Learning the dynamical response of nonlinear non-autonomous dynamical systems with deep operator neural networks

Abstract

We propose using operator learning to approximate the dynamical response of non-autonomous systems, such as nonlinear control systems. Unlike classical function learning, operator learning maps between two function spaces, does not require discretization of the output function, and provides flexibility in data preparation and solution prediction. Particularly, we apply and redesign the Deep Operator Neural Network (DeepONet) to recursively learn the solution trajectories of the dynamical systems. Our approach involves constructing and training a DeepONet that approximates the system’s local solution operator. We then develop a numerical scheme that recursively simulates the system’s long/medium-term dynamic response for given inputs and initial conditions, using the trained DeepONet. We accompany the proposed scheme with an estimate for the error bound of the associated cumulative error. Moreover, we propose a data-driven Runge–Kutta (RK) explicit scheme that leverages the DeepONet’s forward pass and automatic differentiation to better approximate the system’s response when the numerical scheme’s step size is small. Numerical experiments on the predator–prey, pendulum, and cart pole systems demonstrate that our proposed DeepONet framework effectively learns to approximate the dynamical response of non-autonomous systems with time-dependent inputs.