Dynamics of reservoir computing for crises prediction

TL;DR

The paper investigates how reservoir computing can anticipate discontinuous transitions in chaotic time series by linking the learned RC map's fixed-point structure to the underlying bifurcation dynamics. It shows that RC not only reproduces trajectories but also mirrors crisis mechanisms such as collisions of unstable fixed points with chaotic attractors and the associated algebraic scaling of transient lifetimes, for both the logistic and Gauss maps, and extends to the Hénon map in higher dimensions. The authors analyze fixed points via directional fibers and compare RC return maps with analytical predictions, revealing that RC captures the correct scaling exponent $\gamma = 1/2$ near the boundary crisis. The work advances understanding of data-driven crisis prediction and highlights both the potential and limitations of RC in high-dimensional settings.

Abstract

Reservoir computing has emerged as a powerful framework for time series modelling and forecasting including the prediction of discontinuous transitions. However, the mechanism behind its success is not yet fully understood. This letter elucidates the functioning of reservoir computing by examining its successful prediction of boundary and attractor merging crises. We investigate in detail how reservoirs's internal dynamics mimic the actual system, that enables it to accurately reproduce the scaling exponent near boundary crisis. We establish this across distinct systems, exemplified by the logistic and Gauss maps. The study contributes to the broader understanding of the internal dynamics that enable learning algorithms to anticipate critical transitions.

Figures (17)

• Figure 1: Schematic of the trained RC map having the bifurcation parameter as an additional input channel. The weight matrices $W_{in}$ and $W_b$ map the input $u$ and $\varepsilon$, respectively, into a high dimensional reservoir space. $\mathcal{A}$ denotes the adjacency matrix of the reservoir (400 $\times$ 400 in our case), while $W_{out}$, learned during training, projects the reservoir states to give the output. Analysis of the fixed points of the trained RC map, reveals how it captures the critical transition successfully.
• Figure 2: (a) Time series of the trained reservoir map for $\mu_c < \mu = 3.9988$ showing the transient dynamics before convergence to a stable fixed point. (b) Fixed points (FP) of the trained RC map as the bifurcation parameter is varied. Blue crosses and red circles correspond to the stable and unstable fixed points, respectively.
• Figure 3: (a) Crises in a logistic map. The dashed blue lines show two fixed points $x^* = 0$ and $x^* = 1 - 1/\mu$. The first vertical dashed red line at $\mu = 3.68$ corresponds to attractor merging crises (AMC). The second vertical dashed line at $\mu = 4$ shows boundary crisis (BC). (b) Crises in trained RC map due to collision of two unstable fixed points with the chaotic attractor. Collision close to $\mu = 3.68$ is responsible for attractor merging crisis and the collision close to $\mu = 4$ causes boundary crisis.
• Figure 4: Return map of trained reservoir map (red crosses) along with the logistic map (blue dots). The green box denotes the bounded chaotic region, outside which the trajectories escape.
• Figure 5: Histogram of 50000 iterates of reservoir map shows a uniform $\rho(x)$ similar to logistic map (shown in the inset).