A general multi-order feature extractor for reservoir computing via simplicial complexes

The readout-only training mechanism establishes reservoir computing (RC) as a prominent lightweight prediction model, but simultaneously compromises its ability to effectively capture the specific higher-order interaction inherent to complex dynamical systems. To address this issue, a novel RC input layer inspired by simplicial complexes is proposed. As the direct interface to the input time series, this input layer acts as a potential multi-order feature extractor for explicitly and directly modeling the complex interactions within dynamical systems. Specifically, the novel input layer is initialized as a set of random simplices with varying dimensions, each of which is responsible for representing the interaction features of the corresponding order. Like its original counterpart, the presented input layer requires no training, thereby fully preserving the hallmark low training cost of RC. Furthermore, a causality-based quantification method is developed to measure the multi-order information richness of RC. Numerical experiments are then conducted to systematically analyze how the simplex distribution in the new input layer and key RC hyperparameters affect the quantified richness metrics. Finally, the proposed input layer can be extended to various RC variants, and its effectiveness in enhancing RCs’ prediction performance is validated through prediction tasks involving both chaotic systems and real-world datasets.