Geometric deep learning extends deep learning to incorporate information about the geometry and topology data, especially in complex domains like graphs. Despite the popularity of message passing in this field, it has limitations such as the need for graph rewiring, ambiguity in interpreting data, and over-smoothing. In this paper, we take a different approach, focusing on leveraging geometric information from simplicial complexes embedded in $\mathbb{R}^n$ using node coordinates. We use differential k-forms in \mathbb{R}^n to create representations of simplices, offering interpretability and geometric consistency without message passing. This approach also enables us to apply differential geometry tools and achieve universal approximation. Our method is efficient, versatile, and applicable to various input complexes, including graphs, simplicial complexes, and cell complexes. It outperforms existing message passing neural networks in harnessing information from geometrical graphs with node features serving as coordinates.

翻译：几何深度学习将深度学习扩展到整合几何与拓扑数据的信息，尤其在图形等复杂领域中。尽管消息传递在这一领域广受欢迎，但它存在局限性，例如需要重新布线图形、数据解释模糊以及过度平滑等问题。在本文中，我们采用不同的方法，专注于通过节点坐标利用 $\mathbb{R}^n$ 中嵌入的单纯复形的几何信息。我们使用 $\mathbb{R}^n$ 中的微分 $k$-形式来创建单纯形的表示，从而无需消息传递即可提供可解释性和几何一致性。这种方法还使我们能够应用微分几何工具并实现通用逼近。我们的方法高效且通用，可应用于各种输入复形，包括图形、单纯复形和胞复形。在利用节点特征作为坐标的几何图形信息方面，它优于现有的消息传递神经网络。