Pretrained Approximators for Low-Thrust Trajectory Cost and Reachability

This paper introduces machine learning surrogates that accurately and efficiently approximate low-thrust trajectory costs and reachability by leveraging a scaling law and a self-similar transformation to generalize across diverse orbital environments without retraining.

The Gist

Imagine you are a space mission planner trying to figure out the best way to fly a spaceship from Earth to an asteroid using a very weak, but very efficient, engine (like a slow-drifting ion thruster).

In the old days, figuring out the perfect path for every single mission was like trying to solve a massive, complex math puzzle from scratch every time you wanted to go somewhere new. It took supercomputers days to calculate just one route. If you wanted to check a thousand different asteroids, you'd be waiting years.

This paper introduces a new "smart assistant" (a machine learning model) that acts like a seasoned space pilot who has memorized millions of routes. Instead of solving the math puzzle every time, the assistant instantly predicts how much fuel you'll need and how long the trip will take.

Here is a breakdown of how they built this assistant and why it works so well, using simple analogies:

1. The "Scaling Law" Discovery: Bigger is Better

The researchers noticed something interesting: the more "practice problems" they gave the computer, and the "smarter" they made the computer's brain (by adding more layers of neurons), the better it got at predicting routes.

• The Analogy: Think of it like learning to play chess. If you play 10 games, you get okay. If you play 10,000 games against a master, you get very good. They found that there was no "ceiling" to how much better the computer could get; as long as they fed it more data and gave it a bigger brain, it kept improving linearly.

2. The "Homotopy Ray" Method: Training on the Edge

To train this assistant, they needed a massive dataset of space routes. But if you just randomly pick start and end points in space, most of them are impossible to reach with a low-thrust engine. It would be like asking a student to solve math problems where 99% of the answers are "impossible."

• The Analogy: Instead of random guessing, they used a method called the "Homotopy Ray." Imagine you have a rubber band stretched between two points (a valid, easy route). You slowly pull the rubber band tighter and tighter until it's about to snap. That "snapping point" is the edge of what is possible.
• They generated millions of routes by starting with easy ones and slowly stretching them toward the limit. This ensured the computer learned the most critical, difficult, and useful routes—the ones right on the edge of feasibility—rather than wasting time on impossible ones.

3. The "Universal Translator": Seeing the Same Pattern Everywhere

One of the biggest problems with previous AI models was that they were like specialists who only knew how to fly to Mars. If you asked them about Jupiter, they failed.

• The Analogy: The researchers realized that the physics of space travel is "self-similar." A trip from Earth to a nearby asteroid looks mathematically identical to a trip from Jupiter to a moon, just scaled up or down in size and time.
• They created a "Universal Translator" for the data. Before feeding the numbers to the AI, they stripped away the specific details (like "this is 1 million kilometers away") and converted everything into relative ratios (like "this is 10 times the starting distance").
• The Result: The AI learned the shape of the problem, not just the specific numbers. This means the same AI model trained on Earth-Mars data can instantly predict routes for Earth-Jupiter or even around different planets without needing to be retrained. It's like teaching someone to drive a car; once they know the rules of the road, they can drive a Ford or a Toyota without a new lesson.

4. What the AI Actually Does

The team built two specific "brains":

• The Fuel Calculator: Given a start point, an end point, and a time limit, it predicts exactly how much fuel you will burn.
• The Time Calculator: Given a start point, an end point, and a fuel budget, it predicts the fastest possible time to get there.

5. Proof It Works

They didn't just claim it worked; they tested it in three ways:

• Public Challenge: They tested it on a dataset made by other scientists. Their AI was significantly more accurate than previous methods, especially for the tricky, low-fuel routes.
• The "Asteroid Hopping" Game: They used it for a famous space mission competition (GTOC4) where the goal is to visit as many asteroids as possible in a set time. The AI helped design a route that was highly efficient.
• The "Porkchop" Map: In mission planning, engineers draw "porkchop plots" (maps showing the best launch dates and travel times). Traditionally, drawing one of these maps takes days of supercomputer time. The AI generated these maps in a fraction of a second, allowing planners to instantly see the "sweet spots" for launching missions.

Summary

This paper presents a "pre-trained" AI tool that acts as a universal shortcut for space travel planning. By training on a massive, smartly generated dataset and using a "translation" system to ignore irrelevant details, the AI can instantly tell mission planners how much fuel and time a low-thrust journey will take, regardless of the destination or the planet. It turns a process that used to take days of heavy calculation into a split-second prediction, making it much easier to design ambitious future space missions.

Technical Summary

Technical Summary: Pretrained Approximators for Low-Thrust Trajectory Cost and Reachability

Problem Statement

Low-thrust electric propulsion is a critical technology for modern space missions, offering high specific impulse and flexibility. However, its application fundamentally shifts trajectory design from impulsive maneuvers to continuous control laws derived from solving nonlinear optimal control problems. While high-fidelity optimal control methods exist, their computational cost is prohibitive for early-phase mission design and global optimization, where designers must evaluate millions of candidate transfers under varying boundary conditions.

Existing approximation techniques, such as analytical (Edelbaum-type) or semi-analytical (Lambert-based) methods, are computationally efficient but often restricted to specific dynamical regimes. Database-driven approaches offer higher accuracy but degrade outside their training domains. Recent machine learning (ML) surrogate models (Value Networks) show promise but face limitations: they are typically trained on narrowly defined scenarios, lack generalization across different orbital regimes or central bodies, and suffer from a scarcity of public datasets and pretrained models, hindering independent validation.

