Neural Non-Canonical Hamiltonian Dynamics for Long-Time Simulations
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Learning Hamiltonian dynamics from data has proven effective for improving the long-term stability of neural models, but most existing approaches focus on canonical systems. Many physical systems, however, are governed by non-canonical Hamiltonian dynamics, where both the geometry and the numerical integration become more challenging. Building on the neural symplectic form approach of Chen et al. (2021) and on degenerate variational integrators introduced by Ellison et al. (2018), we study the learning of non-canonical Hamiltonian dynamics with an emphasis on long-time numerical behavior. We show that combining structure-preserving learning with structure-preserving integration can lead to unexpected numerical instabilities, even when the learned vector field is accurate. To overcome this issue, we propose two possible strategies: a regularized vector-field learning approach that improves numerical robustness, and a scheme-based learning approach that directly fits a time-discrete non-canonical dynamics. These strategies are validated experimentally on benchmark problems, including the Lotka–Volterra system and the guiding-center model from plasma physics. Our results demonstrate significantly improved long-time stability compared to standard neural ODE approaches, highlighting the importance of jointly designing learning and integration strategies for non-canonical Hamiltonian systems.
