Dynamic behaviors of distribution networks are of great importance for the power system analysis. Nowadays, due to the integration of the renewable energy generation, energy storage, plug-in electric vehicles, and distribution networks turn from passive systems to active ones. Hence, the dynamic behaviors of active distribution networks (ADNs) are much more complex than the traditional ones. The research interests how to establish an accurate model of ADNs in modern power systems are drawing a great deal of attention. In this paper, motivated by the similarities between power system differential algebraic equations and the forward calculation flows of recurrent neural networks (RNNs), a long short-term memory (LSTM) RNN-based equivalent model is proposed to accurately represent the ADNs. First, the adoption reasons of the proposed LSTM RNN-based equivalent model are explained, and its advantages are analyzed from the mathematical point of view. Then, the accuracy and generalization performance of the proposed model is evaluated using the IEEE 39-Bus New England system integrated with ADNs in the study cases. It reveals that the proposed LSTM RNN-based equivalent model has a generalization capability to capture the dynamic behaviors of ADNs with high accuracy.