A Prefixed Patch Time Series Transformer for Two-Point Boundary Value Problems in Three-Body Problems
Akira Hatakeyama, Shota Ito, Toshihiko Yanase, Naoya Ozaki
TL;DR
The paper tackles the challenge of solving two-point boundary value problems in the circular restricted three-body problem ($CR3BP$), where traditional Lambert-based methods fail. It introduces Prefixed PatchTST, a Patch Time Series Transformer that uses prefix tokens encoding initial and terminal states to generate trajectories connecting boundary conditions, with training data constructed via forward propagation. Through a lunar-flyby–driven dataset and Optuna-driven hyperparameter tuning, the approach demonstrates that generated trajectories can approximate true paths with position errors typically below $10^4$ km over a 90-day horizon, while preserving key velocity and vertical ($z$) components. This work offers a practical initial-guess tool for preliminary cislunar trajectory design and highlights the potential of boundary-prefixing in time-series transformers for nonlinear dynamical BVPs, while identifying oscillations and sensitivity as targets for further data and architectural refinements.
Abstract
Two-point boundary value problems for cislunar trajectories present significant challenges in circler restricted three body problem, making traditional analytical methods like Lambert's problem inapplicable. This study proposes a novel approach using a prefixed patch time series Transformer model that automates the solution of two-point boundary value problems from lunar flyby to arbitrary terminal conditions. Using prefix tokens of terminal conditions in our deep generative model enables solving boundary value problems in three-body dynamics. The training dataset consists of trajectories obtained through forward propagation rather than solving boundary value problems directly. The model demonstrates potential practical utility for preliminary trajectory design in cislunar mission scenarios.