用微分方程构建连续型单纯形神经网络，提升稳定性与抗噪能力。
Continuous Simplicial Neural Networks
- 基于单纯形上的偏微分方程设计连续神经网络架构
- 在噪声环境中性能优于现有离散SNN，且更稳定
- 适合处理高阶交互数据，如轨迹预测与网格分析
单纯形复形为建模结构化数据中的高阶交互提供了强大框架，特别适用于轨迹预测和网格处理。然而，现有的单纯形神经网络（SNN）主要依赖离散滤波技术，存在局限性。相比之下，单纯形上的偏微分方程（PDE）能更合理地捕捉连续动态。本文提出连续单纯形神经网络（COSIMO），其架构源自单纯形上的PDE。我们从理论和实验上验证了COSIMO在单纯形扰动下的稳定性，并研究了几何深度学习中的过平滑现象，发现COSIMO对这一问题的控制优于离散SNN。在真实数据集上的实验表明，COSIMO在复杂和噪声环境中表现媲美当前最优SNN。代码已开源：https://github.com/ArefEinizade2/COSIMO。
原文摘要 · Abstract (English)
Simplicial complexes provide a powerful framework for modeling higher-order interactions in structured data, making them particularly suitable for applications such as trajectory prediction and mesh processing. However, existing simplicial neural networks (SNNs), whether convolutional or attention-based, rely primarily on discrete filtering techniques, which can be restrictive. In contrast, partial differential equations (PDEs) on simplicial complexes offer a principled approach to capture continuous dynamics in such structures. In this work, we introduce continuous simplicial neural network (COSIMO), a novel SNN architecture derived from PDEs on simplicial complexes. We provide theoretical and experimental justifications of COSIMO's stability under simplicial perturbations. Furthermore, we investigate the over-smoothing phenomenon, a common issue in geometric deep learning, demonstrating that COSIMO offers better control over this effect than discrete SNNs. Our experiments on real-world datasets demonstrate that COSIMO achieves competitive performance compared to state-of-the-art SNNs in complex and noisy environments. The implementation codes are available in https://github.com/ArefEinizade2/COSIMO.
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