Equivalence of machine learning models in modeling chaos

Recent advances have demonstrated that machine learning models are effective methods for predicting chaotic systems. Although short-term chaos prediction can be successfully realized by seemingly different machine learning models, an intriguing question of their correlation is still unknown. Here, we focus on three commonly used machine learning models that are reservoir computing, long-short term memory networks, and deep belief networks, respectively. We find that these selected models present almost identical long-term statistical properties as that of a learned chaotic system. Specifically, we show that these machine learning models have the same correlation dimension and recurrence time. Furthermore, by sharing a common signal, we realize synchronization, cascading synchronization, and coupled synchronization among machine learning models. Our findings reveal the equivalence of machine learning models in characterizing and modeling chaotic systems.