Improving model-free prediction of chaotic dynamics by purifying the incomplete input

Despite the success of data-driven machine learning in forecasting complex nonlinear dynamics, predicting future evolution based on incomplete historical data remains challenging. Reservoir Computing (RC), a widely adopted approach, suffers from incomplete past observations since it typically requires complete data for accurate predictions. In this paper, a novel data processing scheme is introduced to improve the predictive performance of the RC when the input time series or dynamic trajectories are incomplete, for example, a portion of elements or states are randomly omitted or removed. It is a purification strategy, in which the input data are purified by selecting data or data sequences that are one step ahead of the segments of missing data. The selected data are positioned in turn in a new input, which is no longer indexed by the temporal order in the original time series. This approach matches the one-step-head nature of the convention RC and is thus very simple and efficient, without changing anything in the architecture of RC and avoiding sophisticated pretreatment on the incomplete input. It has been successfully employed to predict the chaotic dynamics in the Logistic map, Lorenz and Rössler systems, when the machine is trained by the purified input. The effect of the missing data on the predictive performance of the RC is also discussed. The results suggest that the purification of input can significantly improve its efficiency of predictive performance.