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

TL;DR

This work investigates how to tailor reservoir computer nonlinearity to the intrinsic nonlinearity of data, using a minimal, deterministic RC framework with a tunable nonlinearity parameter. By introducing a fractional Halvorsen data generator and performing extensive minRC sweeps, the authors show that short-term predictions are maximized when the reservoir nonlinearity matches the data’s nonlinearity, and that the smallest data nonlinearity often dominates when multiple nonlinearities are present. They also demonstrate a practical method to estimate the minimal nonlinearity in unknown time series via nonlinearity sweeps and FT surrogate controls, validated on synthetic systems and financial data. Finally, the concept is transferred to classical RC by augmenting reservoir states with fractional nonlinearities, yielding improvements in hardware-limited scenarios where increasing reservoir size is impractical. Together, these results offer a principled route to design RCs that reflect the data’s complexity and to enhance RC performance with fractional nonlinear features in real-world, resource-constrained settings.

Abstract

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, 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 reservoir'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 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. Finally, 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 model.

Figures (11)

• Figure 1: The results of our grid search for the calculation of the largest Lyapunov exponent for different parameters of $a$ and $\xi_i$ are shown. For this plot all exponents of the fractional Halvorsen system are the same and fixed at $\xi_i$. White color inside the black boundary indicates a diverging trajectory for that parameter combination, while a white color outside the boundary indicates that the parameter combination has not been explored. For each parameter combination we simulate $50\,000$ steps and discard the first $10\,000$ steps as transient behavior. In total, we performed $44\,625$ experiments.
• Figure 2: The performance for different hyperparameters of minimal RCs in the classical setup containing all nonlinearities in the generalized states up to $\eta_\textrm{max}$ predicting the Lorenz system is shown. The performance of successful runs using the forecast horizon is measured in multiple of Lyapunov times. For each realization we use $1\,000$ data points for training, out of which 10 are used for synchronization, and a step size of $\Delta t=0.025$. Each tile shows the average performance of at least 35 realizations and in total we performed $98\,297$ experiments.
• Figure 3: The short and long-term performance of minimal RCs reproducing the Lorenz system is presented. We performed twenty experiments per parameter combination, so the upper plot reports the mean and standard deviation of those twenty runs, while the lower one shows the successful reproductions out of those. The green line shows the true nonlinearity of the Lorenz system. We show the results of $18\,240$ experiments.
• Figure 4: The mean relative forecast horizon for different nonlinear exponents $\eta$ of minimal RCs predicting the fractional Halvorsen system is shown. Each gray line represents the mean for a different $\xi_i$. For each parameter and exponent we perform seven runs and calculate the mean forecast horizon, which we normalize against the peak value. However, we omit the error bars in the interest of readability. We show the results of $62\,130$ experiments.
• Figure 5: This plot shows the absolute forecast horizon in for different exponents $\xi_i$ in data and $\eta$ in the model. It uses the same underlying data of the fractional Halvorsen system as setup in Fig. \ref{['fig:halvorsen-equal-xi-lines']}. Each tile shows the mean forecast horizon of five realizations and the green line indicates $\eta = \xi_i$.