Elliptic Partial Differential Equations (PDEs) play a central role in computing the equilibrium conditions of physical problems (heat, gravitation, electrostatics, etc.). Efficient solutions to elliptic PDEs are also relevant to computer graphics since they encode global smoothness with local control leading to stable, well-behaved solutions. The Poisson equation is a linear elliptic PDE that serves as a prototypical candidate to assess newly-proposed solvers. Solving the Poisson equation on an arbitrary 3D domain, say a 3D scan of a turbine's blade, is computationally expensive and scales quadratically with discretization. Traditional workflows in research and industry exploit variants of the finite element method (FEM), but some key benefits of using Monte Carlo (MC) methods have been identified. Our key idea is to exploit a sparse and approximate solution (via FEM or MC) to the Poisson equation towards inferring an adaptive discretization in one shot. We achieve this by training a lightweight neural network that generalizes across shapes and boundary conditions. Our algorithm, Learned Adaptive Mesh Generation (LAMG), maps from a coarse solution to a sizing field that defines a local (adaptive) spatial resolution. This output space, rather than directly predicting a high-resolution solution, is a unique aspect of our approach. We use standard methods to generate tetrahedral meshes that respect the sizing field, and obtain the solution via one FEM computation on the adaptive mesh. That is, our neural network serves as a surrogate model of a computationally expensive method that requires multiple (iterative) FEM solves. We demonstrate the versatility, controllability, robustness and efficiency of LAMG via systematic experimentation.

翻译：椭圆型偏微分方程在计算物理问题（热传导、引力、静电学等）的平衡条件中起着核心作用。椭圆型偏微分方程的高效求解也与计算机图形学密切相关，因为它们通过局部控制编码全局光滑性，从而产生稳定且行为良好的解。泊松方程作为一种线性椭圆型偏微分方程，是评估新求解器的典型候选对象。在任意三维域（例如涡轮叶片的3D扫描模型）上求解泊松方程计算成本高昂，且计算量随离散化程度呈二次方增长。传统的研究和工业流程利用有限元法的各种变体，但使用蒙特卡罗方法的一些关键优势已被发现。我们的核心思想是利用泊松方程的稀疏近似解（通过有限元法或蒙特卡罗法）一次性推断出自适应离散化方案。我们通过训练一个轻量级神经网络来实现这一目标，该网络能够泛化到不同形状和边界条件。我们的算法——学习型自适应网格生成，将粗略解映射到定义局部（自适应）空间分辨率的尺寸场。输出空间并非直接预测高分辨率解，而是我们方法的一个独特方面。我们使用标准方法生成符合尺寸场的四面体网格，并通过在自适应网格上进行一次有限元计算获得最终解。换言之，我们的神经网络充当了需要多次（迭代）有限元求解的高计算成本方法的代理模型。我们通过系统化实验展示了学习型自适应网格生成在通用性、可控性、鲁棒性和效率方面的优势。