Good and bad predictions: Assessing and improving the replication of chaotic attractors by means of reservoir computing

Abstract

The prediction of complex nonlinear dynamical systems with the help of machine learning techniques has become increasingly popular. In particular, reservoir computing turned out to be a very promising approach especially for the reproduction of the long-term properties of a nonlinear system. Yet, a thorough statistical analysis of the forecast results is missing. Using the Lorenz and Rössler system we statistically analyze the quality of prediction for different parametrizations -both the exact short-term prediction as well as the reproduction of the long-term properties (the "climate") of the system as estimated by the correlation dimension and largest Lyapunov exponent. We find that both short and longterm predictions vary significantly among the realizations. Thus special care must be taken in selecting the good predictions as predictions which deliver better short-term prediction also tend to better resemble the long-term climate of the system. Instead of only using purely random Erdös-Renyi networks we also investigate the benefit of alternative network topologies such as small world or scale-free networks and show which effect they have on the prediction quality. Our results suggest that the overall performance with respect to the reproduction of the climate of both the Lorenz and Rössler system is worst for scale-free networks. For the Lorenz system there seems to be a slight benefit of using small world networks while for the Rössler system small world and Erdös-Renyi networks performed equivalently well. In general the observation is that reservoir computing works for all network topologies investigated here.

Forecasting chaotic systems with very low connectivity reservoir computers

“…This method of evaluating forecasting ability is flawed for our purposes. Previous results have shown that a low ε 1 is not a good indicator of whether a reservoir computer has learned the climate of a system 3,4 and we also observe the same effect here. Figure 4 depicts two common ways for an RC to fail to replicate the true Lorenz attractor, shown in Fig.…”

“…More recently, RCs were used to learn the climate of a chaotic system; 3,4 that is, it learns the long-term features of the system, such as the system's attractor. Reservoir computers have also been realized physically as networks of autonomous logic on an FPGA 5 or as optical feedback systems, 6 both of which can perform chaotic system forecasting at a very high rate.…”

“…Commonly, the reservoir is created as a network of interacting nodes with a random topology. Many types of topologies have been investigated, from Erdös-Rényi networks and small-world networks 4 to simpler cycle and line networks. 7 Optimizing the RC performance for a specific task is accomplished by adjusting some large-scale network properties, known as hyperparameters, while constraining others.…”

Predictability of Power Grid Frequency

“…Secondly, additional features such as wind power generation can be included to better understand the impact of unforeseen pertubations, which are not captured in our univariate forecast. Thirdly, many alternative forecasting methods are available, from artificial neural networks (ANN) [11] and recurrent neural networks (RNN) [35] to classical methods of time series prediction [32]. However, a fully comprehensive review of all available methods was beyond the scope of this study and will be left for the future.…”

Predicting bursting in a complete graph of mixed population through reservoir computing

“…The readout or output layer is the only part where the weights are trained to produce a desired output which should be closer to the target data. Researchers have also devoted to find the optimal parameters of an ESN for accurate detection of target data [13,[19][20][21][22][23][24][25].…”