用神经网络学出可微的体积微分方程解，速度快且能处理复杂几何。
Variational Green's Functions for Volumetric PDEs
- 将格林函数拆成解析项和学习修正项，应对尖锐奇点。
- 支持泊松、屏敝泊松等方程，边界条件自动满足。
- 适合需要快速可微解的物理模拟与几何分析场景。
格林函数是偏微分方程基本解的核心表征，对形状分析到物理模拟均至关重要，但在任意几何离散上计算代价高昂。本文提出变分格林函数（VGF），一种针对线性自伴微分算子（包括泊松、屏敝泊松和双调和方程）的格林函数学习方法，通过神经场构建平滑可微的表示。为解决格林函数固有的尖锐奇点，方法将其分解为解析自由空间项与可学习修正项。基于变分框架自然施加诺伊曼边界条件，并通过输出端投影层实现狄利克雷边界条件。所得格林函数计算高效、对源位置可微，且能根据几何参数进行条件化建模。
原文摘要 · Abstract (English)
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.
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