Multi-scaling reservoir computing learns noise-induced transitions with Lévy noise

Noise-induced transitions, where random fluctuations drive systems between multiple stable states, are pivotal in stochastic dynamical systems across physics, biology, and engineering. These transitions become intricate in the presence of non-Gaussian noise or when systems exhibit complex dynamics beyond fixed points. We leverage the recent multi-scaling reservoir computing framework to learn such noise-induced transitions, focusing on a bistable system driven by Lévy noise and a limit-cycle system under Gaussian noise. We find that the multi-scaling reservoir computing can generate data capturing transition statistics when trained by a trajectory with a few dozen transitions, where the predictions of transition intervals and probability distributions align closely with test data. This showcases its ability to capture stochastic transitions with abrupt noisy shifts and oscillatory dynamics in a probabilistic manner. These findings highlight the potential of reservoir computing in studying general stochastic systems.