Echo state network implementation for chaotic time series prediction

The implementation of an Echo State Neural Network (ESNN) for chaotic time series prediction is introduced. First, the ESNN is simulated using floating-point arithmetic and afterwards fixed-point arithmetic. The synthesis of the ESNN is done in a field-programmable gate array (FPGA), in which the activation function of the neurons' outputs is a hyperbolic tangent one, and is approximated with a new design of quadratic order b-splines and four integer multipliers. The FPGA implementation of the ESNN is applied to predict four chaotic time series associated to the Lorenz, Chua, Lu, and Rossler chaotic oscillators. The experimental results show that with 50 hidden neurons, the fixed-point arithmetic is good enough when using 15 or 16 bits in the fractional part: using more bits does not reduce the mean-squared error prediction. The neurons are limited to four inputs in the hidden layer to achieve a more efficient hardware implementation, guaranteeing a prediction of more than 10 steps ahead.