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arxiv: 2605.26790 · v2 · pith:TJJAK7NQnew · submitted 2026-05-26 · 💻 cs.LG · physics.space-ph

Pretrained Approximators for Low-Thrust Trajectory Cost and Reachability

Zhong Zhang , Giacomo Acciarini , Dario Izzo , Hexi Baoyin , Francesco Topputo This is my paper

Pith reviewed 2026-06-29 19:36 UTC · model grok-4.3

classification 💻 cs.LG physics.space-ph
keywords low-thrust trajectorymachine learning surrogatescaling lawtrajectory optimizationneural networkself-similar transformationfuel consumptiontransfer time
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The pith

Neural networks accurately approximate low-thrust trajectory fuel costs and transfer times while generalizing across orbits via self-similar transforms.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that expensive optimal control computations for low-thrust trajectories can be replaced by fast machine-learning surrogates. Performance improves linearly with the logarithm of training set size and network parameters, with no saturation observed in the tested range. A homotopy-ray dataset generation method paired with a self-similar transformation lets the same model handle arbitrary semi-major axes, inclinations, and central bodies. The resulting approximators are shown to work on single- and multi-revolution transfers, public benchmarks, asteroid flyby competitions, and rendezvous missions.

Core claim

Low-thrust trajectory optimization follows a scaling law in which approximation accuracy for optimal fuel consumption and minimum transfer time rises linearly with the log of dataset size and model capacity; a self-similar transformation applied to homotopy-ray data enables the identical neural model to generalize to new semi-major axes, inclinations, and central bodies without retraining.

What carries the argument

The self-similar transformation that normalizes the homotopy-ray dataset so a single network can generalize across varying semi-major axes, inclinations, and central bodies.

If this is right

  • The same pretrained model predicts costs for both single- and multi-revolution transfers in asteroid flyby and rendezvous missions.
  • Accuracy continues to rise with additional data and larger networks inside the explored regime.
  • One model serves multiple mission classes and central bodies without retraining.
  • Open release of models and datasets enables direct use in existing trajectory design pipelines.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Embedding these surrogates inside iterative optimizers would cut the cost of repeated trajectory evaluations by orders of magnitude.
  • The scaling behavior suggests that further gains remain available simply by increasing compute rather than redesigning the architecture.
  • Analogous surrogate-plus-scaling approaches could be tested on other optimal-control problems that currently rely on repeated expensive solves.

Load-bearing premise

The homotopy-ray strategy produces training examples whose coverage is broad enough for the observed scaling law to continue and for generalization to hold on unseen orbital parameters and central bodies.

What would settle it

Measuring whether approximation error stops decreasing or begins to rise when the training set is enlarged by another factor of ten or when the model is tested on transfers around a central body absent from the training distribution.

Figures

Figures reproduced from arXiv: 2605.26790 by Dario Izzo, Francesco Topputo, Giacomo Acciarini, Hexi Baoyin, Zhong Zhang.

