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

Echo-State Networks (ESNs) are known to reproduce trajectories and main properties of chaotic attractors. Here, we investigate the question of whether ESNs can reproduce rare events in a system with a stationary distribution with slowly decaying bounded tails. In particular, we consider the chaotic four-dimensional competitive Lotka--Volterra system, and quantify extremes using the Generalized Extreme Value distribution. In stationary simulations, the ESN reproduces the stationary distributions and rare event statistics with high accuracy. In addition, the ESN also reproduces rare event statistics for non-equilibrium ensemble simulations for initial conditions relatively close to the underlying attractor. However, the performance of the ESN deteriorates as the ensemble spread increases and trajectories explore regions of phase space not represented in the training data.