Pretrained Approximators for Low-Thrust Trajectory Cost and Reachability

Neural Network Training Dataset for Low-Thrust Trajectory Approximation Description This dataset contains > 300,000,000 optimal low-thrust trajectory samples designed for training neural network approximators for fuel-optimal and time-optimal transfers. It was developed in the context of the paper "Pretrained Approximators for Low-Thrust Trajectory Cost and Reachability" by Zhang et al., submitted at the Journal of Guidance, Navigation and Control. The dataset contains information used in the context of neural predictions of: Fuel consumption for single-revolution fuel-optimal transfers. Transfer time for single-revolution and multi-revolution time-optimal transfers. Starting from sampled boundary conditions (rs, vs, rt, vt), initial mass ms, specific impulse Isp, and maximum thrust Tmax (fixed at 0.1 N): Single-Revolution Fuel-Optimal Dataset For a prescribed time-of-flight tof, the minimum fuel transfer (maximum final mass mf) is computed by solving the corresponding fuel-optimal control problem using Pontryagin's minimum principle. Each optimal transfer is recorded with: Complete boundary states (position and velocity). Initial mass, time of flight, and specific impulse. True final mass from optimal control solution. Neural network approximation mfms_approx. The dataset contains approximately 100,000,000 samples spanning a wide range of: Orbital elements: semi-major axes 0.9-4.0 AU, eccentricities 0-1, inclinations 0-180° Propulsion characteristics: specific impulse 2000-5000 s, initial accelerations 10⁻⁵ to 10⁻³ m/s² Transfer geometries: diverse boundary conditions generated via the Homotopy Ray Method Multi-Revolution Fuel-Optimal Dataset The multi-revolution fuel optimal dataset (0-3 revolutions) is also generated with a similar strategy as the single-revolution one. For data storage limitations, we make available about 12,000,000 samples, spanning the same wide range of transfer geometries, propulsion characteristics, and orbital elements. Each optimal transfer is recorded with: Complete boundary states (position and velocity). Initial mass, time of flight, and specific impulse. True final mass from optimal control solution. Neural network approximation mfms_approx. Single-Revolution Time-Optimal Dataset For given boundary conditions, the minimum transfer time is computed by solving the time-optimal control problem. The dataset records: Complete boundary states. Initial mass and specific impulse. True minimum time dt_true from the optimal control solution. Neural network prediction dt_predict demonstrating surrogate accuracy. Neural network approximation mfms_approx. This dataset also contains approximately 100,000,000 samples, densely populating the reachability boundary where transfers become infeasible due to propulsion limitations. Multi-Revolution Time-Optimal Dataset For multi-revolution transfers (0-3 revolutions), the time-optimal problem is formulated with relaxed terminal constraints (orbit-to-orbit rather than point-to-point). The dataset includes: Complete boundary states Initial mass and specific impulse True minimum time dt_true Neural network prediction dt_predict Neural network approximation mfms_approx This dataset contains approximately 100,000,000 samples distributed across revolution counts (0-3), capturing the complex multi-modal structure of multi-revolution optimal control. Data Generation Methodology All trajectories are computed using an indirect optimal control method based on Pontryagin's minimum principle. The two-point boundary value problems (TPBVPs) are solved via