In this study, we focus on learning Hamiltonian systems, which involves predicting the coordinate (q) and momentum (p) variables generated by a symplectic mapping. Based on Chen & Tao (2021), the symplectic mapping is represented by a generating function. To extend the prediction time period, we develop a new learning scheme by splitting the time series (q_i, p_i) into several partitions. We then train a large-step neural network (LSNN) to approximate the generating function between the first partition (i.e. the initial condition) and each one of the remaining partitions. This partition approach makes our LSNN effectively suppress the accumulative error when predicting the system evolution. Then we train the LSNN to learn the motions of the 2:3 resonant Kuiper belt objects for a long time period of 25000 yr. The results show that there are two significant improvements over the neural network constructed in our previous work (Li et al. 2022): (1) the conservation of the Jacobi integral, and (2) the highly accurate predictions of the orbital evolution. Overall, we propose that the designed LSNN has the potential to considerably improve predictions of the long-term evolution of more general Hamiltonian systems.

翻译：本研究聚焦于学习哈密顿系统，即预测由辛映射生成的坐标(q)和动量(p)变量。基于Chen & Tao (2021)的研究，辛映射由生成函数表示。为延长预测时间周期，我们通过将时间序列(q_i, p_i)划分为多个分区，提出一种新学习方案。进而训练一个大步长神经网络(LSNN)，用于近似第一个分区（即初始条件）与其余每个分区之间的生成函数。这种分区方法使我们的LSNN能有效抑制预测系统演化时的累积误差。随后我们训练LSNN学习2:3共振柯伊伯带天体在25000年长周期内的运动。结果表明，相较于前期工作中构建的神经网络(Li et al. 2022)，该模型在两个方面有显著提升：(1) 雅可比积分的守恒性，(2) 轨道演化的高精度预测。总体而言，我们提出的LSNN有望大幅提升对更广泛哈密顿系统长期演化的预测能力。