Time and State Dependent Neural Delay Differential Equations

TL;DR

This paper introduces Neural State-Dependent Delay Differential Equations (SDDDE), a flexible neural network framework for modeling complex systems with time- and state-dependent delays, outperforming existing models in various delayed dynamical systems.

Contribution

The paper presents a novel Neural SDDDE framework that generalizes Neural DDEs to handle multiple, state- and time-dependent delays, improving modeling accuracy and interpretability.

Findings

Outperforms existing models on delayed dynamical systems

Handles multiple and state-dependent delays effectively

Provides a flexible, interpretable modeling approach

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}.