ABC Fractional Water Supply Model With Deep Neural Network Model Approximation

A substantial body of research has examined water supply systems driven by interdependent natural sources, typically modeled through systems of differential equations. However, limited attention has been given to the fundamental mathematical properties of such systems when applied to water supply forecasting. In this manuscript, we investigate the interaction among the three principal water sources, surface water, groundwater, and recharge water, using a mass-balance framework formulated as a fractional water supply forecasting model based on the Atangana–Baleanu derivative in the Caputo sense. The resulting system consists of three coupled, nonlinear, and nonautonomous fractional differential equations. Using fixed-point theory, we establish existence and uniqueness results for the proposed system, and we analyze its Ulam stability through nonlinear techniques. Approximate solutions are obtained via the fractional-order Adams–Bashforth iterative method. Numerical simulations are performed for several fractional orders and under multiple control strategies. All compartments exhibit convergence and long-term stability, with lower fractional orders reaching steady states more rapidly. In addition, it should be noted that no independent held-out real-world dataset is used for deep neural network (DNN) evaluation; the reported DNN performance measures approximation accuracy with respect to solver-generated numerical data. The results demonstrate that the proposed fractional framework provides improved flexibility in capturing system dynamics compared to classical models. Future work will focus on parameter estimation using real-world data and extending the model to more complex and multiscale systems.