Coarse-graining Hamiltonian systems using WSINDy.

Area of Science:

• Computational Physics
• Dynamical Systems
• Applied Mathematics

Background:

• Weak form equation learning and surrogate modeling are computationally efficient and robust for discovering dynamics governed by ODEs, PDEs, and SDEs.
• Coarse-graining techniques like homogenization and mean-field descriptions are used for interacting particle systems.
• Hamiltonian dynamics with approximate symmetries, often linked to timescale separation, present challenges in deriving reduced-order models.

Purpose of the Study:

• To extend weak form equation learning for coarse-graining Hamiltonian dynamics with approximate symmetries.
• To demonstrate the capability of WSINDy (Weak-form Sparse Identification of Nonlinear Dynamics) in identifying reduced Hamiltonian systems.
• To provide theoretical justification for the method's effectiveness in Hamiltonian coarse-graining.

Main Methods:

• Utilized WSINDy to identify reduced Hamiltonian systems from data, leveraging its ability to preserve Hamiltonian structure by restricting to a basis of Hamiltonian vector fields.
• Employed a single trajectory for learning the global reduced Hamiltonian, avoiding computationally expensive forward solves.
• Applied the method to nearly-periodic Hamiltonian systems exhibiting approximate symmetries.

Main Results:

• WSINDy successfully identified reduced Hamiltonian systems, even with large perturbations and extrinsic noise.
• The method achieved dimension reduction by at least two, accurately capturing the leading-order dynamics.
• A theoretical contribution proved that first-order averaging preserves Hamiltonian structure in nearly-periodic Hamiltonian systems, justifying the WSINDy approach.

Conclusions:

• Weak form equation learning, specifically WSINDy, is a computationally efficient and robust method for Hamiltonian coarse-graining.
• The approach effectively identifies reduced-order models for systems with approximate symmetries, preserving the underlying Hamiltonian structure.
• The method's efficacy was illustrated through physically relevant examples, including coupled oscillators and charged particle dynamics.