Reference evapotranspiration (ET₀) is a multifaceted hydrological variable influenced by numerous climatic factors that govern water and energy balances [1]. It plays a pivotal role in hydro-climatological processes and directly impacts agricultural production [2]. The importance of evapotranspiration extends across disciplines such as ecology, hydrology, and meteorology [3], serving as a key component in understanding the hydrological cycle and supporting applications including irrigation requirement estimation, irrigation scheduling, water resource management, and assessment of climate change effects [4]-[6]. Various methods exist to measure or estimate evapotranspiration and its components across scales—from plant-level to atmospheric boundary-layer analyses reflecting its multidisciplinary nature. These include direct techniques (e.g., lysimeters) and indirect approaches relying on meteorological data [7]. Direct measurements provide high accuracy but are costly, time-intensive, and limited in spatial coverage for regional applications [8]. Indirect empirical methods based on comprehensive climate data offer reliable, cost-effective alternatives for long-term and large-scale estimation [1][6].

The FAO-56 Penman-Monteith (FAO-56 PM) method stands as the established benchmark for estimating reference ET₀ across diverse climates, widely validated and recommended by international bodies [9][10]. As a physically based approach, it requires extensive climatic inputs (e.g., temperature, humidity, wind speed, radiation), which can pose challenges in data-scarce or heterogeneous environments [11]. In such contexts, machine learning (ML) models—particularly the Adaptive Neuro-Fuzzy Inference System (ANFIS)—provide robust alternatives by effectively capturing nonlinear relationships among variables [12].

This study introduces a novel streamlined neural network (NN) approximation inspired by ANFIS to estimate daily reference ET₀ in the heterogeneous semi-arid region of Khenchela province, northeastern Algeria. The target variable for model training, validation, and evaluation is the reference ET₀ computed using the FAO-56 Penman-Monteith method, as directly provided by the Open-Meteo Historical Weather API (variable: et₀ (fao_evapotranspiration)). This value is derived internally from reanalysis-based meteorological inputs (temperature, relative humidity, wind speed, solar/terrestrial radiation) following the standard FAO-56 equations, assuming unlimited soil water availability for a hypothetical well-watered grass reference crop. Unlike conventional ANFIS architectures that incorporate explicit fuzzy rule layers, the proposed NN employs a tanh-activated hidden layer to implicitly approximate fuzzy-like relationships, reducing computational complexity while maintaining accuracy. This design enhances suitability for data-limited regions and builds upon prior ML-based ET₀ studies [2][13] by offering improved scalability. The methodology is intended to be generalizable to other semi-arid areas, addressing gaps in localized, efficient ET₀ modeling. The NN is evaluated against the FAO-56 PM values from Open-Meteo to assess its effectiveness as a computationally efficient surrogate in heterogeneous terrains.

The neural network approximation inspired by the ANFIS model displayed excellent predictive accuracy for estimating ET in systems with various climatic conditions across the 16 stations (Table 2, Table3, and Table 4), with small variability from local climatological conditions. Three stations (Babar 2, Babar 3, and Khirane) exhibited strong estimation accuracies, as indicated by their RMSE (<0.32), R² (>0.99), and RSR (<0.11) values. Willmott’s index at each station was between 0.92 and 0.93, indicating an excellent agreement between the observed and predicted ET. These results demonstrate the strength of the model’s ability to estimate the ET nonlinearity in these systems. Eight stations (Baghai, Bouhmama, Chechar, Djellal, Khenchela, Remila, Tamza, and Taouzient) performed moderately well, with RMSE values ranging from 0.24 - 0.29 and R² (>0.98). These results resulted in RSR values of approximately 0.12 - 0.15, with Willmott’s index ranging from 0.88 to 0.91, thereby confirming our predictions. At some of these stations, the slightly higher RMSE may have been influenced by moderately greater climatic variability or localized data errors. Three stations (Babar 1, El Hamma, and Kais) exhibited slightly higher RMSE values, up to 0.32, with strong R² values (>0.97). Willmott’s index ranged from 0.84 to 0.89, indicating moderate to good agreement between the model outputs and the observed ET estimations. The Zaoui station showed reliable performance, with RMSE, MAE, and R² (>0.94) still within the acceptable range. The reliability of the model was also demonstrated, where ANFIS estimation with local climatic variation closely predicted ET using the FAO Penman-Monteith equation. The neural network approximation accurately predicted ET at all stations. The ability of the model to account for complex nonlinear relationships was evident at stations with lower RMSE and higher R² values (Babar 2, Babar 3, and Khirane). The RMSE variability across stations appears to be linked to localized factors or data variability within the same station. All stations exhibited low RSR values and high Willmott’s index values, confirming the power and reliability of the model for estimating ET under different meteorological conditions.

