  This work introduces neural Green's operators (NGOs), a novel neural operator
network architecture that learns the solution operator for a parametric family
of linear partial differential equations (PDEs). Our construction of NGOs is
derived directly from the Green's formulation of such a solution operator.
Similar to deep operator networks (DeepONets) and variationally mimetic
operator networks (VarMiONs), NGOs constitutes an expansion of the solution to
the PDE in terms of basis functions, that is returned from a sub-network,
contracted with coefficients, that are returned from another sub-network.
However, in accordance with the Green's formulation, NGOs accept weighted
averages of the input functions, rather than sampled values thereof, as is the
case in DeepONets and VarMiONs. Application of NGOs to canonical linear
parametric PDEs shows that, while they remain competitive with DeepONets,
VarMiONs and Fourier neural operators when testing on data that lie within the
training distribution, they robustly generalize when testing on finer-scale
data generated outside of the training distribution. Furthermore, we show that
the explicit representation of the Green's function that is returned by NGOs
enables the construction of effective preconditioners for numerical solvers for
PDEs.
