Stable training method for echo state networks with output feedbacks

In applications of echo state network (ESN), the Wiener-Hopf solution is usually used to learn the ESN´s output connection weights; however, the solution can hardly ensure the asymptotic stability of the ESNs running in a closed-loop generative mode. The reason is firstly analyzed. A sufficient condition of the asymptotic stability for the closed-loop running ESNs is proposed and proved. In addition, the output connection weight learning problem is translated into an optimization problem with a nonlinear restriction. Particle swarm optimization algorithm is explored to solve the optimization problem. The simulation experiment results show that the output weight adaptation alglrithm we proposed (we call it PSOESN) can not only result in the high-precision prediction outputs of the trained ESN, but also ensure its asymptotic stability.