Diffusion-Based Multiple-Shooting Indirect Optimal Control for Fuel-Optimal Spacecraft Trajectory Generation
Saeid Tafazzol, Ehsan Taheri, Ryne Beeson
eess.SY · cs.LG
Quick verdict
Diffusion transformer warm-starts multiple-shooting HBVPs; gains over random init are convincing but single benchmark, no ablations, baseline fairness open.
Summary
The paper couples a diffusion transformer with multiple-shooting indirect optimal control for fuel-optimal low-thrust transfers. A DDPM with a transformer backbone generates joint state–costate trajectory samples, conditioned on one-step-lag Hamiltonian integrations supplied as context tokens and with boundary-condition components masked from corruption; IPOPT refines samples into solutions of the multiple-shooting HBVP. Training uses ~49,152 extremal trajectories from randomized terminal conditions backward-integrated from zero terminal mass costate, filtered by gradient refinement; the Earth–Mars evaluation case was not in training. On a fixed-time 348.795-day Earth–Mars benchmark (7 states, 4 controls), the method attains 100% convergence in 13 of 15 tested departure windows versus 16.8%–62% for a random-initialization IPOPT baseline, at 18.5 s versus 68.4 s average per trial. