Research On Machine Learning In Nonlinear System Predictio

At present,the nonlinear science involves almost all fields of natural science and social science.Nonlinear effects make complex systems exhibit multiple modes of motion.Chaos is the typical behavior of nonlinear systems.The research on chaos phenomenon has been widely applied in many scientific fields such as physics,mathe-matics,earth science or life science.The investigation on the prediction and application of chaotic systems has become one of the most important frontier topics in nonlinear science.In this dissertation,the machine learning methods is employed to explore the prediction of chaotic systems.It has been shown that machine learning methods can carry out forecasting tasks based upon available time series.The reservoir computing adapted in this work has a lower computational cost among many machine learning methods.It has a good prediction effect especially for chaotic systems that are very difficult to predict.Firstly,the prediction of the phase of chaotic systems is studied.Previous studies on the model-free prediction of chaotic systems by machine learning focused on the time evolution of the dynamical variables of the system as a whole,which include both amplitude and phase.There are situations in the real world where the phase informa-tion is important,such as the ups and downs of species populations in ecology,the polarity of a voltage variable in an electronic circuit,and the concentration of certain chemical above or below the average.Using classic chaotic oscillators and a chaotic food-web system from ecology as examples,we demonstrate that reservoir computing can be exploited for long-term prediction of the phase of chaotic oscillators.The typ-ical prediction horizon can be the order of magnitude longer than that with predicting the entire variable,for which we provide a physical understanding.Secondly,we studied the prediction of phase coherence in coupled chaotic sys-tems.We found that sending data from two coupled oscillators to a single large reser-voir network,that is,a completely mixed integrated input scheme,has the ability to dis-tinguish different degrees of phase coherence(such as phase incoherence,partial and complete coherence capabilities).However,the parallel reservoir calculation scheme,i.e.,the independent input scheme,cannot correctly sense phase coherence.Although the parallel reservoir calculation scheme can be used for short-term prediction,the re-sults indicate that the scheme may be difficult to sense the collective dynamics of the entire system.Therefore,we demonstrate that a properly designed reservoir computing machine can reliably sense phase synchronization between a pair of coupled chaotic os-cillators with implications to the design of the parallel reservoir scheme for predicting large chaotic systems.As it is well known,the understanding on working mechanism of reservoir com-puting has always been a very important concern.Thus the last topic in this work is to investigate the problems related to ”the explainable machine learning”.Actually,to make the reservoir computer able to predict the state evolution of the chaotic system,the complexity of the former must ”overpower” that of the latter.The complexity of the target system to be predicted can be characterized by the information dimension of the chaotic attractor.Similarly,the complexity of a reservoir computer is determined by its ”internal” dynamic system,which is usually a complex network in a hidden layer.For a complex network,usually,its complexity increases with its size.We found that as the information dimension of the target chaotic system increases,the size of the reservoir network must increase accordingly to ensure its prediction efficiency.We uncover an exponential scaling law between the minimum network size and the infor-mation dimension of the target chaotic attractor.Moreover,we found that different network topologies will not affect this relationship.Therefore,in predicting chaotic systems using reservoir calculations,to make the network size larger than this critical value may enhance the possibility to success.In addition,our research can help re-duce the scope of the network in the process of hardware implementation.In this way,the network structure can be fixed,which is convenient for implementation,and the performance of the network can be ensured at the same time.The traditional method to predict the evolution of a chaotic system generally has only one or two Lyapunov time lengths,while the prediction length by the reservoir computing can reach at least five or six Lyapunov times.In many occasions,the math-ematical description of a system can not be accurately given,one can hardly make prediction.Now,one may accomplish prediction without using equations,but data only.This implies that,in the future,it may be possible to predict the weather through machine learning,rather than through complex atmospheric models.In addition to weather forecasts,machine learning methods can also predict,for example,the devel-opment of the epidemic,early warning of solar storms,and detection of arrhythmias.Therefore,our prediction of the phase of the chaotic system and the exploration of the understanding of the internal mechanism of the reservoir computing have important scientific significance.At the same time,it has also played a certain role in promoting the application and development of machine learning in the field of nonlinear science.