  In this study, we address the challenge of obtaining a Green's function
operator for linear partial differential equations (PDEs). The Green's function
is well-sought after due to its ability to directly map inputs to solutions,
bypassing the need for common numerical methods such as finite difference and
finite elements methods. However, obtaining an explicit form of the Green's
function kernel for most PDEs has been a challenge due to the Dirac delta
function singularity present. To address this issue, we propose the Deep
Generalized Green's Function (DGGF) as an alternative, which can be solved for
in an efficient and accurate manner using neural network models. The DGGF
provides a more efficient and precise approach to solving linear PDEs while
inheriting the reusability of the Green's function, and possessing additional
desirable properties such as mesh-free operation and a small memory footprint.
The DGGF is compared against a variety of state-of-the-art (SOTA) PDE solvers,
including direct methods, namely physics-informed neural networks (PINNs),
Green's function approaches such as networks for Gaussian approximation of the
Dirac delta functions (GADD), and numerical Green's functions (NGFs). The
performance of all methods is compared on four representative PDE categories,
each with different combinations of dimensionality and domain shape. The
results confirm the advantages of DGGFs, and benefits of Generalized Greens
Functions as an novel alternative approach to solve PDEs without suffering from
singularities.
