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.

翻译：本研究旨在解决线性偏微分方程（PDEs）中格林函数算子的求解难题。格林函数因其能直接建立输入与解的映射关系而备受关注，避免了有限差分法、有限元法等常见数值方法的使用。然而，由于狄拉克δ函数奇异性的存在，显式推导大多数PDEs的格林函数核一直面临挑战。为此，我们提出深度广义格林函数（DGGF）作为替代方案，可利用神经网络模型高效精准地求解。DGGF在继承格林函数可复用特性的同时，提供了更高效精确的线性PDEs求解途径，并具备无网格操作、低内存占用等额外优势。我们将DGGF与多种最先进（SOTA）的PDE求解器进行对比，包括直接方法（如物理信息神经网络PINNs）、格林函数方法（如狄拉克δ函数高斯逼近网络GADD）以及数值格林函数（NGFs）。针对四类代表性PDEs（涵盖不同维度组合与求解域形状），我们比较了所有方法的性能。实验结果证实了DGGF的优势，并展示了广义格林函数作为无奇异性PDE求解新型替代方案的价值。