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Time and State Dependent Neural Delay Differential Equations

Thibault Monsel, Onofrio Semeraro, Lionel Mathelin, Guillaume Charpiat

TL;DR

This work revisits 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 and is competitive and outperforms other continuous-class models on a wide variety of delayed dynamical systems.

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}.

Time and State Dependent Neural Delay Differential Equations

TL;DR

This work revisits 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 and is competitive and outperforms other continuous-class models on a wide variety of delayed dynamical systems.

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}.
Paper Structure (29 sections, 17 equations, 9 figures, 8 tables, 2 algorithms)

This paper contains 29 sections, 17 equations, 9 figures, 8 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Neural State-Dependant DDEs
  3. Methods
  4. Experiments
  5. Description of the test cases
  6. Time-dependent delay system
  7. State-dependent time-delay system
  8. Delayed Diffusion Equation
  9. Evaluation
  10. Results
  11. Testset prediction
  12. Increasing trajectory fed for Neural Laplace
  13. Conclusion and Future Work
  14. Overview in DDE integration
  15. Definition
  16. ...and 14 more sections

Figures (9)

  • Figure 1: Time Dependent DDE randomly sampled test trajectory plots
  • Figure 2: State Dependent DDE randomly sampled test trajectory plots
  • Figure 3: Diffusion Delay PDE randomly sampled from the testset
  • Figure 5: Time-dependent DDE randomly sampled testset trajectories where 50% of data is fed to Neural Laplace
  • Figure 6: Time Dependent DDE randomly sampled extrapolated trajectory plots
  • ...and 4 more figures