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.

翻译：弱形式稀疏非线性动力学识别算法（WSINDy）已被证明在相互作用粒子系统背景下具有粗粒化能力（https://doi.org/10.1016/j.physd.2022.133406）。本研究将该能力拓展至具有近似对称性的哈密顿动力学粗粒化问题。此类近似对称性通常导致存在降维哈密顿系统，可有效捕捉相关自由度的动力学特性。推导此类约化系统或对其进行数值逼近仍是一项持续挑战。我们证明，WSINDy能够成功识别该约化哈密顿系统，即使存在由近似对称性不精确性及外部噪声引起的大幅扰动。此结果具有重要意义，部分原因在于此类系统的解析推导过程极为复杂。WSINDy通过将搜索空间限制在哈密顿向量场的试验基上，自然地保持了哈密顿结构，且该方法计算高效，通常仅需单条轨迹即可学习完整的约化哈密顿量，并在学习过程中避免正向求解。基于此，我们认为弱形式方程学习尤其适用于哈密顿粗粒化。以近似周期哈密顿系统作为具有近似对称性的典型系统类别，我们证明在观测相关自由度的条件下，WSINDy能够稳健地从原始$(2N)$维系统中识别出正确的$2(N-1)$维或$N$维度主导约化系统。本文提供了具有物理意义的实例，包括耦合振子动力学、银河系恒星运动的Henon-Heiles系统，以及带电粒子动力学。