Time and State Dependent Neural Delay Differential Equations
This work addresses a limitation in modeling delayed systems for fields like physics and engineering, but it is incremental as it builds on existing Neural DDE frameworks.
The authors tackled the problem of modeling systems with discontinuities and delays, which cannot be handled by standard ODEs or Neural ODEs, by introducing Neural State-Dependent Delay Differential Equations (SDDDE) to model multiple, state- and time-dependent delays, showing that it outperforms other continuous-class models on various delayed dynamical systems.
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}.