The maps (Figure 4(a)-(e)) show the spatial patterns of the predicted mean annual ET in the Khenchela region of eastern Algeria. Maps were obtained by interpolating the station observations (Figure 1). From 2000 to 2005, ET demonstrated a downward trend, with a cumulative loss of ~8 - 15% across stations, largely in the northern and central areas. The regional mean ET fell from 143.3 mm yr⁻¹ (2000-2001) to 130.8 mm yr⁻¹ (2003-2004), with a small rebound to 135.8 mm yr⁻¹ in 2005. This is represented by contour contraction: high values in red (>150 mm yr⁻¹) shrink in areas around Khirane and Tebessa, and low values in blue expand to the north of the study area. Linear regression showed statistically significant negative trends at the northern/central stations (Table 2), with slopes of −3.0 mm yr⁻¹ decade⁻¹ at Kais and Khirane (p < 0.05, R² = 0.66). These trends may reflect interannual variability in precipitation or the greenness of vegetation, which is likely linked to episodic drought events (e.g., reduced winter precipitation). The southern stations had weaker, non-significant trends (slopes of −1.7 to −2.1 mm yr⁻¹ decade⁻¹; p > 0.10), possibly indicating more stable control against short-term fluctuations due to more dominant evaporative control. The increase in 2005 (~5 - 10 mm yr⁻¹ increase) coincided with the southern contour expansion, possibly indicating a temporary wet anomaly in the region. These patterns correlate with elevation and soil moisture, with semi-arid regions showing higher ET than arid/hyper-arid zones.

Table 2. Performance metrics for the 16 stations (Training data from 2000 to 2017).

Table 3.Performance metrics for the 16 stations (Validation data from 2018 to 2021).

Table 4.Performance metrics for the 16 stations (Testing data from 2022 to 2024).

Figure 4. (a) Neural network approximation inspired by ANFIS-based mapping of mean annual evapotranspiration for 2000-2005; (b) Neural network approximation inspired by ANFIS-based mapping of mean annual evapotranspiration for 2006-2011; (c) Neural network approximation inspired by ANFIS-based mapping of mean annual evapotranspiration for 2012-2017; (d) Neural network approximation inspired by ANFIS-based mapping of mean annual evapotranspiration for 2018-2023; (e) Neural network approximation inspired by ANFIS-based mapping of mean annual evapotranspiration for 2024.

Sixteen training stations were used: Babar 1 (1), Babar 2 (2), Babar 3 (3), Baghai (4), Bouhmama (5), Chechar (6), Djellal (7), El Hamma (8), Kais (9), Khenchela (10), Khirane (11), M’sara (12), Remila (13), Tamza (14), Taouzient (15), and Zaoui (16).

This study effectively established the usefulness of a neural network inspired by the adaptive neuro-fuzzy inference system (ANFIS) integrated with a Geographic Information System (GIS) to predict and spatially map ET in the heterogeneous climatic context of the Khenchela region in Algeria. Using historical climate data from 16 sites from 2000 to 2024, the neural network model performed well (R² > 0.94, RMSE ranged from 0.23 to 0.50). Specific sites (Babar (2), Babar (3), and Khirane) were accurate (R² > 0.99, RMSE < 0.27), demonstrating the ability of the approximation method to capture localized climatic conditions. In comparison, station Babar (1) had higher predictive errors, signaling ET on spacing, and topographic, arid, and other environmental variations. The neural network model also demonstrated strong generalizability with low root sum errors (RSR < 0.15), close to one Nash-Sutcliffe efficiency (NSE), and Willmott’s index between 0.84 and 0.92, highlighting its reliability over other methods, such as the FAO-56 Penman-Monteith ET₀ values from Open-Meteo (R² > 0.94, RMSE 0.23 - 0.50 mm/day), highlighting its reliability as an efficient approximation in areas with heterogeneous climates or limited computational resources. Spatial mapping using GIS showed that ET spatial variability was related to elevation, rainfall, and soil characteristics, with semi-arid regions showing higher ET values than arid and hyper-arid regions. These results demonstrate that neural network approximation provides a good degree of flexibility.

I sincerely thank Nabil Mega for his excellent work in preparing the GIS visualizations and spatial interpolation maps (Figure 1and Figure 4(a)-(e)). I am grateful to Abdelmonen Miloudi for carrying out the formal analysis, model validation, and compilation of the performance metrics presented in Tables 2-4. I also express my deep appreciation to António Canatário Duarte and Abderahamane Khechekhouche for their thorough critical reviews, valuable suggestions, and insightful contributions to strengthening the manuscript.