Methodology

1. Data Generation: The Homotopy Ray Method

To address data scarcity and distribution bias, the authors propose a Homotopy Ray Method for generating large-scale, mission-oriented datasets.

• Strategy: Instead of uniform random sampling, which yields many infeasible transfers, the method starts with feasible Keplerian orbital solutions. It then progressively deforms these solutions along a random "ray" in the boundary-condition space (perturbing initial and terminal velocities) toward the reachability boundary.
• Continuation: A homotopy parameter controls the deformation. Costates from the previous solution serve as initial guesses for the shooting method at each step, ensuring rapid convergence.
• Targeting: This strategy densely populates the feasible region and the critical boundary of reachability (where transfers become infeasible or require maximum fuel), regions most relevant for optimization but often underrepresented in random datasets.
• Scale: The method generated a dataset of approximately 100 million trajectory samples, covering single- and multi-revolution transfers, a wide range of orbital elements (eccentricity 0–1, inclination 0–180°), and diverse propulsion parameters.

2. Problem Formulation and Indirect Methods

The dataset is constructed by solving two optimal control problems using an indirect method (Pontryagin's Minimum Principle) transformed into Two-Point Boundary Value Problems (TPBVPs):

• Fuel-Optimal: Minimizes propellant consumption.
• Time-Optimal: Minimizes transfer time.
• Techniques: The authors employ modified equinoctial elements (MEE) to avoid singularities, logarithmic homotopy to smooth bang-bang controls, and normalization of costate vectors to improve convergence. For time-optimal multi-revolution transfers, an "orbit-to-orbit" constraint (relaxing the final true longitude) is used to avoid bifurcations and ensure robust convergence.

3. Dimensionality Reduction and Self-Similar Transformation

To enable a single model to generalize across different mission scenarios without retraining, the authors introduce a self-similar transformation that exploits physical invariances:

• Rotational Invariance: The coordinate system is rotated such that the departure position aligns with the x-axis and the arrival position lies in the xy-plane, removing three redundant degrees of freedom.
• Dimensional Invariance: Variables are non-dimensionalized using reference length ($L_0 = \|r_0\|$) and time ($T_0 = \sqrt{L_0^3/\mu}$). This removes explicit dependence on absolute scales and the gravitational parameter $\mu$.
• Input Features: The input vector is reduced from 17 variables to 11 independent, dimensionless features. Crucially, Lambert solver solutions (two-impulse transfer characteristics) are embedded as input features to provide strong physical priors.

4. Neural Network Architecture

• Single-Revolution: Standard deep neural networks with 9 hidden layers and 128 neurons per layer.
• Multi-Revolution: To handle increased nonlinearity and control switching, Residual Multilayer Perceptrons (ResMLP) with skip connections are employed.
• Scaling Law: The authors observe that approximation error decreases linearly with the logarithm of both dataset size and model parameters, with no evidence of saturation in the explored regime. This guided the decision to scale up the dataset size rather than indefinitely increasing model complexity.

Key Contributions

1. Scaling Law Observation: The paper provides empirical evidence that low-thrust trajectory approximation follows a scaling law, where performance improves systematically with increased data and model capacity, justifying the construction of massive datasets.
2. Homotopy Ray Data Generation: A novel strategy to generate high-quality, boundary-focused datasets that capture the critical regions of the design space (low-fuel and reachability boundaries) efficiently.
3. Self-Similar Generalization: The introduction of a transformation allowing a single pretrained model to generalize across different semi-major axes, inclinations, central bodies, and propulsion parameters, eliminating the need for scenario-specific retraining.
4. Open-Source Release: The authors release the trained models, the large-scale dataset, and the implementation code (C++, Python, MATLAB) to the community.

Results and Performance

The models were validated on three fronts:

1. Public Dataset Benchmark: Tested on the dataset from Acciarini et al. [34]. The proposed model achieved a Mean Absolute Error (MAE) of 9.73 m/s (1.21% relative error) for fuel prediction, significantly outperforming the reference model (65.49 m/s, 8.26%), particularly in the low-thrust regime ($\Delta v < 1000$ m/s).
2. GTOC4 Multi-Flyby Mission: Applied to a 10-year, 49-asteroid flyby mission. The model accurately predicted segment-wise $\Delta v$ requirements (MAE ~10 m/s), enabling near-optimal trajectory assessment and demonstrating utility in complex sequential mission planning.
3. Porkchop Plot Generation: Used for rapid screening of Earth-asteroid rendezvous missions. The surrogate model generated cost landscapes (porkchop plots) in negligible time compared to days required by high-fidelity solvers, accurately identifying feasible low-cost windows and handling cases where traditional solvers failed to converge due to local optima.

Accuracy Metrics:

• Single-Revolution: Mean Relative Error (MRE) of 0.014% for fuel ($\Delta v$) and 0.63% for time ($\Delta t$).
• Multi-Revolution: MRE of 0.003% for fuel and 0.54% for time.

Significance and Claims

The paper claims that this work shifts the paradigm of low-thrust trajectory evaluation from a per-scenario "re-solve and re-train" workflow to a task-reusable pretrained model. By leveraging scaling laws and physical symmetries, the framework bridges the gap between computationally expensive optimal control and rapid mission design exploration.

The authors emphasize that their approach lowers the barrier to entry for non-specialists (e.g., planetary scientists), enabling them to perform rapid reachability and cost assessments for next-generation interplanetary missions without requiring deep expertise in optimal control theory. The models are presented as versatile, high-fidelity surrogates capable of handling a wide variety of mission design parameters and orbital environments without retraining.