Comparative assessment of deep belief network and hybrid adaptive neuro-fuzzy inference system model based on a meta-heuristic optimization algorithm for precise predictions of the potential evapotranspiration

Accurately predicting potential evapotranspiration (PET) is crucial in water resource management, agricultural planning, and climate change studies. This research aims to investigate the performance of two machine learning methods, the adaptive network-based fuzzy inference system (ANFIS) and the deep belief network (DBN), in forecasting PET, as well as to explore the potential of hybridizing the ANFIS approach with the Snake Optimizer (ANFIS-SO) algorithm. The study utilized a comprehensive dataset spanning the period from 1983 to 2020. The ANFIS, ANFIS-SO, and DBN models were developed, and their performances were evaluated using statistical metrics, including R², \({R}_{adj}^2\) $R_{\mathit{adj}}^2$ , NSE, WI, STD, and RMSE. The results showcase the exceptional performance of the DBN model, which achieved R² and \({R}_{adj}^2\) $R_{\mathit{adj}}^2$ values of 0.99 and NSE and WI scores of 0.99 across the nine stations analyzed. In contrast, the standard ANFIS method exhibited relatively weaker performance, with R² and \({R}_{adj}^2\) $R_{\mathit{adj}}^2$ values ranging from 0.52 to 0.88. However, the ANFIS-SO approach demonstrated a substantial improvement, with R² and \({R}_{adj}^2\) $R_{\mathit{adj}}^2$ values ranging from 0.94 to 0.99, suggesting the value of optimization techniques in enhancing the model’s capabilities. The Taylor diagram and violin plots with box plots further corroborated the comparative analysis, highlighting the DBN model’s superior ability to closely match the observed standard deviation and correlation and its consistent and low-variance predictions. The ANFIS-SO method also exhibited enhanced performance in these visual representations, with an RMSE of 0.86, compared to 0.95 for the standard ANFIS. The insights gained from this study can inform the selection of the most appropriate modeling technique, whether it be the high-precision DBN, the flexible ANFIS, or the optimized ANFIS-SO approach, based on the specific requirements and constraints of the application.