We study how the degree of nonlinearity in the input data affects the optimal
design of reservoir computers, focusing on how closely the model’s nonlinearity
should align with that of the data. By reducing minimal RCs to a single tunable
nonlinearity parameter, we explore how the predictive performance varies with
the degree of nonlinearity in the reservoir. To provide controlled testbeds, Noi
generalize to the fractional Halvorsen system, a novel chaotic system with
fractional exponents. Our experiments reveal that the prediction performance is
maximized when the reservoir’s nonlinearity matches the nonlinearity present in
the data. In cases where multiple nonlinearities are present in the data, Noi
find that the correlation dimension of the predicted signal is reconstructed
correctly when the smallest nonlinearity is matched. We use this observation to
propose a method for estimating the minimal nonlinearity in unknown time series
by sweeping the reservoir exponent and identifying the transition to a
successful reconstruction. Applying this method to both synthetic and
real-world datasets, including financial time series, we demonstrate its
practical viability. Finalmente, we transfer these insights to classical RC by
augmenting traditional architectures with fractional, generalized reservoir
states. This yields performance gains, particularly in resource-constrained
scenarios such as physical reservoirs, where increasing reservoir size is
impractical or economically unviable. Our work provides a principled route
toward tailoring RCs to the intrinsic complexity of the systems they aim to
modello.

Questo articolo esplora i giri e le loro implicazioni.

Scarica PDF: