Solving Hamiltonian Systems Based on a Data-driven Deep Learning Algorithm

Hamiltonian systems possess important properties, such as the preservation of the symplectic structure and the conservation of energy. Traditional numerical iteration methods generally violate these properties. This paper proposes a data-driven deep learning algorithm to solve separable Hamiltonian systems. It is computationally efficient and scalable. The proposed algorithm tries to hold the balance between symplectic and energy-conserving. Additionally, it can work efficiently with few data points, and the obtained solution is continuous. The algorithm consists of three main steps: (i) obtaining data points from a solver, (ii) data augmentation, and (iii) ensuring energy conservation. The algorithm was evaluated on the simple harmonic oscillator, nonlinear oscillator, and Lotka-Volterra systems. The proposed algorithm has demonstrated successful performance across all these systems despite the few data. In conclusion, it has been experimentally demonstrated that the proposed method is effective even with few data points. Furthermore, the proposed algorithm can effectively work regardless of the solver chosen in all examples.