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 associated with timescale separation. 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 symmetry-invariant dependent variables. 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 intrinsic perturbations while remaining robust to extrinsic noise. This is significant in part due to the nontrivial means by which such systems are derived analytically. WSINDy also naturally preserves the Hamiltonian structure by restricting to a trial basis of Hamiltonian vector fields. The methodology is computational efficient, often requiring only a single trajectory to learn the global reduced Hamiltonian, and avoiding forward solves in the learning process. Using nearly-periodic Hamiltonian systems as a prototypical class of systems with approximate symmetries, we show that WSINDy robustly identifies the correct leading-order system, with dimension reduced by at least two, upon observation of the relevant degrees of freedom. We also provide a contribution to averaging theory by proving that first-order averaging at the level of vector fields preserves Hamiltonian structure in nearly-periodic Hamiltonian systems. 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粗粒化中具有应用潜力（https://doi.org/10.1016/j.physd.2022.133406）。本研究将该能力扩展至粗粒化具有时间尺度分离近似对称性的哈密顿动力学问题。此类近似对称性通常导致存在降维哈密顿系统，可用于有效捕捉对称性不变依赖变量的动力学行为。推导此类降维系统或对其进行数值近似仍是一个持续挑战。研究表明，WSINDy能在存在大固有扰动的情况下成功识别该降维哈密顿系统，同时保持对外部噪声的鲁棒性。这一突破意义重大，部分原因在于此类系统的分析推导手段极为复杂。WSINDy通过将搜索空间限制在哈密顿向量场试验基中，自然保留了哈密顿结构。该方法的计算效率高，通常仅需单条轨迹即可学习全局降维哈密顿量，且学习过程中无需前向求解。以近周期哈密顿系统作为具近似对称性的典型系统类别，我们证明WSINDy在观测相关自由度后，能稳健识别正确的主导阶系统，其维度至少降低两维。此外，我们对平均理论作出贡献，证明在近周期哈密顿系统中，向量场层级的一阶平均保持哈密顿结构。本文还提供了具物理意义的实例，包括耦合振子动力学、用于描述星系内恒星运动的厄农-海莱斯系统以及带电粒子动力学。