用神经网络学习带奇点的格林函数，加速微分方程求解。
Learning Singularity-Encoded Green's Functions with Application to Iterative Methods
- 将格林函数嵌入高一维空间，用神经网络处理奇异性与高维问题。
- 在2维和4维问题上成功解析奇点，显著加速传统迭代求解器。
- 适合需要高效求解椭圆型微分方程的研究者，尤其关注预条件或混合求解。
格林函数为椭圆型偏微分方程的理论分析与数值方法提供了内在联系，但其通常无闭式表达，需通过代理模型指导求解器设计。然而，由于维度加倍和固有奇异性，格林函数的数值计算仍具挑战性。本文提出一种无监督的奇点编码学习方法：将格林函数的先验估计作为增广变量嵌入一阶更高维空间，并采用神经网络参数化以应对维度增加；通过将训练后的神经网络解投影回原域，所提出的深度代理模型利用其谱偏差，可加速经典迭代方法，既可用作预条件子，也可作为混合求解器的一部分。在二维与四维格林函数的数值实验中，该方法验证了对奇点的良好解析能力及对迭代求解器的显著加速效果。
原文摘要 · Abstract (English)
Green's function provides an inherent connection between theoretical analysis and numerical methods for elliptic partial differential equations, and general absence of its closed-form expression necessitates surrogate modeling to guide the design of effective solvers. Unfortunately, numerical computation of Green's function remains challenging due to its doubled dimensionality and intrinsic singularity. In this paper, we present a novel singularity-encoded learning approach to resolve these problems in an unsupervised fashion. Our method embeds the Green's function within a one-order higher-dimensional space by encoding its prior estimate as an augmented variable, followed by a neural network parametrization to manage the increased dimensionality. By projecting the trained neural network solution back onto the original domain, our deep surrogate model exploits its spectral bias to accelerate conventional iterative schemes, serving either as a preconditioner or as part of a hybrid solver. The effectiveness of our proposed method is empirically verified through numerical experiments with two and four dimensional Green's functions, achieving satisfactory resolution of singularities and acceleration of iterative solvers.
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