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The Weak-form Sparse Identification of Nonlinear Dynamics algorithm (WSINDy)has been demonstrated to offer coarse-graining capabilities in the context ofinteracting particle systems (https://doi.org/10.1016/j.physd.2022.133406). Inthis work we extend this capability to the problem of coarse-grainingHamiltonian dynamics which possess approximate symmetries associated withtimescale separation. Such approximate symmetries often lead to the existenceof a Hamiltonian system of reduced dimension that may be used to efficientlycapture the dynamics of the symmetry-invariant dependent variables. Derivingsuch reduced systems, or approximating them numerically, is an ongoingchallenge. We demonstrate that WSINDy can successfully identify this reducedHamiltonian system in the presence of large intrinsic perturbations whileremaining robust to extrinsic noise. This is significant in part due to thenontrivial means by which such systems are derived analytically. WSINDy alsonaturally preserves the Hamiltonian structure by restricting to a trial basisof Hamiltonian vector fields. The methodology is computational efficient, oftenrequiring only a single trajectory to learn the global reduced Hamiltonian, andavoiding forward solves in the learning process. Using nearly-periodicHamiltonian systems as a prototypical class of systems with approximatesymmetries, we show that WSINDy robustly identifies the correct leading-ordersystem, with dimension reduced by at least two, upon observation of therelevant degrees of freedom. We also provide a contribution to averaging theoryby proving that first-order averaging at the level of vector fields preservesHamiltonian structure in nearly-periodic Hamiltonian systems. We providephysically relevant examples, namely coupled oscillator dynamics, theH\'enon-Heiles system for stellar motion within a galaxy, and the dynamics ofcharged particles.

Coarse-Graining Hamiltonian Systems Using WSINDy