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.

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