A Numerical Simulation Study of Complex Multi-Source Groundwater Based on PKAN

Groundwater flow problems involve complex nonlinear and spatiotemporal characteristics, where traditional numerical methods (e.g., finite element, finite difference) often encounter challenges such as low computational efficiency and insufficient accuracy when dealing with complex boundary conditions and heterogeneous media. To address these issues, this study proposes a novel physics-informed Kolmogorov–Arnold network (PKAN) framework that combines the unique variable decomposition mechanism of KAN networks with physical constraints. The framework introduces three key innovations: (1) implementing KAN network’s univariate function decomposition to enhance the network’s ability to express nonlinear features; (2) designing a pre-training network mechanism to effectively handle complex boundary conditions; and (3) innovatively incorporating a distance function to achieve natural transition from boundary to interior solutions. The results demonstrate that in one-dimensional heterogeneous medium transient simulation, PKAN achieves superior prediction accuracy (R2 = 0.9966, RMSE = 0.0313) compared to traditional PINN (R2 = −0.7194, RMSE = 0.7001). In two-dimensional multi-well pumping system simulations, PKAN (R2 = 0.917, RMSE = 0.077) similarly exhibits exceptional performance (PINN: R2 = −0.3043, RMSE = 0.3067). Notably, in handling local strong gradient problems, PKAN accurately captures cone of depression characteristics and precisely reproduces inter-well interference effects, with maximum error only one-fourth that of traditional PINN. Sensitivity analysis reveals that a configuration of 50 × 50 uniform sampling points combined with four hidden layers and 64 neurons per layer achieves optimal balance between computational efficiency and simulation accuracy. These findings demonstrate PKAN’s breakthrough in groundwater numerical simulation, offering a novel approach for the efficient solution of complex hydrogeological problems.