Time and State Dependent Neural Delay Differential Equations

• URL: http://arxiv.org/abs/2306.14545v2
• Date: Thu, 26 Sep 2024 08:29:00 GMT
• Title: Time and State Dependent Neural Delay Differential Equations
• Authors: Thibault Monsel, Onofrio Semeraro, Lionel Mathelin, Guillaume Charpiat,
• Abstract summary: Delayed terms are encountered in the governing equations of a large class of problems ranging from physics and engineering to medicine and economics. We introduce Neural State-Dependent DDE, a 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.
• Score: 0.5249805590164901
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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}.

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