Data-driven estimation of neural network Hamiltonian in wave kinetic theory

Modeling the dynamics of complex hierarchical turbulent phenomena observed in fusion reactors, such as the interaction between microscopic turbulent motion and macroscopic zonal flows, as a Hamiltonian dynamical system with a few degrees of freedom, called wave kinetic equations, is useful for understanding and controlling turbulent flows. In this study, we propose a data-driven method using neural networks to estimate a Hamiltonian that models the dynamics of turbulent wave packets following the wave kinetic equation. Specifically, we show that the Hamiltonian cannot be uniquely estimated from time series data of turbulent intensity distribution, and we propose a method to stably estimate the Hamiltonian by focusing on the system's symmetry and suppressing its indefiniteness. The effectiveness of the proposed method is verified by simulation data corresponding to turbulent phenomena in a fusion reactor.