Structure-Preserving Method for Reconstructing Unknown Hamiltonian Systems From Trajectory Data

We present a numerical approach for approximating unknown Hamiltonian systems using observational data. A distinct feature of the proposed method is that it is structure-preserving, in the sense that it enforces the conservation of the reconstructed Hamiltonian. This is achieved by directly approximating the underlying unknown Hamiltonian, rather than the right-hand side of the governing equations. We present the technical details of the proposed algorithm and its error estimate in a special case, along with a practical denoising procedure to cope with noisy data. A set of numerical examples is presented to demonstrate the structure-preserving property and effectiveness of the algorithm.