Learning how complex dynamical systems evolve over time is a key challenge in system identification. For safety critical systems, it is often crucial that the learned model is guaranteed to converge to some equilibrium point. To this end, neural ODEs regularized with neural Lyapunov functions are a promising approach when states are fully observed. For practical applications however, partial observations are the norm. As we will demonstrate, initialization of unobserved augmented states can become a key problem for neural ODEs. To alleviate this issue, we propose to augment the system's state with its history. Inspired by state augmentation in discrete-time systems, we thus obtain neural delay differential equations. Based on classical time delay stability analysis, we then show how to ensure stability of the learned models, and theoretically analyze our approach. Our experiments demonstrate its applicability to stable system identification of partially observed systems and learning a stabilizing feedback policy in delayed feedback control.

翻译：学习复杂的动态系统如何随时间演变是系统识别的关键挑战。 对于安全关键系统来说,学习到的模型通常至关重要,保证会与某种平衡点趋同。为此,神经组织与神经系统 Lyapunov 函数的正规化是国家完全观察到的一种有希望的方法。然而,对于实际应用来说,部分观测是常规的。正如我们将要证明的那样,未观测到的增强状态的初始化会成为神经组织的一个关键问题。为了缓解这一问题,我们提议用其历史来强化系统状态。在离散时间系统中的状态增强的激励下,我们因此获得神经延迟差异方程式。根据传统的时间延迟稳定分析,我们然后展示如何确保所学模式的稳定性,并从理论上分析我们的方法。我们的实验表明它对于系统稳定识别部分观测到的系统以及学习延迟反馈控制的稳定反馈政策是可行的。