shooting methods with modified equinoctial elements (MEE). The Homotopy Ray Method is employed to efficiently generate large-scale datasets that focus on mission-relevant regions: Start from a feasible Keplerian trajectory Define a homotopy direction in boundary velocity space Progressively increase perturbations while continuing from previous solutions Densely sample trajectories near the reachability boundary This approach ensures the dataset naturally contains: Low fuel-consumption transfers critical for mission optimization Trajectories near reachability boundaries where optimal control is most challenging Smooth, well-conditioned samples suitable for neural network training Neural Approximations The mfms_approx field provides neural estimates of the final-to-initial mass ratio (for fuel optimal problems) or the time of flight (for time optimal problems). These approximations: Offer fast, closed-form estimates without solving optimal control problems. Use features like Lambert solutions to enhance the performances of the model. More details on the neural models, the chosen features, and input preparation can be found in the publication. Data Format All datasets are provided as txt files with the following columns: Single-Revolution Fuel-Optimal xs [m], ys [m], zs [m], vxs [m/s], vys [m/s], vzs [m/s], xt [m], yt [m], zt [m], vxt [m/s], vyt [m/s], vzt [m/s], ms [kg], tof [d], Isp [s], mf_true [kg], mfms_approx Multi-Revolution Fuel-Optimal xs [m], ys [m], zs [m], vxs [m/s], vys [m/s], vzs [m/s], xt [m], yt [m], zt [m], vxt [m/s], vyt [m/s], vzt [m/s], ms [kg], tof [d], Isp [s], mf_true [kg], mfms_approx Single-Revolution Time-Optimal xs [m], ys [m], zs [m], vxs [m/s], vys [m/s], vzs [m/s], xt [m], yt [m], zt [m], vxt [m/s], vyt [m/s], vzt [m/s], ms [kg], tof [d], Isp [s], dt_true [d], dt_predict [d], mfms_approx Multi-Revolution Time-Optimal xs [m], ys [m], zs [m], vxs [m/s], vys [m/s], vzs [m/s], xt [m], yt [m], zt [m], vxt [m/s], vyt [m/s], vzt [m/s], ms [kg], Isp [s], dt_true [d], dt_predict [d], mfms_approx Note: In the multi-revolution time-optimal case, the source state (xs,ys,zs,vxs,vys,vzs), and the target state (xt,yt,zt,vxt,vyt,vzt) are defined at the same epoch. This differs from the single-revolution cases, where the target state is defined after a time shift of tof. Applications This dataset enables: Training neural network surrogates for rapid low-thrust trajectory evaluation Preliminary mission design requiring millions of trajectory evaluations Global trajectory optimization in multi-target missions (e.g., GTOC problems) Benchmarking of analytical approximation methods Reachability analysis for electric propulsion missions Citation If you use this dataset in your research, please cite: Zhang, Z., Acciarini, G., Izzo, D., Baoyin, H., & Topputo, F. (2026). Pretrained Approximators for Low-Thrust Trajectory Cost and Reachability. To appear in Journal of Guidance, Navigation and Control And this dataset (see zenodo bibtex associated with this entry). Related Resources Previous dataset: Acciarini, G., Izzo, D., & Beauregard, L. (2024). Optimal low thrust transfers among asteroid belt asteroids (Version v2) [Data set]. Zenodo. https://doi.org/10.5281/zenodo.11502524. Link: https://zenodo.org/records/11502524). Related work: Computing low-thrust transfers in the asteroid belt, a comparison between astrodynamical manipulations and a machine learning approach, by Acciarini G., Izzo D., and Beauregard L. presented at the 29th International Symposium on Space Flight Dynamics (ISSFD) in 2024 (https://doi.org/10.48550/arXiv.2405.18918). Contact For questions, issues, or additional information: Zhong Zhang: [e-mail] Giacomo Acciarini: [e-mail] Dario Izzo: [e-mail]用于低推力轨迹近似的神经网络训练数据集 描述 本数据集包含超过3亿条最优低推力轨迹样本，专为训练神经网络近似器以实现燃料最优与时间最优转移而构建。