Discovering governing equations of complex network dynamics is a fundamental challenge in contemporary science with rich data, which can uncover the hidden 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 patterns of changes in complex system states by combining the excellent fitting capability of deep learning with the equation inference ability of pre-trained symbolic regression. We perform extensive and intensive experimental verifications on more than ten representative scenarios from fields such as physics, biochemistry, ecology, and epidemiology. The results demonstrate the remarkable effectiveness and efficiency of our tool compared to 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 further scientific discoveries.