Learning Hamiltonian dynamics with reservoir computing

Reconstructing the Kolmogorov-Arnold-Moser (KAM) dynamics diagram of Hamiltonian system from the time series of a limited number of parameters is an outstanding question in nonlinear science, especially when the Hamiltonian governing the system dynamics is unknown. Here we demonstrate that this question can be addressed by the machine learning approach knowing as reservoir computing (RC). Specifically, we show that without prior knowledge about the Hamilton equations of motion, the trained RC is able to not only predict the short-term evolution of the system state, but also replicate the long-term ergodic properties of the system dynamics. Furthermore, using the architecture of parameter-aware RC, we show that the RC trained by the time series acquired at a handful parameters is able to reconstruct the entire KAM dynamics diagram with a high precision by tuning a control parameter externally. The feasibility and efficiency of the learning techniques are demonstrated in two classical nonlinear Hamiltonian systems, namely, the double-pendulum oscillator and the standard map. Our study indicates that, as a complex dynamical system, RC is able to learn from data the Hamiltonian.

Automated summary

The study demonstrates that machine learning approach using reservoir computing technique can reconstruct the KAM dynamics diagram of Hamiltonian system, even when the Hamiltonian equations of motion governing the system dynamics are unknown. This method can not only predict the short-term evolution of the system state, but also replicate the entire KAM dynamics diagram with high precision by tuning a control parameter externally.