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

We study how the degree of nonlinearity in the input data affects the optimal design of reservoir computers (RCs), 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 model. To provide controlled testbeds, we generalize to the fractional Halvorsen system, a novel chaotic system with fractional exponents. Our experiments reveal that the prediction performance is maximized when the model's nonlinearity matches the nonlinearity present in the data. In cases where multiple nonlinearities are present in the data, we 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 model 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. Additionally, we briefly study the SINDy framework as a complementary approach for identifying nonlinearities in data. Finally, we transfer these insights to classical RCs, 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 model.