Differential equations are a ubiquitous tool to study dynamics, ranging from physical systems to complex systems, where a large number of agents interact through a graph with non-trivial topological features. Data-driven approximations of differential equations present a promising alternative to traditional methods for uncovering a model of dynamical systems, especially in complex systems that lack explicit first principles. A recently employed machine learning tool for studying dynamics is neural networks, which can be used for data-driven solution finding or discovery of differential equations. Specifically for the latter task, however, deploying deep learning models in unfamiliar settings - such as predicting dynamics in unobserved state space regions or on novel graphs - can lead to spurious results. Focusing on complex systems whose dynamics are described with a system of first-order differential equations coupled through a graph, we show that extending the model's generalizability beyond traditional statistical learning theory limits is feasible. However, achieving this advanced level of generalization requires neural network models to conform to fundamental assumptions about the dynamical model. Additionally, we propose a statistical significance test to assess prediction quality during inference, enabling the identification of a neural network's confidence level in its predictions.

翻译：微分方程是研究动力学的普遍工具，其应用范围从物理系统到复杂系统。在复杂系统中，大量智能体通过具有非平凡拓扑特征的图结构相互作用。基于数据的微分方程近似方法为揭示动力系统模型提供了传统方法之外的有前途替代方案，特别是在缺乏显式第一性原理的复杂系统中。近期用于动力学研究的机器学习工具是神经网络，它可用于数据驱动的微分方程求解或发现。然而，专门针对后一项任务，在陌生场景（如预测未观测状态空间区域或新型图结构上的动力学）中部署深度学习模型可能导致虚假结果。聚焦于其动力学由通过图耦合的一阶微分方程组描述的复杂系统，我们证明：将模型泛化能力扩展到传统统计学习理论极限之外是可行的。但实现这种高级泛化需要神经网络模型符合关于动力学模型的基本假设。此外，我们提出了一种统计显著性检验方法，用于在推理过程中评估预测质量，从而能够识别神经网络对其预测结果的置信水平。