# Structure-preserving Method for Reconstructing Unknown Hamiltonian Systems from Trajectory Data

## TL;DR

This paper introduces a structure-preserving numerical method for reconstructing unknown Hamiltonian systems from trajectory data, ensuring conservation laws are maintained and effectively handling noisy data.

## Contribution

The method directly approximates the Hamiltonian function, preserving the system's structure and conservation properties, which is a novel approach compared to existing techniques.

## Key findings

- Successfully reconstructs Hamiltonian systems from data
- Maintains conservation of the Hamiltonian in reconstructions
- Effective noise handling with a de-noising procedure

## Abstract

We present a numerical approach for approximating unknown Hamiltonian systems using observation data. A distinct feature of the proposed method is that it is structure-preserving, in the sense that it enforces 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 de-noising procedure to cope with noisy data. A set of numerical examples are then presented to demonstrate the structure-preserving property and effectiveness of the algorithm.