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.

该研究介绍了一种新颖的神经算子网络架构，即神经格林算子 (NGOs)，它学习了参数化线性偏微分方程组的解算子。研究发现，NGOs在测试分布范围内与DeepONets、VarMiONs和傅里叶神经算子竞争力强，在测试分布范围外生成更细粒度的数据时，能够稳健地泛化。此外，研究还发现NGOs返回的格林函数的显式表示可用于构建有效的偏微分方程数值求解器的预处理器。

参数化偏微分方程的神经格林算子