Discontinuities and delayed terms are encountered in the governing equations of a large class of problems ranging from physics, engineering, medicine to economics. These systems are impossible to be properly modelled and simulated with standard Ordinary Differential Equations (ODE), or any data-driven approximation including 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 complicated systems. In this work we revisit the recently proposed Neural DDE by introducing Neural State-Dependent DDE (SDDDE), a general and flexible framework featuring multiple and state-dependent delays. The developed framework is auto-differentiable and runs efficiently on multiple backends. We show that our method is competitive and outperforms other continuous-class models on a wide variety of delayed dynamical systems.

翻译：在从物理学、工程学、医学到经济学等广泛问题的控制方程中，经常出现间断项和时滞项。这些系统无法使用标准常微分方程（ODE）或任何数据驱动近似方法（包括神经ODE）进行恰当的建模和模拟。为解决这一问题，通常引入潜变量在更高维空间中求解系统动力学，并通过投影到原始空间获得解。然而，这种解法缺乏物理可解释性。相比之下，时滞微分方程（DDE）及其数据驱动近似方法自然成为刻画此类复杂系统的合适候选方案。本文通过引入神经状态依赖时滞微分方程（SDDDE）重新审视了近期提出的神经DDE，构建了一个支持多时滞和状态依赖时滞的通用灵活框架。该框架具有自动微分能力，可在多种后端高效运行。我们证明，该方法在各类时滞动力系统中具有竞争力，且性能优于其他连续类模型。