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Learning dynamical systems by recurrent neural networks from orbits.

M Kimura¹, R Nakano

¹NTT Communication Science Laboratories, 2-4 Hikaridai, Seika-cho, Kyoto, Japan

Neural Networks : the Official Journal of the International Neural Network Society

|March 29, 2003

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This study shows recurrent neural networks (RNNs) can approximate dynamical systems (DSs). Affine neural dynamical systems (A-NDSs) offer a method for RNNs to precisely approximate any DS, aiding generalization.

Area of Science:

Dynamical Systems Theory
Machine Learning
Recurrent Neural Networks

Background:

Recurrent Neural Networks (RNNs) are explored for approximating dynamical systems (DSs).
The challenge lies in RNNs producing and identifying DSs from observed data.

Purpose of the Study:

To investigate RNNs' capability to approximate dynamical systems.
To explore the generalization problem in RNNs for DS approximation.
To determine if RNN-generated DSs can be identified from observed orbits.

Main Methods:

Proving RNNs without hidden units uniquely produce specific DS classes.
Introducing Neural Dynamical Systems (NDSs) for RNNs with hidden units.
Developing Affine Neural Dynamical Systems (A-NDSs) as a concrete NDS example.

Related Experiment Videos

Proving any DS can be finitely approximated by A-NDSs.

Main Results:

RNNs without hidden units uniquely generate certain DSs.
Any dynamical system can be approximated to any precision by an Affine Neural Dynamical System (A-NDS).
A geometric criterion for RNN generalization is derived and extended to networks with hidden units.

Conclusions:

Affine Neural Dynamical Systems (A-NDSs) are proposed as the practical means for RNNs to approximate target dynamical systems.
The derived geometric criterion facilitates the generalization of RNNs in learning dynamical systems.