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.

翻译：我们提出了一种深度算子网络（DeepONet）框架，用于从数据中学习连续时间非线性控制系统的动态响应。为此，我们首先构建并训练一个用于近似控制系统局部解算子的DeepONet。然后，我们设计了一种数值方案，该方案通过递归使用训练好的DeepONet，模拟给定控制输入和初始条件下控制系统的长期/中期动态响应。我们为所提方案给出了相关累积误差的误差界估计。此外，我们设计了一种数据驱动的显式龙格-库塔（RK）格式，该格式利用DeepONet的前向传播和自动微分，在数值方案步长足够小时，能更好地近似系统的响应。在捕食者-猎物系统、摆系统和推车-杆系统上的数值实验证实，我们的DeepONet框架能够有效学习近似非线性控制系统的动态响应。