K. Gustavsson

L. Storm, K. Gustavsson, B. Mehlig

Echo-state networks are simple models of discrete dynamical systems driven by a time series. By selecting network parameters such that the dynamics of the network is contractive, characterized by a negative maximal Lyapunov exponent, the network may synchronize with the driving signal. Exploiting this synchronization, the echo-state network may be trained to autonomously reproduce the input dynamics, enabling time-series prediction. However, while synchronization is a necessary condition for prediction, it is not sufficient. Here, we study what other conditions are necessary for successful time-series prediction. We identify two key parameters for prediction performance, and conduct a parameter sweep to find regions where prediction is successful. These regions differ significantly depending on whether full or partial phase space information about the input is provided to the network during training. We explain how these regions emerge.

X. Jiang, J. Qiu, K. Gustavsson et al.

Artificial gliders are designed to disperse as they settle through a fluid, requiring precise navigation to reach target locations. We show that a compact glider settling in a viscous fluid can navigate by dynamically adjusting its centre-of-mass. Using fully resolved direct numerical simulations (DNS) and reinforcement learning, we find two optimal navigation strategies that allow the glider to reach its target location accurately. These strategies depend sensitively on how the glider interacts with the surrounding fluid. The nature of this interaction changes as the particle Reynolds number Re$_p$ changes. Our results explain how the optimal strategy depends on Re$_p$. At large Re$_p$, the glider learns to tumble rapidly by moving its centre-of-mass as its orientation changes. This generates a large horizontal inertial lift force, which allows the glider to travel far. At small Re$_p$, by contrast, high viscosity hinders tumbling. In this case, the glider learns to adjust its centre-of-mass so that it settles with a steady, inclined orientation that results in a horizontal viscous force. The horizontal range is much smaller than for large Re$_p$, because this viscous force is much smaller than the inertial lift force at large Re$_p$. *These authors contributed equally.