Deterministic Reservoir Computing for Chaotic Time Series Prediction

TL;DR

This work tackles time-series prediction by replacing random reservoir mappings with deterministic high-dimensional mappings in reservoir computing. It introduces two deterministic variants, TCRC-LM (Logistic Mapping) and TCRC-CM (Chebyshev Mapping), and a non-linear Lobachevsky activation to enhance predictive power, especially for chaotic sequences such as Mackey-Glass. Across non-chaotic and chaotic regimes, the proposed DM approaches often outperform traditional ESNs, with notable gains in chaotic cases when using the lambda activation. While the methods incur higher single-run computation, they avoid the variability and repetition of randomized mappings, offering a deterministic and competitive alternative for high-fidelity time-series forecasting.

Abstract

Reservoir Computing was shown in recent years to be useful as efficient to learn networks in the field of time series tasks. Their randomized initialization, a computational benefit, results in drawbacks in theoretical analysis of large random graphs, because of which deterministic variations are an still open field of research. Building upon Next-Gen Reservoir Computing and the Temporal Convolution Derived Reservoir Computing, we propose a deterministic alternative to the higher-dimensional mapping therein, TCRC-LM and TCRC-CM, utilizing the parametrized but deterministic Logistic mapping and Chebyshev maps. To further enhance the predictive capabilities in the task of time series forecasting, we propose the novel utilization of the Lobachevsky function as non-linear activation function. As a result, we observe a new, fully deterministic network being able to outperform TCRCs and classical Reservoir Computing in the form of the prominent Echo State Networks by up to $99.99\%$ for the non-chaotic time series and $87.13\%$ for the chaotic ones.

Figures (5)

• Figure 1: TCRC architecture; black arrows ($\rightarrow$) refer to the multiplied tokens, red arrows ($\rightarrow$) refer to the learned mapping from the state space of each layer ${}_ls^{(t)}$ to the output $\hat{y}^{(t)}$.
• Figure 2: Comparative depiction of state space size $N^{\textrm{res}}$ of Next-Gen RC with the non-linear function utilized by gauthier2021next and $N^{\textrm{tc}}$ of TCRC.
• Figure 3: Exemplary TCRC-ELM architecture; black arrows ($\rightarrow$) refer to the multiplied tokens, green arrows ($\rightarrow$) refer to the randomly drawn weights of $W^{\mathrm{in}}$, red arrows ($\rightarrow$) refer to the learned mapping from the state space $\hat{s}^{(t)}$ to the output $\hat{y}^{(t)}$.
• Figure 4: Exemplary visualizations of the values of the deterministic mappings.
• Figure 5: Plot of the activated value for inputs in the range $[-10,10]$.