Discovering governing equations of complex network dynamics is a fundamental challenge in contemporary science with rich data, which can uncover the mysterious patterns and mechanisms of the formation and evolution of complex phenomena in various fields and assist in decision-making. In this work, we develop a universal computational tool that can automatically, efficiently, and accurately learn the symbolic changing patterns of complex system states by combining the excellent fitting ability from deep learning and the equation inference ability from pre-trained symbolic regression. We conduct intensive experimental verifications on more than ten representative scenarios from physics, biochemistry, ecology, epidemiology, etc. Results demonstrate the outstanding effectiveness and efficiency of our tool by comparing with the state-of-the-art symbolic regression techniques for network dynamics. The application to real-world systems including global epidemic transmission and pedestrian movements has verified its practical applicability. We believe that our tool can serve as a universal solution to dispel the fog of hidden mechanisms of changes in complex phenomena, advance toward interpretability, and inspire more scientific discoveries.