Complex network perspective on modelling chaotic systems via machine learning?

Tong-Feng Weng(翁同峰) Xin-Xin Cao(曹欣欣) and Hui-Jie Yang(楊會(huì)杰)

1Institute of Information Economy and Alibaba Business College,Hangzhou Normal University,Hangzhou 311121,China

2College of Science,University of Shanghai for Science and Technology,Shanghai 200093,China

3Business School,University of Shanghai for Science and Technology,Shanghai 200093,China

Keywords: reservoir computing approach,complex networks,chaotic systems

Recently,machine learning technique,as a critical branch of artificial intelligence,has attracted increasing attention.[1,2]A great variety of machine learning methods, such as Bayes learning,[3]neural networks,[2]and decision trees,[4]are proposed for dealing with the problems related to artificial intelligence. Among them, an intriguing one named reservoir computing approach has received considerable attention in time series domain.[5,6]A growing number of studies have demonstrated that this approach is competent for forecasting low-dimensional chaotic systems,[6]inferring unmeasured variables,[7]and even predicting spatiotemporally chaotic systems.[8]In this sense, reservoir computing approach provides an effective way for modelling and characterizing chaotic systems.

Beyond short-term prediction, there is a growing industry in revealing long-term behaviors of the trained reservoir system, for example, Lyapunov exponents,[9]correlation dimension,[10,11]and even attractor.[12]However,these works mainly study a trained reservoir system from a dynamical angle, while a wide range of statistics in complex network domain have been overlooked. In fact, complex network theory provides a new paradigm to understand and characterize dynamical systems.[13–15]It offers a range of powerful tools to describe a great variety of nonlinear systems. For example,recurrence networks in terms of recurrence allow us to classify dynamics and to detect dynamical transitions.[14]Therefore,network theory will bring us a new perspective on study longterm behaviors of the trained reservoir system in modelling chaotic systems.

In this paper, we study long-term behaviors of a trained reservoir system by virtue of network measurements for which we transform its prediction trajectory into recurrence networks. We find that a great variety of network statistics,such as degree distribution,the clustering coefficient,and the mixing pattern,induced for the trained reservoir system are almost the same as that of a considered chaotic system. Remarkably,we show that when learning an observed system in distinct dynamical regimes,their resultant network structures are consequently different. These distinctions can appropriately detect and identify the dynamical transitions in the learned system.Numerical results on two benchmark chaotic systems(i.e.,the R¨ossler system and the H′enon map)further support our findings.

We begin with introducing the basic framework of reservoir computing(RC)approach.Its architecture is usually composed of three components: an input layer coupled with an input vectoru(t),a reservoir network in the middle layer consisting ofNnodes,and an output layer coupled with an output vectory(t),as illustrated in Fig.1(a).Here,we follow Jaeger’s design and define the evolving equations of the reservoir vectorrand the output vectoryas follows:[6]

where I is an identity matrix andλis the ridge regresion parameter. Thek-th columns of the matricesXandYare[bout;s(k);r(k)] ands(k+1), respectively. After training stage,wheny(t+1)is adopted asu(t+1),the reservoir computing system can run autonomously based on Eqs. (1) and(2).

Fig.1. (a)Schematic illustration of a reservoir computer with the input vector u(t)and the output vector y(t).(b)The prediction of the trained reservoir computer and the actual trajectories of the R¨ossler system.

We first apply the reservoir computing approach to the R¨ossler system in the chaotic regime given by

We calculate a numerical solution for this system based on the fourth-order Runge–Kutta method and obtain 2×104points with time step ?t=0.1. For eliminating transient states, we discard the leading 5000 observations. We then use the first 2600 points with the input vectoru= (x,y,z) for training.Here, we choose the reservoir parametersn=2600,α=0.5,N=500,andλ=1×10?8. It is shown that the trained reservoir system produces short-term prediction correctly, as described in Fig.1(b). Since the sensitive to the initial condition of the R¨ossler system, the prediction data gradually deviates from the actual R¨ossler trajectory, as expected. Nonetheless,we notice that aftert ≈110,the long-term profile of the reservoir system resemble that of the R¨ossler system. This phenomenon hints that besides predicting short-term dynamics,the long-term behavior of a considered system seems to be consequently captured in the trained reservoir system.

We now explore the long-term behavior of a trained RC model from complex network perspective for which a wide range of network metrics can be employed. In fact, network science has recently been applied to understand and characterize chaotic systems of interest.[14,20]A number of approaches for transforming time series into networks have been reported,such as cycle network,[13]visibility graph,[21]and recurrence network.[14,22]Here,we adopt the recurrence method for mapping time series of chaotic systems and their learned RC models into networks. In particular, for a time series given by{Si}Ni=1,the transformed network is

whereΘ(·) is the Heaviside function,εis a threshold value,andδijis the Kronecker delta. Empirically, we choose a sufficiently small threshold value for which the resultant recurrence network is connected.[14]In the constructed network,we observe its local, intermediate, and global network properties in terms of degree distributionP(k),the clustering coefficientC,the distribution of shortest path lengthP(d)and the assortativity coefficientr, respectively. We then apply these network measurements to study the previous R¨ossler system and the corresponding RC model. Here, we select the threshold valueε=0.13 and the length of time seriesl=5000. Interestingly,we find that the degree distributionP(k)of the RC model presents the similar profile as that of the R¨ossler system,see Fig.2(a).This phenomenon also occurs on the distribution of shortest path length,as illustrated in Fig.2(b).These results reveal that the topological feature of an observed chaotic system is preserved in the trained RC model. This is further supported by observing the average clustering coefficient and the assortativity coefficient for whichC=0.619 andr=0.841 for the R¨ossler system,whileC=0.624 andr=0.829 for the RC model. Remarkably,when examing the spatial distribution of the degree in(x,y)plane,they present almost identical pattern,see Figs.2(c)and 2(d). Clearly,high values ofklie in the central region,and low values almost filled the other region. Our findings reveal that the RC model is identical with a chaotic system of interest from complex network perspective.

