用延迟微分方程建模部分可观测系统的记忆效应，提升预测精度。

Recent advances in learning dynamical systems from data have shown significant promise. However, many existing methods assume access to the full state of the system -- an assumption that is rarely satisfied in practice, where systems are typically monitored through a limited number of sensors, leading to partial observability. To address this challenge, we draw inspiration from the Mori-Zwanzig formalism, which provides a theoretical connection between hidden variables and memory terms. Motivated by this perspective, we introduce a constant-lag Neural Delay Differential Equations (NDDEs) framework, providing a continuous-time approach for learning non-Markovian dynamics directly from data. These memory effects are captured using a finite set of time delays, which are identified via the adjoint method. We validate the proposed approach on a range of datasets, including synthetic systems, chaotic dynamics, and experimental measurements, such as the Kuramoto-Sivashinsky equation and cavity-flow experiments. Results demonstrate that NDDEs compare favourably with existing approaches for partially observed systems, including long short-term memory (LSTM) networks and augmented neural ordinary differential equations (ANODEs). Overall, NDDEs offer a principled and data-efficient framework for modelling non-Markovian dynamics under partial observability. An open-source implementation accompanies this article.