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Using Echo-State Networks to Reproduce Rare Events in Chaotic Systems

Anton Erofeev, Balasubramanya T. Nadiga, Ilya Timofeyev

TL;DR

The paper addresses the challenge of reproducing rare events in chaotic dynamical systems by using Echo-State Networks (ESNs) to learn the flow map of the $d=4$ chaotic competitive Lotka-Volterra system and to reproduce its invariant measure via Generalized Extreme Value (GEV) analysis. The ESN is trained on long LV trajectories and then run autonomously with a fixed time-step, demonstrating generalized synchronization and robust tail reproduction for all variables. Results show the ESN can predict trajectories for about $T_{lyap} \approx 49.3$ and produce histograms that closely match the LV system, with GEV fits indicating finite tails and accurate tail shapes ($\xi$) between LV and ESN across multiple averaging windows. This suggests ESNs can generate long, statistically faithful chaotic trajectories suitable for applications such as climate studies and reservoir physics, with open-source code and data available for replication.

Abstract

We apply the Echo-State Networks to predict the time series and statistical properties of the competitive Lotka-Volterra model in the chaotic regime. In particular, we demonstrate that Echo-State Networks successfully learn the chaotic attractor of the competitive Lotka-Volterra model and reproduce histograms of dependent variables, including tails and rare events. We use the Generalized Extreme Value distribution to quantify the tail behavior.

Using Echo-State Networks to Reproduce Rare Events in Chaotic Systems

TL;DR

The paper addresses the challenge of reproducing rare events in chaotic dynamical systems by using Echo-State Networks (ESNs) to learn the flow map of the chaotic competitive Lotka-Volterra system and to reproduce its invariant measure via Generalized Extreme Value (GEV) analysis. The ESN is trained on long LV trajectories and then run autonomously with a fixed time-step, demonstrating generalized synchronization and robust tail reproduction for all variables. Results show the ESN can predict trajectories for about and produce histograms that closely match the LV system, with GEV fits indicating finite tails and accurate tail shapes () between LV and ESN across multiple averaging windows. This suggests ESNs can generate long, statistically faithful chaotic trajectories suitable for applications such as climate studies and reservoir physics, with open-source code and data available for replication.

Abstract

We apply the Echo-State Networks to predict the time series and statistical properties of the competitive Lotka-Volterra model in the chaotic regime. In particular, we demonstrate that Echo-State Networks successfully learn the chaotic attractor of the competitive Lotka-Volterra model and reproduce histograms of dependent variables, including tails and rare events. We use the Generalized Extreme Value distribution to quantify the tail behavior.

Paper Structure

This paper contains 4 sections, 10 equations, 4 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Dynamic Equations and the Reservoir
  3. Numerical Results
  4. Conclusions

Figures (4)

  • Figure 1: Log of the $L^2$ norm $||r_1(t) - r_2(t)||_2$ between two reservoirs with different initial conditions. One time-step is $\Delta t=2$. We observe a fast exponential decay of the $L^2$ norm, indicating that the initial conditions are not affecting the utility of predictions for individual trajectories.
  • Figure 2: Prediction of individual time-series with $\Delta t=2$. This demonstrates that the ESN predicts trajectories of the LV system for approximately 8 Lyapunov times.
  • Figure 3: Histograms of dependent variables $x_i$, $i=0,\ldots,3$ in long simulations of the LV model in \ref{['lv']} and simulations of the ESN. There is a perfect overlap between the histogram generated by the LV system in \ref{['lv']} and predicted by the ESN. Zoom-in on the tails of these histograms is presented in Figure \ref{['fig1a']}.
  • Figure 4: Zoom-in on tails of the histograms in Figure \ref{['fig1']}.