神经微分方程在复杂网络上泛化能力受节点度异质性影响显著。
When do neural ordinary differential equations generalize on complex networks?
- 基于巴尔巴西-巴泽尔形式的向量场设计神经ODE
- 度异质性是决定泛化性能的关键因素，优于聚类系数
- 适用于动态系统建模与缺失数据场景，适合复杂系统研究者
神经常微分方程（neural ODEs）能从时序数据中有效学习动力系统，但其在图结构数据上的行为仍不明确，尤其在训练图与测试图规模或结构不同时。本文研究了遵循巴尔巴西-巴泽尔形式的神经ODE（nODE），在五种常见动力系统合成数据上训练。通过S¹-模型生成具有现实且可调结构的图，发现度异质性和动力系统类型是决定nODE跨图规模与属性泛化能力的主要因素。该特性也影响其对不动点的捕捉能力及在缺失数据下的表现。平均聚类仅起次要作用。研究揭示nODE在理解复杂系统中的潜力，但也凸显真实图中度异质性和聚类带来的挑战。
原文摘要 · Abstract (English)
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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