- source
- (Gauthier et al. 2021)
- tags
- Reservoir computing

## Summary

This paper bases itself on the demonstration that some reservoir computers (echo-state networks) are mathematically identical to nonlinear vector autoregression (NVAR) machines (Bollt 2021). A NVAR is just a regression over a feature vector composed of \(k\) time-delay observations of the dynamical system to be learned and nonlinear functions of these observations.

The authors introduce **Next-Generation Reservoir computing** (NG-RC) which is essentially a NVAR. Instead of the standard RC setup, they regress the outer layer over a vector

\[\mathbb{O}_{\text{total} }= c + \mathbb{O}_{\text{lin}} + \mathbb{O}_{\text{nonlin}}\]

where \(c\) is a constant, \(\mathbb{O}_{\text{lin}}\) is the vector of time delayed input observations and \(\mathbb{O}_{\text{nonlin}}\) is a vector of non-linear transformations of these observations.

Then, the output layer computes the NVAR output as a linear transformation of the feature vector, through \(\mathbf{Y}_i = \mathbf{W}_{\text{out}} \mathbb{O}_{\text{total}, i}\).

For a training dataset \(\mathbf{Y}_d\), output \(\mathbf{Y}\) is matched to it by solving a least-square linear regression problem.

The authors then evaluate the NG-RC model on two chaotic dynamical systems and show some performance scores and a few train/test trajectories comparisons.

## Comments

There doesn’t seem to be any comparison against standard RC, which is what this model is supposed to replace.

The benefits of reservoir computing probably lie beyond echo-state networks. With different reservoirs, one can harness the computations of complex non-linear dynamical systems which might not be possible using only recurrent neural networks.

## Bibliography

- Daniel J. Gauthier, Erik Bollt, Aaron Griffith, Wendson A. S. Barbosa. . "Next Generation Reservoir Computing".
*Nature Communications*12 (1). Nature Publishing Group:5564. DOI. - Erik Bollt. . "On Explaining the Surprising Success of Reservoir Computing Forecaster of Chaos? the Universal Machine Learning Dynamical System with Contrast to VAR and DMD".
*Chaos (woodbury, N.Y.)*31 (1):013108. DOI.