神经普通微分方程何时能在复杂网络上泛化？
When do neural ordinary differential equations generalize on complex networks?
AI总结：
研究探讨了神经ODEs在复杂网络上的泛化能力，发现度异质性和动态系统类型是影响泛化的主要因素，同时指出平均聚类作用较弱。
摘要：
神经普通微分方程（神经ODEs）能够有效地从时间序列数据中学习动态系统，但它们在图结构数据上的行为仍不清晰，尤其是在应用于与训练时遇到的图大小或结构不同的图时。我们研究了向量场遵循Barabási-Barzel形式的神经ODEs（nODEs），在五个常见的图上动态系统合成数据上进行训练。使用S¹模型生成具有现实且可调结构的图，我们发现度异质性和动态系统类型是决定nODEs在不同图大小和属性上泛化能力的主要因素。这扩展到nODEs捕捉固定点和在缺失数据中维持性能的能力。平均聚类在决定nODEs性能中的作用次之。我们的发现强调nODEs作为理解复杂系统强大方法，但也突显了现实图中度异质性和聚类带来的挑战。
英文摘要：
Neural ordinary differential equations (neural ODEs) can effectively learn dynamical systems from time series data, but their behavior on graph-structured data remains poorly understood, especially when applied to graphs with different size or structure than encountered during training. We study neural ODEs ($\mathtt{nODE}$s) with vector fields following the Barabási-Barzel form, trained on synthetic data from five common dynamical systems on graphs. Using the $\mathbb{S}^1$-model to generate graphs with realistic and tunable structure, we find that degree heterogeneity and the type of dynamical system are the primary factors in determining $\mathtt{nODE}$s' ability to generalize across graph sizes and properties. This extends to $\mathtt{nODE}$s' ability to capture fixed points and maintain performance amid missing data. Average clustering plays a secondary role in determining $\mathtt{nODE}$ performance. Our findings highlight $\mathtt{nODE}$s as a powerful approach to understanding complex systems but underscore challenges emerging from degree heterogeneity and clustering in realistic graphs.
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