In this paper, we develop a neural network-based approach for time-series prediction in unknown Hamiltonian dynamical systems. Our approach leverages a surrogate model and learns the system dynamics using generalized coordinates (positions) and their conjugate momenta while preserving a constant Hamiltonian. To further enhance long-term prediction accuracy, we introduce an Autoregressive Hamiltonian Neural Network, which incorporates autoregressive prediction errors into the training objective. Additionally, we employ Bayesian data assimilation to refine predictions in real-time using online measurement data. Numerical experiments on a spring-mass system and highly elliptic orbits under gravitational perturbations demonstrate the effectiveness of the proposed method, highlighting its potential for accurate and robust long-term predictions.

翻译：本文提出了一种基于神经网络的未知哈密顿动力系统时间序列预测方法。我们的方法利用代理模型，通过广义坐标（位置）及其共轭动量来学习系统动力学，同时保持哈密顿量恒定。为了进一步提高长期预测精度，我们引入了自回归哈密顿神经网络，将自回归预测误差纳入训练目标。此外，我们采用贝叶斯数据同化技术，利用在线测量数据实时优化预测。在弹簧-质量系统以及引力摄动下的高椭圆轨道上进行的数值实验验证了所提方法的有效性，突显了其在实现准确、鲁棒的长期预测方面的潜力。