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

Neural Green's Operators for Parametric Partial Differential Equations

This work is concerned with the use of Gaussian surrogate models for Bayesian
inverse problems associated with linear partial differential equations. A
particular focus is on the regime where only a small amount of training data is
available. In this regime the type of Gaussian prior used is of critical
importance with respect to how well the surrogate model will perform in terms
of Bayesian inversion. We extend the framework of Raissi et. al. (2017) to
construct PDE-informed Gaussian priors that we then use to construct different
approximate posteriors. A number of different numerical experiments illustrate
the superiority of the PDE-informed Gaussian priors over more traditional
priors.

本研究关注使用高斯代理模型处理与线性偏微分方程相关的贝叶斯反问题，重点关注只有少量训练数据可用的情况下使用的高斯先验类型对于近似后验的性能影响的扩展研究。实验表明 PDE - 信息高斯先验优于传统先验。