Efficient and accurate evaluation of free-surface Green's function is the key to hydrodynamic problems solved by boundary element method (BEM). However, so far, there is still no unified numerical method that can accurately approximate all kinds of free-surface Green's functions. In theory, machine learning can be used to approximate any function with high accuracy. In the present study, neural networks are used for the numerical approximation of pulsating source Green's function, and the corresponding optimization algorithms for gradient descent are adopted. Regularization is used to prevent overfitting. Double-precision numerical results obtained by Romberg quadrature are used as training set and validation set. To improve the accuracy of present numerical approximation, the calculation domain of both Green's function and its gradient are divided into 4 zones, and various network structures are adopted in each zone. Finally, a machine model, called ZeroGF, is obtained by machine learning that can predict Green's function and its derivatives. The numerical results show that ZeroGF owns at least 4 digits of accuracy in above 99% area of all zones. BEM program incorporating with ZeroGF is validated in the hydrodynamic calculation of the hemisphere, Wigley III and the Barge. Good accuracy and reliability of ZeroGF is shown.