Dynamical systems produced by recurrent neural networks
Masahiro Kimura, Ryohei Nakano · Systems and Computers in Japan · 2000
Concerning the learning problems of recurrent neural networks (RNNs), this paper deals with the problem of approximating a dynamical system (DS) by an RNN as one extension of the problem of approximating trajectories by an RNN. In particular, we systematically investigate how an RNN can produce a DS on the visible state space to approximate a target DS. First, it is proved that RNNs without hidden units uniquely produce a certain class of DSs. Next, a neural dynamical system (NDS) is proposed as such a DS that an RNN with hidden units can produce on the visible state space, and affine neural dynamical systems (A-NDSs) are constructed as concrete examples of NDSs. Moreover, we prove that any DS on a Euclidean space can be finitely approximated by some A-NDS with any precision, and propose adopting an A-NDS as such a DS that an RNN with hidden units produces to approximate a target DS. © 2000 Scripta Technica, Syst Comp Jpn, 31(4): 77–86, 2000