Many network architectures exist for learning on meshes, yet their constructions entail delicate trade-offs between difficulty learning high-frequency features, insufficient receptive field, sensitivity to discretization, and inefficient computational overhead. Drawing from classic local-global approaches in mesh processing, we introduce PoissonNet, a novel neural architecture that overcomes all of these deficiencies by formulating a local-global learning scheme, which uses Poisson's equation as the primary mechanism for feature propagation. Our core network block is simple; we apply learned local feature transformations in the gradient domain of the mesh, then solve a Poisson system to propagate scalar feature updates across the surface globally. Our local-global learning framework preserves the features's full frequency spectrum and provides a truly global receptive field, while remaining agnostic to mesh triangulation. Our construction is efficient, requiring far less compute overhead than comparable methods, which enables scalability -- both in the size of our datasets, and the size of individual training samples. These qualities are validated on various experiments where, compared to previous intrinsic architectures, we attain state-of-the-art performance on semantic segmentation and parameterizing highly-detailed animated surfaces. Finally, as a central application of PoissonNet, we show its ability to learn deformations, significantly outperforming state-of-the-art architectures that learn on surfaces.

翻译：尽管存在多种用于网格学习的网络架构，但其构建过程需要在学习高频特征的难度、感受野不足、对离散化的敏感性以及计算开销效率低下之间进行微妙的权衡。借鉴网格处理中经典的局部-全局方法，我们提出了PoissonNet——一种新颖的神经架构，通过构建局部-全局学习方案克服了上述所有缺陷，该方案以泊松方程作为特征传播的主要机制。我们的核心网络模块简洁明了：首先在网格的梯度域中应用学习到的局部特征变换，然后求解泊松系统以在曲面全局范围内传播标量特征更新。我们的局部-全局学习框架保留了特征的全频段频谱，提供了真正全局的感受野，同时保持对网格三角剖分的不敏感性。该架构计算高效，所需计算开销远低于同类方法，从而实现了良好的可扩展性——无论是数据集规模还是单个训练样本的尺寸。这些特性在多项实验中得到验证：与先前的本征架构相比，我们在语义分割和高度细节化动画曲面参数化任务中均达到了最先进的性能表现。最后，作为PoissonNet的核心应用，我们展示了其在学习形变任务上的能力，其性能显著优于现有的曲面学习先进架构。