Title: Time and State Dependent Neural Delay Differential Equations Show Chinese title

Title: 时间与状态相关的神经延迟微分方程

Abstract: Discontinuities and delayed terms are encountered in the governing equations of a large class of problems ranging from physics and engineering to medicine and economics. These systems cannot be properly modelled and simulated with standard Ordinary Differential Equations (ODE), or data-driven approximations such as Neural Ordinary Differential Equations (NODE). To circumvent this issue, latent variables are typically introduced to solve the dynamics of the system in a higher dimensional space and obtain the solution as a projection to the original space. However, this solution lacks physical interpretability. In contrast, Delay Differential Equations (DDEs), and their data-driven approximated counterparts, naturally appear as good candidates to characterize such systems. In this work we revisit the recently proposed Neural DDE by introducing Neural State-Dependent DDE (SDDDE), a general and flexible framework that can model multiple and state- and time-dependent delays. We show that our method is competitive and outperforms other continuous-class models on a wide variety of delayed dynamical systems. Code is available at the repository \href{https://github.com/thibmonsel/Time-and-State-Dependent-Neural-Delay-Differential-Equations}{here}. Show Chinese abstract

Abstract: 在从物理学和工程学到医学和经济学的大量问题中，控制方程中会遇到不连续性和延迟项。 这些系统无法通过标准常微分方程(ODE)或数据驱动的近似方法如神经常微分方程(NODE)进行正确建模和仿真。 为解决这个问题，通常引入隐变量，在更高维空间中求解系统的动力学，并将解作为原始空间的投影。 然而，这种解决方案缺乏物理可解释性。 相反，延迟微分方程(DDE)及其数据驱动的近似方法自然成为表征此类系统的良好候选。 在本工作中，我们重新审视了最近提出的神经DDE，并引入了神经状态依赖DDE(SDDDE)，这是一个通用且灵活的框架，可以对多个状态和时间相关的延迟进行建模。 我们证明，我们的方法在各种延迟动力系统上具有竞争力，并优于其他连续类模型。 代码可在存储库\href{https://github.com/thibmonsel/Time-and-State-Dependent-Neural-Delay-Differential-Equations}{这里}中获得。