  Green's function plays a significant role in both theoretical analysis and
numerical computing of partial differential equations (PDEs). However, in most
cases, Green's function is difficult to compute. The troubles arise in the
following three folds. Firstly, compared with the original PDE, the dimension
of Green's function is doubled, making it impossible to be handled by
traditional mesh-based methods. Secondly, Green's function usually contains
singularities which increase the difficulty to get a good approximation.
Lastly, the computational domain may be very complex or even unbounded. To
override these problems, we leverage the fundamental solution, boundary
integral method and neural networks to develop a new method for computing
Green's function with high accuracy in this paper. We focus on Green's function
of Poisson and Helmholtz equations in bounded domains, unbounded domains. We
also consider Poisson equation and Helmholtz domains with interfaces. Extensive
numerical experiments illustrate the efficiency and the accuracy of our method
for solving Green's function. In addition, we also use the Green's function
calculated by our method to solve a class of PDE, and also obtain
high-precision solutions, which shows the good generalization ability of our
method on solving PDEs.
