Modern neural networks have shown promise for solving partial differential equations over surfaces, often by discretizing the surface as a mesh and learning with a mesh-aware graph neural network. However, graph neural networks suffer from oversmoothing, where a node's features become increasingly similar to those of its neighbors. Unitary graph convolutions, which are mathematically constrained to preserve smoothness, have been proposed to address this issue. Despite this, in many physical systems, such as diffusion processes, smoothness naturally increases and unitarity may be overconstraining. In this paper, we systematically study the smoothing effects of different GNNs for dynamics modeling and prove that unitary convolutions hurt performance for such tasks. We propose relaxed unitary convolutions that balance smoothness preservation with the natural smoothing required for physical systems. We also generalize unitary and relaxed unitary convolutions from graphs to meshes. In experiments on PDEs such as the heat and wave equations over complex meshes and on weather forecasting, we find that our method outperforms several strong baselines, including mesh-aware transformers and equivariant neural networks.

翻译：现代神经网络在对曲面上的偏微分方程求解中展现出潜力，通常通过将曲面离散化为网格并采用网格感知的图神经网络进行学习。然而，图神经网络存在过平滑问题，即节点的特征会逐渐趋同于其邻居节点。为缓解该问题，研究者提出了具有数学约束以保持平滑性的酉图卷积。但在许多物理系统（如扩散过程）中，平滑性会自然增强，此时酉约束可能过于严格。本文系统研究了不同图神经网络在动力学建模中的平滑效应，并证明了酉卷积会损害此类任务的性能。我们提出松弛酉卷积，在平滑保持与物理系统所需的自然平滑之间取得平衡。同时将酉卷积及松弛酉卷积从图扩展到网格。在包含复杂网格的热传导与波动方程等偏微分方程求解以及天气预报实验中，我们的方法均优于包括网格感知Transformer和等变神经网络在内的多个强基线模型。