Green's functions characterize the fundamental solutions of partial differential equations; they are essential for tasks ranging from shape analysis to physical simulation, yet they remain computationally prohibitive to evaluate on arbitrary geometric discretizations. We present Variational Green's Function (VGF), a method that learns a smooth, differentiable representation of the Green's function for linear self-adjoint PDE operators, including the Poisson, the screened Poisson, and the biharmonic equations. To resolve the sharp singularities characteristic of the Green's functions, our method decomposes the Green's function into an analytic free-space component, and a learned corrector component. Our method leverages a variational foundation to impose Neumann boundary conditions naturally, and imposes Dirichlet boundary conditions via a projective layer on the output of the neural field. The resulting Green's functions are fast to evaluate, differentiable with respect to source application, and can be conditioned on other signals parameterizing our geometry.

翻译：格林函数表征了偏微分方程的基本解；从形状分析到物理模拟，它们在众多任务中至关重要，但在任意几何离散化上计算格林函数仍然具有极高的计算成本。本文提出变分格林函数（VGF）方法，该方法能够学习线性自伴偏微分方程算子（包括泊松方程、屏蔽泊松方程和双调和方程）格林函数的光滑可微表示。为解析格林函数特有的尖锐奇异性，我们的方法将格林函数分解为解析的自由空间分量和学习的修正分量。该方法利用变分基础自然地施加诺伊曼边界条件，并通过神经场输出端的投影层施加狄利克雷边界条件。所得格林函数具有快速求值、对源应用可微的特性，并且可以基于参数化几何的其他信号进行条件化。