  Partial differential equations are often used to model various physical
phenomena, such as heat diffusion, wave propagation, fluid dynamics,
elasticity, electrodynamics and image processing, and many analytic approaches
or traditional numerical methods have been developed and widely used for their
solutions. Inspired by rapidly growing impact of deep learning on scientific
and engineering research, in this paper we propose a novel neural network,
GF-Net, for learning the Green's functions of linear reaction-diffusion
equations in an unsupervised fashion. The proposed method overcomes the
challenges for finding the Green's functions of the equations on arbitrary
domains by utilizing physics-informed approach and the symmetry of the Green's
function. As a consequence, it particularly leads to an efficient way for
solving the target equations under different boundary conditions and sources.
We also demonstrate the effectiveness of the proposed approach by experiments
in square, annular and L-shape domains.
