Learning the dynamics of autonomous nonlinear delay systems

In this paper, we focus on learning the time delay and nonlinearity of autonomous dynamical systems using trainable time delay neural networks. We demonstrate that, with delays trained together with weights and biases, the trained neural networks may approximate the right hand side of delay differential equations. It is shown that data collected from the vicinity a stable equilibrium or limit cycle do not contain rich enough dynamics, therefore the trained networks can have very poor generalization. However, including data about the transient behavior can significantly enhance the performance, and similar improvements can be achieved when data collected near a chaotic attractor is utilized. We also evaluate how the learning performance is affected by the selected loss function and measurement noise. Numerical results are presented for learning examples: Mackey-Glass equation and a predator-prey model.