Figure 1
Figure 1. Figure 1: Model size scaling law in low-thrust approximation. [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Dataset size scaling law in low-thrust approximation. [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Schematic diagram of the homotopy ray method. [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Graph illustrating the relationship between the fuel consumption and the corresponding variation in departure [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The diagram illustrates the rotational invariance. [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: The diagram illustrates the dimensional invariance. [PITH_FULL_IMAGE:figures/full_fig_p012_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: The input and output of the prediction model. [PITH_FULL_IMAGE:figures/full_fig_p015_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Parameter distributions of the multi-revolution transfer training dataset. [PITH_FULL_IMAGE:figures/full_fig_p018_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Impact of terminal constraints on homotopy continuation for time-optimal transfers. Left: point-to-point [PITH_FULL_IMAGE:figures/full_fig_p019_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: The input and output of the multi-rev model. [PITH_FULL_IMAGE:figures/full_fig_p020_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Performance of single and multirev ∆v models in terms of absolute value of the sorted relative error. distributions of key transfer features. Errors are largely insensitive to eccentricity, inclination, and initial mass. However, more challenging transfers, characterized by higher fuel consumption, longer time-of-flight, and larger Lambert cost, tend to appear in the worst-performing cases. The median fin… view at source ↗
Figure 12
Figure 12. Figure 12: Performance of multirev ∆t models in terms of absolute value of the sorted relative error. 0.00 0.25 0.50 0.75 1.00 Eccentricity 0 50 100 150 Inclination (deg) 0.0000 0.0075 0.0050 0.0025 0.0100 Probability Density 0 1 2 0.2 0.4 0.6 0.8 1.0 mf/m0 0 5 10 15 20Probability Density Worst Best 0 5 10 VL1 + VL2 (km/s) 0.00 0.25 0.50 0.75 1.00 0 2500 5000 7500 10000 m 0 (kg ) 0.0000 0.0001 0.0002 0.0003 Probabil… view at source ↗
Figure 13
Figure 13. Figure 13: Distributions of key transfer characteristics for best- and worst-performing test cases: eccentricity, inclination, [PITH_FULL_IMAGE:figures/full_fig_p021_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: presents the performance of the ∆v model across the dataset. Ground-truth ∆v values were sorted in ascending order and divided into consecutive windows of 1,000 samples each. For every window, the average ∆v and corresponding MAE were computed to visualize error trends. 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 True v (m/s) -100 0 100 200 300 400 500 600 700 800 Prediction Error (m/s) Our Model… view at source ↗
Figure 15
Figure 15. Figure 15: compares the predicted versus actual ∆v values across all flyby segments. The model exhibits strong predictive performance, with accurate estimates even for segments requiring less than 100 m/s of velocity increment. This highlights the importance of including low-thrust trajectories in the training dataset to ensure reliability in precision mission design. 0 5 10 15 20 25 30 35 40 45 50 Transfer ID 0 500… view at source ↗
Figure 16
Figure 16. Figure 16: Porkchop plot for the 2012 LA Asteroid. 8 Conclusion This paper has presented a neural network-based framework for rapid and accurate estimation of fuel consumption and transfer feasibility in low-thrust trajectory design. The proposed models advance the performance by enabling broad applicability across a wide variety of mission scenarios without the need for retraining or task-specific dataset generatio… view at source ↗
Figure 17
Figure 17. Figure 17: Porkchop plot for the 2008 ST Asteroid. a) Numerical result b) Prediction result c) Relative error [PITH_FULL_IMAGE:figures/full_fig_p026_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: Porkchop plot for the 2022 OC3 Asteroid. [PITH_FULL_IMAGE:figures/full_fig_p026_18.png] view at source ↗
read the original abstract

Low-thrust trajectory design relies heavily on repeated evaluations of fuel consumption and transfer feasibility, which require expensive optimal control solutions. In this work, we show these quantities can be accurately approximated by machine learning surrogates, enabling fast and scalable evaluation across a wide range of scenarios. By increasing both dataset size and model capacity, we observe that low-thrust trajectory optimization follows a scaling law, with performance improving linearly with the logarithm of training data and network parameters, and no evidence of saturation within the explored regime. Guided by this observation, we construct a large-scale dataset using the proposed homotopy-ray strategy tailored to mission design requirements. A key is the introduction of a self-similar transformation, which allows generalization across semi-major axes, inclinations, and central bodies avoiding retraining. As a result, the same neural approximator can be applied to diverse orbital environments and mission classes. The proposed models accurately predict optimal fuel consumption and minimum transfer time for single- and multi-revolution transfers. Their performance and generalization are demonstrated on a public dataset, a multi-asteroid flyby problem from the Global Trajectory Optimization Competition, and an asteroid rendezvous mission design. The models and datasets are released as open-source to support the space community.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 0 minor

Summary. The manuscript claims that neural network surrogates can accurately approximate optimal fuel consumption and minimum transfer time for low-thrust single- and multi-revolution trajectories. A homotopy-ray dataset generation strategy combined with a self-similar transformation is introduced to enable generalization across semi-major axes, inclinations, and central bodies without retraining. The authors report an empirical scaling law in which approximation performance improves linearly with the logarithm of training set size and network parameters, with no saturation observed in the explored regime. Results are demonstrated on a public dataset, a GTOC multi-asteroid flyby problem, and an asteroid rendezvous mission, with models and datasets released as open source.

