Learning Time Delay Systems with Neural Ordinary Differential Equations

TL;DR

This paper introduces a neural ordinary differential equation approach with trainable delays to learn the dynamics of time delay systems from data, successfully capturing complex behaviors like bifurcations.

Contribution

It presents a novel neural network framework that incorporates trainable delays and relates delay differential equations to ODEs for system learning.

Findings

Successfully learned Mackey-Glass system dynamics from data

Reproduced bifurcation diagrams matching original system

Demonstrated effectiveness on chaotic time delay systems

Abstract

A novel way of using neural networks to learn the dynamics of time delay systems from sequential data is proposed. A neural network with trainable delays is used to approximate the right hand side of a delay differential equation. We relate the delay differential equation to an ordinary differential equation by discretizing the time history and train the corresponding neural ordinary differential equation (NODE) to learn the dynamics. An example on learning the dynamics of the Mackey-Glass equation using data from chaotic behavior is given. After learning both the nonlinearity and the time delay, we demonstrate that the bifurcation diagram of the neural network matches that of the original system.