Learning Hamiltonian dynamics by reservoir computer

TL;DR

This paper demonstrates that reservoir computers can learn Hamiltonian dynamics from limited data, accurately predicting short-term behavior and reconstructing long-term ergodic properties and KAM diagrams without prior knowledge of the equations.

Contribution

The study introduces a parameter-aware reservoir computer architecture capable of reconstructing the entire KAM dynamics diagram from limited parameter data.

Findings

RC predicts short-term system evolution accurately.

RC replicates long-term ergodic properties.

High-precision reconstruction of KAM diagrams.

Abstract

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.