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This paper introduces a novel approach combining physics-informed neural networks (PINN) with the generalized finite difference method (GFDM) to solve groundwater flow equations. In recent years, with advancements in computational performance and algorithm improvements, artificial intelligence has rapidly developed and been applied to many areas. One neural network architecture that incorporates physical meaning by integrating governing equations, known as the PINN, has gained widespread attention. By utilizing neural networks to approximate functional values and employing automatic differentiation for derivative computation, while incorporating the residual of governing equations (partial differential equations, PDE), boundary conditions, initial conditions, and practical measurement data into the loss function for optimization, the neural network can be trained to solve boundary value problems (BVPs) and initial boundary value problems (IBVPs). However, automatic differentiation is computationally expensive. To address this issue, the GFDM is integrated into the PINN framework to replace automatic differentiation and accelerate the training procedure of the original PINN. The GFDM, which adopts Taylor series expansion and the moving least square method, approximates partial derivatives as a linear accumulation of functional values and weighting coefficients at each node and nearby nodes within the computational domain. Moreover, by employing a space-time (ST) coupling concept, known as ST-GFDM, the time axis can be treated as an additional spatial dimension, facilitating the solution of time-dependent problems more efficiently. This study demonstrates the feasibility and enhanced computational performance of the PINN with ST-GFDM approach for solving BVPs and IBVPs in groundwater flow scenarios. Several numerical examples validate the method’s effectiveness, confirming its accuracy and superior computational performance compared to the original PINN.