Significance. If the reported accuracy and scaling behavior prove robust, the work could enable substantially faster evaluation of low-thrust costs and reachability in mission design loops. The open-source release of models and datasets is a concrete strength that supports reproducibility and community follow-on work.

major comments (2)
  1. [Abstract] Abstract: the headline claim of linear scaling with log(training data) and log(network parameters) and the generalization claim both rest on the assumption that the homotopy-ray dataset plus self-similar mapping provides sufficient coverage; the abstract supplies no quantitative error distributions, validation splits, ablation on the sampling strategy, or coverage metrics (e.g., convex-hull volume or KS distances to test distributions) to substantiate this.
  2. [Results] The central empirical results are fits to a generated dataset; without reported validation splits or ablation studies on the homotopy-ray procedure, it is not possible to determine whether the observed scaling and zero-shot generalization to arbitrary inclinations and revolutions are load-bearing or artifacts of interpolation within the training distribution.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed review and constructive feedback on the abstract and empirical validation. We agree that additional quantitative details on validation, coverage, and ablations would strengthen the presentation of the scaling law and generalization claims. We will revise the manuscript accordingly while maintaining that the self-similar transformation and out-of-distribution tests on GTOC and rendezvous missions provide substantive support beyond pure interpolation.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the headline claim of linear scaling with log(training data) and log(network parameters) and the generalization claim both rest on the assumption that the homotopy-ray dataset plus self-similar mapping provides sufficient coverage; the abstract supplies no quantitative error distributions, validation splits, ablation on the sampling strategy, or coverage metrics (e.g., convex-hull volume or KS distances to test distributions) to substantiate this.

    Authors: We acknowledge the abstract is concise and omits these specifics. The full paper reports mean relative errors below 1% on held-out test points from the public dataset, plus successful zero-shot application to GTOC multi-asteroid and rendezvous cases with different inclinations and revolution counts. We will revise the abstract to include key quantitative error statistics, mention the train/test split used for scaling experiments, and note that coverage is ensured by the homotopy-ray sampling density and self-similar normalization. Revision will be made. revision: yes

  2. Referee: [Results] The central empirical results are fits to a generated dataset; without reported validation splits or ablation studies on the homotopy-ray procedure, it is not possible to determine whether the observed scaling and zero-shot generalization to arbitrary inclinations and revolutions are load-bearing or artifacts of interpolation within the training distribution.

    Authors: The results are generated via the homotopy-ray procedure, but the paper already demonstrates generalization on two external mission scenarios (GTOC flyby and asteroid rendezvous) whose inclination and revolution distributions differ from the training set. The self-similar transformation is derived from orbital mechanics and enables the observed zero-shot behavior. We agree that explicit validation splits (e.g., 80/20) and ablations on ray density and homotopy parameters were not reported and will add them in revision. We maintain the scaling law and generalization are not artifacts, as performance continues to improve with scale and transfers to dissimilar problems succeed. revision: partial

Circularity Check

0 steps flagged

No circularity: empirical ML fits on generated data with independent test evaluation

full rationale

The paper generates a dataset via homotopy-ray sampling plus self-similar transform, trains neural approximators, and reports empirical accuracy plus log-linear scaling on held-out public datasets, GTOC instances, and asteroid rendezvous cases. No equation, prediction, or central claim reduces to a fitted parameter or self-citation by construction; the reported performance is measured against external benchmarks rather than being forced by the training procedure itself. Self-citation, if present, is not load-bearing for the scaling or generalization results.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on the representativeness of the homotopy-ray dataset and the validity of the self-similar transformation; both are introduced without external verification in the abstract.

free parameters (1)
  • neural network weights and biases
    All model parameters are fitted to the generated trajectory dataset.
axioms (1)
  • domain assumption The self-similar transformation maps the optimal-control problem identically across different semi-major axes, inclinations, and central bodies.
    Invoked to justify training once and applying everywhere.

pith-pipeline@v0.9.1-grok · 5758 in / 1243 out tokens · 30202 ms · 2026-06-29T19:36:58.253627+00:00 · methodology

discussion (0)

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Reference graph

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