: Develop Machine Learning and Deep Learning Models with Python Birmingham: Packt Publishing Ltd., 2022.
• RIVERO, M.; OROZCO, S.; SELLSCHOPP, F.S.; LOERA-PALOMO, R. A new methodology to extend the validity of the Hargreaves-Samani model to estimate global solar radiation in different climates: Case study Mexico. Renewable Energy, v. 114, p. 1340-1352, 2017. doi
  » https://doi.org/10.1016/j.renene.2017.08.003
• SAATH, K.C.D.O.; FACHINELLO, A.L. Crescimento da demanda mundial de alimentos e restrições do fator terra no Brasil. Revista de Economia e Sociologia Rural, v. 56, n. 2., p. 195-212, 2018. doi
  » https://doi.org/10.1590/1234-56781806-94790560201
• SABA, T.; REHMAN, A.; ALGHAMDI, J.S. Weather forecasting based on hybrid neural model. Applied Water Science, v. 7, p. 3869-3874, 2017.
• SAGGI, M.K.; JAIN, S. Reference evapotranspiration estimation and modeling of the Punjab Northern India using deep learning. Computers and Electronics in Agriculture, v. 156, p. 387-398, 2019. doi
  » https://doi.org/10.1016/j.compag.2018.11.031
• SANTOS, P.A.B.D.; SCHWERZ, F.; CARVALHO, L.G.D.; BAPTISTA, V.B.D.S.; MARIN, D.B.; et al. Machine learning and conventional methods for reference evapotranspiration estimation using limited-climatic-data scenarios. Agronomy, v. 13, n. 9, 2366, 2023. doi
  » https://doi.org/10.3390/agronomy13092366
• SATTARI, M.T.; APAYDIN, H.; BAND, S.S.; MOSAVI, A.; PRASAD, R. Comparative analysis of kernel-based versus ANN and deep learning methods in monthly reference evapotranspiration estimation. Hydrology and Earth System Sciences, v. 25, n. 2, p. 603-618, 2021. doi
  » https://doi.org/10.5194/hess-25-603-2021
• SENTELHAS, P.C.; GILLESPIE, T.J.; SANTOS, E.A. Evaluation of FAO Penman-Monteith and alternative methods for estimating reference evapotranspiration with missing data in Southern Ontario, Canada. Agricultural Water Management, v. 97, n. 5, p. 635-644, 2010. doi
  » https://doi.org/10.1016/j.agwat.2009.12.001
• SILVA JUNIOR, J.C.; MEDEIROS, V.; GARROZI, C.; MONTENEGRO, A.; GONCALVES, G.E. Random forest techniques for spatial interpolation of evapotranspiration data from Brazilian's Northeast. Computers and Electronics in Agriculture, v. 166, 105017, 2019. doi
  » https://doi.org/10.1016/j.compag.2019.105017
• SILVA, V.P.R.; BELO FILHO, A.F.; SILVA, B.B.; CAMPOS, J.H.B.C. Desenvolvimento de um sistema de estimativa da evapotranspiração de referência. Revista Brasileira de Engenharia Agrícola e Ambiental, v. 9, n. 4, p. 547-553, 2005. doi
  » https://doi.org/10.1590/S1415-43662005000400017
• SPAROVEK, G.; LEITE, C.V.; BARRETTO, A.G.O.P.; MAULE, R.F.; DOURADO NETO, D. Análise territorial e potencial da agricultura irrigada no Brasil. Irrigação & Tecnologia Moderna, n. 106, p. 56-62, 2015.
• SPONTONI, T.A.; VENTURA, T.M.; PALáCIOS, R.S.; CURADO, L.F.; FERNANDES, W.A.; et al. Evaluation and modelling of reference evapotranspiration using different machine learning techniques for a Brazilian tropical savanna. Agronomy, v. 13, n. 8, 2056, 2023. doi
  » https://doi.org/10.3390/agronomy13082056
• TAO, H.; DIOP, L.; BODIAN, A.; DJAMAN, K.; NDIAYE, P.M.; et al. Reference evapotranspiration prediction using hybridized fuzzy model with firefly algorithm: Regional case study in Burkina Faso. Agricultural Water Management, v. 208, p. 140-151, 2018. doi
  » https://doi.org/10.1016/j.agwat.2018.06.018
• TAUSIF, M.; IQBAL, M.W.; BASHIR, R.N.; ALGHOFAILY, B.; ELYASSIH, A.; et al. Federated learning based reference evapotranspiration estimation for distributed crop fields. PloS One, v. 20, n. 2, e0314921, 2025. doi
  » https://doi.org/10.1371/journal.pone.0314921
• TIKHAMARINE, Y.; MALIK, A.; SOUAG-GAMANE, D.; KISI, O. Artificial intelligence models versus empirical equations for modeling monthly reference evapotranspiration. Environmental Science and Pollution Research, v. 27, p. 30001-30019, 2020. doi
  » https://doi.org/10.1007/s11356-020-08792-3
• TORRES, A.; WALKER, W.R.; MCKEE, M. Forecasting daily potential evapotranspiration using machine learning and limited climatic data. Agricultural Water Management, v. 98, n. 4, p. 553-562, 2011. doi
  » https://doi.org/10.1016/j.agwat.2010.10.012
