变分格林函数用于体积分方程
Variational Green's Functions for Volumetric PDEs
AI总结：
本文提出了一种基于变分方法的学习框架，用于高效计算体积分方程的格林函数，通过分解格林函数为解析和学习部分，实现了对不同边界条件的自然处理。
摘要：
格林函数表征偏微分方程的基本解；它们对于从形状分析到物理模拟的任务至关重要，但仍然难以在任意几何离散化上计算。我们提出了变分格林函数（VGF），一种学习线性自共轭PDE算子的格林函数的平滑、可微表示的方法，包括泊松、筛除泊松和双调和方程。为了解决格林函数固有的尖锐奇异性，我们的方法将格林函数分解为一个解析的自由空间成分和一个学习的校正器成分。我们的方法利用变分基础来自然地施加Neumann边界条件，并通过神经场的输出上的投影层施加Dirichlet边界条件。所得到的格林函数易于评估，对源应用可微，并且可以基于其他信号参数化我们的几何。
英文摘要：
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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