Dynamical Anatomy of NARMA10 Benchmark Task

The emulation task of a nonlinear autoregressive moving average model, i.e., the NARMA10 task, has been widely used as a benchmark task for recurrent neural networks, especially in reservoir computing. However, the type and quantity of computational capabilities required to emulate the NARMA10 model remain unclear, and, to date, the NARMA10 task has been utilized blindly. Therefore, in this study, we have investigated the properties of the NARMA10 model from a dynamical system perspective. We revealed its bifurcation structure and basin of attraction, as well as the system's Lyapunov spectra. Furthermore, we have analyzed the computational capabilities required to emulate the NARMA10 model by decomposing it into multiple combinations of orthogonal nonlinear polynomials using Legendre polynomials, and we directly evaluated its information processing capacity together with its dependences on some system parameters. The result demonstrates that the NARMA10 model contains an unstable region in the phase space that makes the system diverge according to the selection of the input range and initial conditions. Furthermore, the information processing capacity of the model varies according to the input range. These properties prevent safe application of this model and fair comparisons among experiments, which are unfavorable for a benchmark task. As a result, we propose a benchmark model that can clearly evaluate equivalent computational capacity using NARMA10. Compared to the original NARMA10 model, the proposed model is highly stable and robust against the input range settings.