Physics-informed neural networks for enhanced reference evapotranspiration estimation in Morocco: Balancing semi-physical models and deep learning

Reference evapotranspiration (ETo) is essential for agricultural water management, crop productivity, and irrigation systems. The Penman-Monteith (PM) equation is the standard method for estimating ETo, but its data-intensive nature makes it impractical, especially in situations where the cost of full standardized weather station is prohibitive, maintenance is inadequate, or data quality and continuity are compromised. To overcome those limitations, various semi-physical (SP) and empirical models with limited weather parameters were developed. In this context, artificial intelligence methods for ETo estimation are gaining more attention, balancing simplicity, minimal data requirements, and high accuracy. However, their data-driven nature raises concerns regarding explainability, trustworthiness, adherence to bio-physical laws, and reliability in operational settings. To address this issue, this paper, inspired by the emerging field of Physics-Informed Neural Networks (PINNs), evaluates the integration of SP models into the loss function during the learning process. The new residual loss combines two losses -the data-driven loss and the loss from SP- through a θ parameter, allowing for a convex combination. In-situ agrometeorological data were collected at four automatic weather stations in Tensift Watershed in Morocco, including air temperature (Ta), solar radiation (Rs), relative humidity (RH), and wind speed (Ws). The study integrates Priestley-Taylor (PT), Makkink (MK), Hargreaves-Samani (HS), and Abtew (AB), under four scenarios of data availability levels: (1) Ta, Rs and RH; (2) Ta and Rs; (3) only Ta; and (4) only Rs. The investigation begins with quality-controlling the data and studying the driving factors of ETo. Next, the SP models were calibrated using the CMA-ES optimization algorithm. The proposed PINN was trained and evaluated, first, for the equal contribution scenario (θ = 0.5) and then for θ in the interval [0, 1] with a step of 0.2, thus analyzing the impact of θ on the PINN performance. For the equal contribution, the results showed that the integration had improved the PINN performance in all scenarios in terms of the RMSE and R², surpassing the fully data-driven model (θ = 0) and the baseline model (θ = 1). Additionally, for all θ within the interval [0.2, 0.8], the PINN required less training to reach optimal values. Finally, the optimal θ values were determined for each scenario using CMA-ES and were 0.258, 0.771, 0.7226 and 0.169 for PT, MK, HS and AB, respectively. While PINNs demonstrated a promising approach for accurate ETo estimation and consequently improved water resource management, the study also represents a step towards implementing controlled, trustworthy, and physics-informed AI in environmental science.