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 ( this https URL ).
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.

通过使用弱形式稀疏非线性动力学算法（WSINDy），我们成功地在存在大的扰动的情况下，识别出具有近似对称性的哈密顿动力学中的约简哈密顿系统。