  We propose a Deep Operator Network~(DeepONet) framework to learn the dynamic
response of continuous-time nonlinear control systems from data. To this end,
we first construct and train a DeepONet that approximates the control system's
local solution operator. Then, we design a numerical scheme that recursively
uses the trained DeepONet to simulate the control system's long/medium-term
dynamic response for given control inputs and initial conditions. We accompany
the proposed scheme with an estimate for the error bound of the associated
cumulative error. Furthermore, we design a data-driven Runge-Kutta~(RK)
explicit scheme that uses the DeepONet forward pass and automatic
differentiation to better approximate the system's response when the numerical
scheme's step size is sufficiently small. Numerical experiments on the
predator-prey, pendulum, and cart pole systems confirm that our DeepONet
framework learns to approximate the dynamic response of nonlinear control
systems effectively.
