  In this paper, we present a deep surrogate model for learning the Green's
function associated with the reaction-diffusion operator in rectangular domain.
The U-Net architecture is utilized to effectively capture the mapping from
source to solution of the target partial differential equations (PDEs). To
enable efficient training of the model without relying on labeled data, we
propose a novel loss function that draws inspiration from traditional numerical
methods used for solving PDEs. Furthermore, a hard encoding mechanism is
employed to ensure that the predicted Green's function is perfectly matched
with the boundary conditions. Based on the learned Green's function from the
trained deep surrogate model, a fast solver is developed to solve the
corresponding PDEs with different sources and boundary conditions. Various
numerical examples are also provided to demonstrate the effectiveness of the
proposed model.
