In marine hydrodynamics, the core of the boundary element method (BEM) lies in the numerical calculation of the free-surface Green’s function. With the rise of artificial intelligence, using neural networks to fit Green’s function has become a new trend, yet most existing studies are confined to fitting Green’s function in infinite water depth. In this paper, a neural network fitting method for a finite-depth Green’s function is proposed. The classical Multilayer Perceptron (MLP) network and the emerging Kolmogorov–Arnold Network (KAN) are employed to conduct global and partition-based fitting experiments. Experiments indicate that the partition-based KAN fitting model achieves higher fitting accuracy, with most regions reaching 4D fitting precision. For large-scale data input, the average time for the model to calculate a single Green’s function value is 0.0868 microseconds, which is significantly faster than the 0.1120 s required by the traditional numerical integration method. These results demonstrate that the KAN can serve as an accurate and efficient model for finite-depth Green’s functions. The proposed KAN-based fitting method not only reduces the computational cost of numerical evaluation of Green’s functions but also maintains high prediction precision, providing an alternative approach to accelerate BEM calculations for floating body hydrodynamic analysis.