Coarse-Graining Hamiltonian Systems Using WSINDy

The Weak-form Sparse Identification of Nonlinear Dynamics algorithm (WSINDy) has been demonstrated to offer coarse-graining capabilities in the context of interacting particle systems ( https://doi.org/10.1016/j.physd.2022.133406 ). In this work we extend this capability to the problem of coarse-graining Hamiltonian dynamics which possess approximate symmetries. Such approximate symmetries often lead to the existence of a Hamiltonian system of reduced dimension that may be used to efficiently capture the dynamics of the relevant degrees of freedom. Deriving such reduced systems, or approximating them numerically, is an ongoing challenge. We demonstrate that WSINDy can successfully identify this reduced Hamiltonian system in the presence of large perturbations imparted from both the inexact nature of the symmetry and extrinsic noise. This is significant in part due to the nontrivial means by which such systems are derived analytically. WSINDy naturally preserves the Hamiltonian structure by restricting to a trial basis of Hamiltonian vector fields, and the methodology is computational efficient, often requiring only a single trajectory to learn the full reduced Hamiltonian, and avoiding forward solves in the learning process. In this way, we argue that weak-form equation learning is particularly well-suited for Hamiltonian coarse-graining. Using nearly-periodic Hamiltonian systems as a prototypical class of systems with approximate symmetries, we show that WSINDy robustly identifies the correct leading-order reduced system of dimension $2(N-1)$ or $N$ from the original $(2N)$-dimensional system, upon observation of the relevant degrees of freedom. We provide physically relevant examples, namely coupled oscillator dynamics, the H\'enon-Heiles system for stellar motion within a galaxy, and the dynamics of charged particles.