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

Learning Stable Deep Dynamics Models for Partially Observed or Delayed Dynamical Systems