Large-step neural network for learning the symplectic evolution from partitioned data
This work addresses long-term evolution predictions for Hamiltonian systems, such as celestial mechanics, with incremental improvements in accuracy and error suppression.
The study tackled learning Hamiltonian systems by developing a large-step neural network (LSNN) that uses partitioned data to approximate generating functions, extending prediction time periods and reducing accumulative errors. Results on 2:3 resonant Kuiper belt objects over 25000 years showed improved conservation of the Jacobi integral and highly accurate orbital evolution predictions compared to prior work.
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.