Fig. 2. Comparison of the R¨ossler system and its learned RC model with respect to different network statistics: (a)the degree distribution and(b)the distribution of shortest path length. Colour-coded representation of the degree k in the (x,y) plane for (c) the R¨ossler system and (d) the reservoir computer model.

We further confirm this interesting finding on the H′enon map given by

wherea=1.4 andb=0.3. We generate the 2×104observations from this map and use the first 2600 points with the inputu=(x,y)for training. Here,we set the reservoir parametersn=2600,α=0.25,N=500, andλ=1×10?8. After the training stage,we generate a trajectory of lengthl=5000 of this reservoir system and transform it into recurrence network. Here, we choose the threshold valueε=0.03. Figures 3(a) and 3(b) show the degree distribution and the distribution of shortest path length for the H′enon map and the RC model, respectively. Clearly, it is shown that profiles ofP(k)versus kfor the H′enon map and the RC model present a similar tendency. This phenomenon is also established on the distribution of shortest path length. Moreover,we further observe that the spatial distribution of the degree in phase space for the RC model is almost identical with that for the H′enon map, see Figs. 3(c) and 3(d). These findings further confirm that beside prediction, the trained reservoir system captures long-term behavior of an observed chaotic system in terms of network statistics.

Finally, we show that the long-term behaviors of the trained RC model are different when learning distinct dynamical systems and these distinctions can identify the dynamical transitions of the complex system of interest. Here, we take the previous H′enon map as a benchmark example. We selectb=0.3 anda ∈[1,1.4]with a step size ?a=0.005. With the increase ofa,the H′enon map undergoes from period-doubling route to chaos. For everyahere, we record 2×104successive values after discarding the leading 5000 data(to eliminate transient states). The bifurcation diagram gives an intuitive feeling of the dynamical transition of the H′enon map, see the top panel of Fig. 4(a). For each record, we use the first 2600 points with the inputu=(x,y),α=0.25,andN=500 for training the RC model. After training stage, we generate 1×104data points from each trained RC system. Interestingly,we show that the trained RC systems can reproduce the bifurcation diagram of the H′enon map exactly, as illustrated in Fig.4(a). We then calculate the clustering coefficientsCof recurrence networks constructed from the H′enon map and the RC model. We find that they present an identical tendency,see Fig. 4(b). Meanwhile, we notice that they are sensitive to the presence of dynamical transitions of H′enon map indicating by the largest Lyapunov exponentλmax. Here, we calculate the largest Lyapunov exponent using the TISEAN software package.[23]Specifically, the maximal values ofC(i.e.,C=1)are calculated from the periodic regime,whereas the chaotic behavior results in a relatively smaller value.These results reveal that the topological feature of the trained RC model is not only the same with that of its learned system, but also can be used to dicriminate different dynamical regimes. This is further supported by observing the mean degree〈k〉and the assortativity coefficientr, where they match almost exactly between the H′enon map and the trained RC model, as illustrated in Figs. 4(c) and 4(d). Our findings uncover that from a complex network perspective,the RC model is indistinguishable from that of an observed chaotic system.

Fig.3. Comparison of the H′enon map and its learned RC model with respect to different network statistics: (a)degree distribution and(b)distribution of shortest path length. Colour-coded representation of the degree k in phase space for(c)the H′enon map and(d)the reservoir computer model.

Fig. 4. (a) The bifurcation diagrams of the H′enon map (top panel) and the associated RC model(bottom panel). (b)The maximum Lyapunov exponent λmax and the clustering coefficient C versus the parameter a. (c) The mean degree 〈k〉 and (d) the assortativity coefficient r of recurrence networks obtained with different a.

In summary,we studied the reservoir computing approach for modelling chaotic systems from a complex network perspective. By transforming their trajectories into recurrence networks, we find that a great variety of network measurements, such as degree distribution, the clustering coefficient,and the assortativity coefficient are almost identical between the trained reservoir system and its learned chaotic system of interest. Remarkably,we show that some statistics commonly used in network science generated from the RC model are sensitive to dynamical transitions and can be in turn used to detect dynamical changes in chaotic systems. Our findings are confirmed on two classical dynamical systems. Our work reveals that from a complex network perspective,reservoir computing approach provides an alternative way for modelling chaotic systems rather than conventional dynamical equations. Moreover, for convenience, the transformation method we have considered here is the recurrence network. The investigation of a broad range of transformation methods, for example cycle network,[13,24]and ordinal partition network,[25]calls for additional research effects.