Hierarchical delay-memory echo state network: A model designed for multi-step chaotic time series prediction

Abstract Predicting for long-term dynamics of complex systems from observations is a challenging topic in the field of time series modeling and analysis, and is continually under research. Noteworthily, multi-step prediction requires accurate learning of dynamics and correlations between historical data for predicting future behavior. In this paper, we proposed a modified recurrent neural network named hierarchical delay-memory echo state network (HDESN) for solving the task of multi-step chaotic time series prediction. The HDESN uses multiple reservoirs with delay-memory capabilities, which can simultaneously discover and explore the information of short-term and long-term memory hidden in the historical sequence, and extract the valuable evolution patterns through deep topology and hierarchical processing. Moreover, to ensure high-quality prediction results and reduce the computational burden as much as possible, we further design a phase-space representation strategy which can calculate a compact topology and delay-memory coefficient according to the chaotic characteristics of the data. Compared with other improved ESN-based models, the proposed HDESN does not have a larger memory capacity to capture potential evolution law hidden in the complex system layer by layer, but can also adaptively determine a suitable network architecture to reflect the mapping relations in chaotic phase space. The experimental results on two benchmark chaotic systems and a real-world meteorological dataset demonstrate that the proposed HDESN model obtains satisfactory performance in multi-step chaotic time series prediction.