  Reconstructing the 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 are unknown. Here, we demonstrate that this question can be addressed
by the machine learning approach knowing as reservoir computer (RC).
Specifically, we show that without prior knowledge about the Hamilton's
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, by the architecture of
parameter-aware RC, we also 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.