• TOP, J.; JANSSEN, S.; BOOGAARD, H.; KNAPEN, R.; SIMSEK-SENEL, G. Cultivating FAIR principles for agri-food data. Computers and Electronics in Agriculture, v. 196, 106909, 2022. doi
  » https://doi.org/10.1016/j.compag.2022.106909
• TRAORE, S.; LUO, Y.; FIPPS, G. Deployment of artificial neural network for short-term forecasting of evapotranspiration using public weather forecast restricted messages. Agricultural Water Management, v. 163, p. 363-379, 2016. doi
  » https://doi.org/10.1016/j.agwat.2015.10.009
• TRAORE, S.; LUO, Y.; FIPPS, G. Gene-expression programming for short-term forecasting of daily reference evapotranspiration using public weather forecast information. Water Resource Management, v. 31, p. 4891-4908, 2017. doi
  » https://doi.org/10.1007/s11269-017-1784-5
• TREDER, W.; KLAMKOWSKI, K.; WóJCIK, K.; TRYNGIEL-GAć, A. Evapotranspiration estimation using machine learning methods. Journal of Horticultural Research, v. 31, n. 2, p. 35-44, 2023. doi
  » https://doi.org/10.2478/johr-2023-0033
• VáSQUEZ, C.; CéLLERI, R.; CóRDOVA, M.; CARRILLO-ROJAS, G. Improving reference evapotranspiration (ETo) calculation under limited data conditions in the high Tropical Andes. Agricultural Water Management, v. 262, 107439, 2022. doi
  » https://doi.org/10.1016/j.agwat.2021.107439
• VENTURA, T.M.; MARTINS, C.A.; FIGUEIREDO, J.M.; OLIVEIRA, A.G.; MONTANHER, J.R.P. MANNGA: A robust method for gap filling meteorological data. Revista Brasileira de Meteorologia, v. 34, n. 2, p. 315-323, 2019. doi
  » https://doi.org/10.1590/0102-77863340035
• WANKHADE, K.; RANE, D.; THOOL, R. A new feature selection algorithm for stream data classification. In: 2013 International Conference on Advances in Computing, Communications and Informatics (ICACCI). IEEE, p. 1843-1848, 2013. doi
  » https://doi.org/10.1109/ICACCI.2013.6637462
• WASEEM, M.H.; NADEEM, A.; SAJJAD, M.; KHAN, I.R.; AZIZ, W.; et al. Reinforcing artificial neural networks through traditional machine learning algorithms for robust classification of cancer. Computers, Materials & Continua, v. 75, n. 2, p. 4293-4315, 2023. doi
  » https://doi.org/10.32604/cmc.2023.036710
• WU, M.; FENG, Q.; WEN, X.; DEO, R.C.; YIN, Z.; et al. Random forest predictive model with uncertainty analysis capability for estimation of evapotranspiration in an arid oasis region. Hydrology Research, v. 51, n. 4, p. 648-665, 2020. doi
  » https://doi.org/10.2166/nh.2020.012
• WU, T.; ZHANG, W.; JIAO, X.; GUO, W.; HAMOUD, Y.A. Comparison of five boosting-based models for estimating daily reference evapotranspiration with limited meteorological variables. PLoS One, v. 15, n. 6, e0235324, 2020. doi
  » https://doi.org/10.1371/journal.pone.0235324
• XIAN, Z.; WU, H.; SIQING, S.; SHAOQUAN, Z. Application of genetic algorithm-neural network for the correction of bad data in power system. In: 2011 International Conference on Electronics, Communications and Control (ICECC). IEEE, p. 1894-1897, 2011. doi
  » https://doi.org/10.1109/ICECC.2011.6067574
• XUE, S.; YANG, M.; LI, C.; NIE, J. Meteorological prediction using support vector regression with genetic algorithms. In: 2009 First International Conference on Information Science and Engineering, p. 4931-4935, 2009. doi
  » https://doi.org/10.1109/ICISE.2009.735
• YASSIN, M.A.; ALAZBA, A.A.; MATTAR, M. Artificial neural networks versus gene expression programming for estimating reference evapotranspiration in arid climate. Agricultural Water Management, v. 163, p. 110-124, 2016. doi
  » https://doi.org/10.1016/j.agwat.2015.09.009
• RAZIEI, T.; PEREIRA, L.S. Estimation of ETo with Hargreaves-Samani and FAO-PM temperature methods for a wide range of climates in Iran. Agricultural Water Management, v. 121, p. 1-18, 2013. doi
  » https://doi.org/10.1016/j.agwat.2012.12.019
• ZHAO, L.; ZHAO, X.; ZHOU, H.; WANG, X.; XING, X. Prediction model for daily reference crop evapotranspiration based on hybrid algorithm and principal components analysis in southwest China. Computers and Electronics in Agriculture, v. 190, 106424, 2021. doi
  » https://doi.org/10.1016/j.compag.2021.106424

Publication Dates

• Publication in this collection
  07 Apr 2025
• Date of issue
  2025

History

• Received
  23 Feb 2024
• Accepted
  19 Feb 2025