Learning hidden dynamics using echo state network

In nonlinear dynamics, training a model using data from system determined by a few parameter values or initial values to predict the hidden dynamics under different parameter values or initial values is a significant issue. Echo state network is a specialized type of recurrent neural network extensively employed for dynamics prediction. However, addressing this issue remains inherently challenging with the original echo state network. This study introduces local information flow echo state network (LIF-ESN) to overcome these challenges. LIF-ESN relies on local input information and emphasizes the significance of the reservoir initial state. Based on LIF-ESN, a parameter-aware reservoir is constructed to predict dynamics under different parameter values. Furthermore, we propose an initial value-aware scheme and integrate it into LIF-ESN to predict dynamics under different initial values. Three numerical experiments, including accurate prediction of evolutionary behavior on a second-order delay differential equation and a fourth-order differential equation, as well as attraction basin prediction of the singularity in a three-dimensional dynamical system, demonstrate the effectiveness of the proposed method.