动力学模型中的平滑误差及如何避免
Smoothness Errors in Dynamics Models and How to Avoid Them
AI总结：
本文研究了不同GNN在动力学建模中的平滑效应，证明了单位ary卷积对这类任务有害，并提出放松的单位ary卷积以平衡平滑性保留与物理系统需求。
摘要：
现代神经网络在求解表面偏微分方程中表现出潜力，通常通过将表面离散化为网格并使用网格感知图神经网络进行学习。然而，图神经网络存在过平滑问题，即节点特征逐渐趋同。单位ary图卷积通过数学约束保持平滑性，被提出以解决此问题。尽管如此，在许多物理系统如扩散过程中，平滑性会自然增加，单位性可能过于约束。本文系统研究了不同GNN的平滑效应，并证明单位ary卷积对这类任务有害。我们提出放松的单位ary卷积以平衡平滑性保留与物理系统需求。我们还将单位ary和放松的单位ary卷积从图扩展到网格。在热方程和波方程等复杂网格上的PDE以及天气预测实验中，我们发现我们的方法优于包括网格感知变压器和等变神经网络在内的多个强基线。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
