We study the discrete-time recurrent neural network that derived from the Leaky-integrator model and its application to compression of infra-red spectrum. Our results show that the discrete-time Leaky-integrator recurrent neural network (RNN) model can be used to approximate the continuous-time model and inherit its dynamical characters if a proper step size is chosen. Moreover, the discrete-time Leaky-integrator RNN model is absolutely stable. By developing the double discrete integral method and employing the state space search algorithm for the discrete-time recurrent neural network model, we demonstrate with quality spectra regenerated from the compressed data how to compress the infra-red spectrum effectively. The information we stored is the parameters of the system and its initial states. The method offers an ideal setting to carry out the recurrent neural network approach to chaotic cases of data compression.