RESUMO
Reservoir computing (RC) has been widely applied to predict the chaotic dynamics in many systems. Yet much broader areas related to nonsmooth dynamics have seldom been touched by the RC community which have great theoretical and practical importance. The generalization of RC to this kind of system is reported in this paper. The numerical work shows that the conventional RC with a hyperbolic tangent activation function is not able to predict the dynamics of nonsmooth systems very well, especially when reconstructing attractors (long-term prediction). A nonsmooth activation function with a piecewise nature is proposed. A kind of physics-informed RC scheme is established based on this activation function. The feasibility of this scheme has been proven by its successful application to the predictions of the short- and long-term (reconstructing chaotic attractor) dynamics of four nonsmooth systems with different complexity, including the tent map, piecewise linear map with a gap, both noninvertible and discontinuous compound circle maps, and Lozi map. The results show that RC with the new activation function is efficient and easy to run. It can make perfectly both short- and long-term predictions. The precision of reconstructing attractors depends on their complexity. This work reveals that, to make efficient predictions, the activation function of an RC approach should match the smooth or nonsmooth nature of the dynamical systems.
RESUMO
Predicting future evolution based on incomplete information of the past is still a challenge even though data-driven machine learning approaches have been successfully applied to forecast complex nonlinear dynamics. The widely adopted reservoir computing (RC) can hardly deal with this since it usually requires complete observations of the past. In this paper, a scheme of RC with (D+1)-dimension input and output (I/O) vectors is proposed to solve this problem, i.e., the incomplete input time series or dynamical trajectories of a system, in which certain portion of states are randomly removed. In this scheme, the I/O vectors coupled to the reservoir are changed to (D+1)-dimension, where the first D dimensions store the state vector as in the conventional RC, and the additional dimension is the corresponding time interval. We have successfully applied this approach to predict the future evolution of the logistic map and Lorenz, Rössler, and Kuramoto-Sivashinsky systems, where the inputs are the dynamical trajectories with missing data. The dropoff rate dependence of the valid prediction time (VPT) is analyzed. The results show that it can make forecasting with much longer VPT when the dropoff rate θ is lower. The reason for the failure at high θ is analyzed. The predictability of our RC is determined by the complexity of the dynamical systems involved. The more complex they are, the more difficult they are to predict. Perfect reconstructions of chaotic attractors are observed. This scheme is a pretty good generalization to RC and can treat input time series with regular and irregular time intervals. It is easy to use since it does not change the basic architecture of conventional RC. Furthermore, it can make multistep-ahead prediction just by changing the time interval in the output vector into a desired value, which is superior to conventional RC that can only do one-step-ahead forecasting based on complete regular input data.
