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We study how the degree of nonlinearity in the input data affects the optimaldesign of reservoir computers, focusing on how closely the model's nonlinearityshould align with that of the data. By reducing minimal RCs to a single tunablenonlinearity parameter, we explore how the predictive performance varies withthe degree of nonlinearity in the reservoir. To provide controlled testbeds, wegeneralize to the fractional Halvorsen system, a novel chaotic system withfractional exponents. Our experiments reveal that the prediction performance ismaximized when the reservoir's nonlinearity matches the nonlinearity present inthe data. In cases where multiple nonlinearities are present in the data, wefind that the correlation dimension of the predicted signal is reconstructedcorrectly when the smallest nonlinearity is matched. We use this observation topropose a method for estimating the minimal nonlinearity in unknown time seriesby sweeping the reservoir exponent and identifying the transition to asuccessful reconstruction. Applying this method to both synthetic andreal-world datasets, including financial time series, we demonstrate itspractical viability. Finally, we transfer these insights to classical RC byaugmenting traditional architectures with fractional, generalized reservoirstates. This yields performance gains, particularly in resource-constrainedscenarios such as physical reservoirs, where increasing reservoir size isimpractical or economically unviable. Our work provides a principled routetoward tailoring RCs to the intrinsic complexity of the systems they aim tomodel.

Tailored minimal reservoir computing: on the bidirectional connection between nonlinearities in the reservoir and in data