本数据集源自Zhang等人投稿至《制导、导航与控制期刊》（*Journal of Guidance, Navigation and Control*）的论文《面向低推力轨迹代价与可达性的预训练近似器》（*Pretrained Approximators for Low-Thrust Trajectory Cost and Reachability*）。 本数据集涵盖用于神经网络预测的以下信息： - 单圈燃料最优转移的燃料消耗量 - 单圈与多圈时间最优转移的转移时长 数据集输入参数为采样得到的边界条件$(r_s, v_s, r_t, v_t)$、初始质量$m_s$、比冲$I_{sp}$以及最大推力$T_{max}$（固定为0.1 N）。 单圈燃料最优数据集 针对指定飞行时长$t_{of}$，通过求解基于庞特里亚金极小值原理（Pontryagin's minimum principle）的燃料最优控制问题，可得到最小燃料转移（即最大最终质量$m_f$）。每条最优转移轨迹记录以下信息： - 完整边界状态（位置与速度） - 初始质量、飞行时长与比冲 - 最优控制解得到的真实最终质量 - 神经网络近似结果`mfms_approx` 本数据集包含约1亿条样本，覆盖广泛的参数范围： - 轨道要素：半长轴0.9~4.0天文单位（AU）、偏心率0~1、轨道倾角0~180° - 推进特性：比冲2000~5000 s、初始加速度$10^{-5}~10^{-3} m/s^2$ - 转移几何：通过同伦射线法（Homotopy Ray Method）生成的多样化边界条件 多圈燃料最优数据集 多圈燃料最优数据集（0~3圈）采用与单圈数据集类似的策略生成。受限于数据存储规模，本次公开约1200万条样本，覆盖相同范围的转移几何、推进特性与轨道要素。 每条最优转移轨迹记录以下信息： - 完整边界状态（位置与速度） - 初始质量、飞行时长与比冲 - 最优控制解得到的真实最终质量 - 神经网络近似结果`mfms_approx` 单圈时间最优数据集 针对给定边界条件，通过求解时间最优控制问题可得到最小转移时长。本数据集记录以下内容： - 完整边界状态 - 初始质量与比冲 - 最优控制解得到的真实最小转移时长$dt_true$ - 体现替代模型精度的神经网络预测结果$dt_predict$ - 神经网络近似结果`mfms_approx` 本数据集包含约1亿条样本，密集覆盖了因推进限制导致转移不可行的可达性边界区域。 多圈时间最优数据集 针对多圈转移（0~3圈），时间最优问题采用松弛终端约束形式（轨道到轨道而非点对点约束）。本数据集包含以下信息： - 完整边界状态 - 初始质量与比冲 - 真实最小转移时长$dt_true$ - 神经网络预测结果$dt_predict$ - 神经网络近似结果`mfms_approx` 本数据集包含约1亿条样本，覆盖0~3圈的转移圈数，捕捉了多圈最优控制的复杂多模态结构。 数据生成方法论 所有轨迹均采用基于庞特里亚金极小值原理的间接最优控制方法计算。两点边值问题（two-point boundary value problems, TPBVPs）通过采用改进分点要素（modified equinoctial elements, MEE）的打靶法求解。 本研究采用同伦射线法高效生成大规模数据集，聚焦任务相关区域： - 从可行的开普勒轨迹出发 - 在边界速度空间定义同伦方向 - 从前序解继续迭代，逐步增加扰动幅度 - 在可达性边界附近密集采样轨迹 该方法确保数据集自然包含以下内容： - 对任务优化至关重要的低燃料消耗转移轨迹 - 最优控制难度最高的可达性边界附近轨迹 - 适合神经网络训练的平滑、良态样本 神经近似 `mfms_approx`字段提供了神经网络估计结果：针对燃料最优问题为最终质量与初始质量之比，针对时间最优问题为飞行时长。这些近似方法具备以下优势： - 无需求解最优控制问题即可快速得到闭式估计 - 利用兰伯特解（Lambert solutions）等特征提升模型性能 更多关于神经网络模型、选用特征与输入预处理的细节可参见上述论文。 数据格式 所有数据集均以文本文件（txt）形式提供，各数据集的列信息如下： 单圈燃料最优数据集 `xs [m], ys [m], zs [m], vxs [m/s], vys [m/s], vzs [m/s], xt [m], yt [m], zt [m], vxt [m/s], vyt [m/s], vzt [m/s], ms [kg], tof [d], Isp [s], mf_true [kg], mfms_approx` 多圈燃料最优数据集 列信息与单圈燃料最优数据集一致：`xs [m], ys [m], zs [m], vxs [m/s], vys [m/s], vzs [m/s], xt [m], yt [m], zt [m], vxt [m/s], vyt [m/s], vzt [m/s], ms [kg], tof [d], Isp [s], mf_true [kg], mfms_approx` 单圈时间最优数据集 `xs [m], ys [m], zs [m], vxs [m/s], vys [m/s], vzs [m/s], xt [m], yt [m], zt [m], vxt [m/s], vyt [m/s], vzt [m/s], ms [kg], tof [d], Isp [s], dt_true [d], dt_predict [d], mfms_approx` 多圈时间最优数据集 `xs [m], ys [m], zs [m], vxs [m/s], vys [m/s], vzs [m/s], xt [m], yt [m], zt [m], vxt [m/s], vyt [m/s], vzt [m/s], ms [kg], Isp [s], dt_true [d], dt_predict [d], mfms_approx` 注意：在多圈时间最优场景中，源状态$(x_s,y_s,z_s,v_{xs},v_{ys},v_{zs})$与目标状态$(x_t,y_t,z_t,v_{xt},v_{yt},v_{zt})$定义于同一历元；而单圈场景中，目标状态定义于飞行时长$t_{of}$对应的时间偏移后。 应用场景 本数据集可用于以下研究方向： - 训练神经网络替代模型，实现低推力轨迹的快速评估 - 需要数百万次轨迹评估的初步任务设计 - 多目标任务中的全局轨迹优化（如GTOC问题） - 解析近似方法的基准测试 - 电推进任务的可达性分析 引用 若您在研究中使用本数据集，请引用以下文献： Zhang, Z., Acciarini, G., Izzo, D., Baoyin, H., & Topputo, F. (2026). Pretrained Approximators for Low-Thrust Trajectory Cost and Reachability. 即将发表于《Journal of Guidance, Navigation and Control》（《制导、导航与控制期刊》） 同时请引用本数据集对应的Zenodo BibTeX条目。 相关资源 1. 前期数据集：Acciarini, G., Izzo, D., & Beauregard, L. (2024). 小行星带小行星间最优低推力转移（版本v2）[数据集]. Zenodo. https://doi.org/10.5281/zenodo.11502524. 链接：https://zenodo.org/records/11502524 2. 相关研究：Acciarini G., Izzo D. 与 Beauregard L. 于2024年第29届国际空间飞行动力学研讨会（ISSFD）发表的《小行星带低推力转移计算：天体动力学操纵与机器学习方法对比》（https://doi.org/10.48550/arXiv.2405.18918） 联系方式 如有疑问、问题或需要更多信息，请联系： - 张忠：[e-mail] - Giacomo Acciarini：[e-mail] - Dario Izzo：[e-mail]