SDC-GON：用于求解偏微分方程的奇异分解与一致性正则化格林算子网络
SDC-GON: Singular Decomposition and Consistency-Regularized Green's Operator Networks for Solving Partial Differential Equations
- Saskatchewan Polytechnic（萨斯喀彻温理工学院）
- University of Alberta（阿尔伯塔大学）
- University of Regina（里贾纳大学）
机构由 AI 辅助整理，请以论文原文为准。
中文总结 AI 辅助
提出SDC-GON，通过将格林函数分解为解析奇异部分与网络学习的平滑修正，并引入自一致性损失，统一解决奇异逼近与梯度一致性问题，在多个偏微分方程基准上以更小网络取得更优精度。
中文摘要 AI 辅助
基于格林函数的算子逼近为在不同边界条件和源项下求解线性偏微分方程提供了一条高效途径。一旦格林函数被学习到，新配置的解即可通过积分获得，而无需重新求解微分方程。现有的格林函数学习方法面临两个结构性的挑战。第一个挑战是格林函数在源点附近的奇异行为，这给神经网络带来了困难的逼近负担。第二个挑战是学习到的格林函数与其梯度之间缺乏显式的一致性，尽管这两个量都直接出现在积分解表示中。本工作提出了SDC-GON，一种奇异分解与一致性正则化格林算子网络，在统一框架内解决了这两个挑战。格林函数被分解为一个解析已知的奇异部分和一个由网络学习的平滑修正项，使得神经逼近仅针对响应核的正则部分。一个自一致性损失强制平滑修正项的梯度与自动微分梯度之间保持一致。该方法在二维泊松、三维热传导、非均匀反应扩散和Stokes基准上进行了评估，在所有情况下均一致优于所比较的基线方法。在非均匀管道基准上，SDC-GON以更小的网络架构实现了$3.70\ imes10^{-4}$的测试误差，而相同宽度基线的误差为$9.60\ imes10^{-4}$，更大配置的误差为$4.63\ imes10^{-4}$，这表明结构改进比增加模型规模更有效。
英文摘要
Green's function based operator approximation offers an efficient route for solving linear partial differential equations under varying boundary conditions and source terms. Once the Green's function is learned, solutions for new configurations are obtained through integration rather than by solving the differential equation again. Existing Green's function learning methods face two structural challenges. The first is the singular behavior of the Green's function near the source point, which places a difficult approximation burden on neural networks. The second is the absence of explicit consistency between the learned Green's function and its gradient, although both quantities enter the integral solution representation directly. This work proposes SDC-GON, a Singular Decomposition and Consistency-Regularized Green's Operator Network that addresses both challenges within a unified framework. The Green's function is decomposed into an analytically known singular component and a smooth correction learned by the network, so that the neural approximation targets only the regular part of the response kernel. A self-consistency loss enforces agreement between the gradient and the autodifferentiation gradient of the smooth correction. The method is evaluated on two dimensional Poisson, three dimensional heat conduction, heterogeneous reaction diffusion, and Stokes benchmarks, consistently outperforming the compared baselines across all cases. On the heterogeneous pipe benchmark, SDC-GON achieves a testing error of $3.70\times10^{-4}$ with a smaller network architecture, compared with $9.60\times10^{-4}$ for the same-width baseline and $4.63\times10^{-4}$ for a larger configuration, demonstrating that structural improvements are more effective than increasing model size.
