Aircrafts working with soft robotic arms aroused extensive concerns in recent years. In this paper, a physics-informed neural network (PINN) is used to learn the dynamics of continuum soft robots. In detail, we incorporate Euler Lagrange equation into the deep neural network, providing improved physical interpretability and learning efficiency. Afterwards, a finite-time stable sliding mode controller is designed to achieve fast convergence, more accurate trajectory tracking and anti-disturbance ability. Theoretical analysis proves the closed-loop stability, and comparative simulations verify its superiority in transient and steady-state performance.