Predicting dynamical systems with echo state networks with different topology

The prediction of complex nonlinear dynamical systems with the help of machine learning techniques has become increasingly popular. In particular, the so-called echo state networks (ESN) turned out to be a very promising approach especially for the reproduction of the long-term properties of the system. The heart of ESN is a network of nodes that is fed with input data and is connected with an output layer. So far only random Erdös-Renyi networks are used. However, there is a variety of conceivable other network topologies that may have an influence on the results. As a first step, we statistically analyze the quality of prediction - both the exact short term prediction as well as the reproduction of the long-term properties of the system as estimated by the correlation dimension and largest Lyapunov exponent for random, small world and scale-free networks. Using the Lorenz and the Rössler system we find significant variations of the results. The longterm prediction is worse for the scale-free network, where the differences between the network types are more pronounced in the Rössler system. Random and scale-free networks perform similar in both cases with slight advantages for the small world network. Our results suggest that the network topology has significant influence on the performance of ESN. Studying the results for different networks in detail also gives new insights about the complexity of the underlying dynamical system