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.

翻译：本文提出了一种深度代理模型，用于学习矩形域内反应-扩散算子相关的格林函数。我们采用U-Net架构有效捕捉目标偏微分方程（PDEs）中从源项到解的映射关系。为实现无需依赖标注数据的模型高效训练，我们设计了一种新颖的损失函数，该函数借鉴了求解PDE的传统数值方法思想。此外，通过硬编码机制确保预测的格林函数与边界条件完全匹配。基于训练所得深度代理模型习得的格林函数，我们开发了一种快速求解器，可求解具有不同源项和边界条件的相应PDE。文中还提供了多种数值算例，以验证所提模型的有效性。