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]