Effective Training Principles of Physical Reservoirs

This work addresses a hardware constraint in reservoir computing: the limited size of the readout layer imposed by systems with a physical readout. We investigate a strategy to accommodate this constraint based on random projection, which compresses high-dimensional reservoir states into a lower-dimensional subspace while preserving key properties of the source space and information- processing capabilities. To evaluate this approach, we compare a small, standalone time delay reservoir against a larger configuration whose output is projected down to match the same restricted readout dimension. Using task-independent metrics, we demonstrate that the distribution of information-processing capacities may differ between the two configurations, even at identical readout sizes. Furthermore, we perform a comprehensive hyperparameter scan to assess how both systems behave under varying physical regimes. Finally, we benchmark this approach on the standard NARMA10 task, showing that the random projection framework can yield superior performance compared to a standalone constrained reservoir, within specific compression range. These results provide a scalable pathway to bypass physical readout bottlenecks in hardware-based reservoir computing.

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.

Reservoir computing has emerged as an efficient machine learning framework for predicting time series generated by dynamical systems. In contrast to other machine and deep learning approaches, a reservoir computing trains only the output layer via linear regression, leaving the reservoir (recurrent layer) untrained. This simplification makes reservoir computers easier to train and more amenable to experimentation. However, because current reservoirs consist of networks of randomly connected nodes and require the optimization of numerous hyperparameters, a framework that precisely explains how reservoir computing operates and how it can be optimized remains missing. Here, we propose a frequency-based reservoir inspired by the brain's oscillatory dynamics and its hierarchy of timescales. The frequency-based reservoir can be interpreted as an ensemble of independent oscillatory units, each processing a portion of the input's frequency content. This allows us to understand the reservoir's internal behavior by modeling it as a single unit driven by an external input. Borrowing from the theory of a nonlinear oscillator forced by complex periodic inputs, we found that units of the frequency-based reservoir selectively amplify and store specific input frequencies, which are then used for prediction. The frequency-based reservoir performs as well as or better than equivalent random reservoirs. Furthermore, the frequency-based approach can be optimized to improve short-term prediction, a property that random reservoirs lack. Finally, we show that the frequency-based reservoir can also predict complex spatiotemporal dynamics. Our results show that reservoir computing can be designed using brain properties and theoretical insights borrowed from the physics of forced nonlinear oscillators.