Dynamical Analysis Of Few Nodes Reservoir Computing

Learning system dynamics from time series is one of the important methods to investigate complex systems.Recurrent neural network(RNN)is an effective and widely used method for learning complex systems.However,the training of RNN is usually very difficult and computational demanding due to the explosion or disappearance of gradients in the recursive structure.Compared with recurrent neural network,reservoir computing(RC)network has simple structure,fast learning speed,low computing cost,and is useful for learning and forecasting time series.But the success and power of RC lack systematic research and explanation.This thesis considers the influence of parameters on the learning effect of uncoupled RC through model building,simulation and theoretical analysis.In the first part of this paper,we consider RC learning the periodic sequence of Logistic map,and RC’s learning process is analyzed as a nonlinear dynamic procedure to study the influence of parameters on the RC learning effect.RC shows rich dynamical features in the learning process.In particular,even for the simple periodic sequence,RC cannot learn well,when the parameter selection is inappropriate.And even for complex chaotic sequences,the simplest single-node reservoir computing can predict about 20 steps under certain parameters.By analyzing stability on the single-node RC’s self-feedback equation,we can know that the single node RC can accurately learn the Logistic map of period Ⅰ,period Ⅱ and period Ⅲ,and the corresponding parameters can be given analytically,which are consistent with the simulation results.Increasing the number of nodes in the reservoir can learn more complex Logistic map sequence.When multi-node RC learn Logistic map of periodic sequence,we can select the optimal parameters of each node to achieve the optimal learning ability of the RC,and further verify the correctness of the theory.The second part of this paper expands the research of the first part,and explore RC learning the chaotic sequence of Logistic map.When the single-node RC learns the Logistic chaotic sequence,the corresponding time sequence is completely different,even both the output sequence is chaotic.When the single-node RC learns the Logistic chaotic sequence,even if the chaotic sequence is output,the corresponding time sequence is completely different.By studying the Logistic map dynamics form,similar dynamics parameter index is set to explore the relationship between output and input sequence.Scanning the parameter area,the linear relationship of the corresponding parameters win,b are fitted by numerical when the learning effect is better.With the increased RC size,the region of appropriate parameters for correct states increases rapidly,and the analysis results are consistent with the simulations.When the number of nodes in the reservoir increases to 6,the average of the transient Lyapunov index of the output sequence is the same as the mean of the input chaotic sequence.Continued to increase the reservoir nodes,the dynamic stability of the reservoir computing system tends to be saturated.In this paper,from single to few nodes,are considered for learning various states of logistic maps.The learning process is abstracted as a nonlinear dynamic process to study the influence of parameters on RC learning effect.In addition to focusing on the corresponding parameter region when the system learning effect is good,it also shows that when the learning effect is bad,the system is unstable under the corresponding parameters.For the few-node RC,the relationship between the parameters of reservoir nodes and the learning ability is studied,and the transient Lyapuov index of the output sequence of the 6-node RC is the same as the input sequence.The dynamical analysis of reservoir computing provides the basis for its success in learning and predicting various dynamical behaviors.