Learning Time Delay Systems with Neural Ordinary Differential Equations
This addresses the problem of modeling complex time delay systems for researchers in dynamical systems and machine learning, but it is incremental as it builds on existing NODE methods with a specific adaptation for delays.
The paper tackled learning the dynamics of time delay systems from sequential data by proposing a neural network with trainable delays to approximate delay differential equations, relating them to neural ordinary differential equations (NODE) through discretization. The result demonstrated that after learning both nonlinearity and time delay from chaotic data of the Mackey-Glass equation, the bifurcation diagram of the neural network matched that of the original system.
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.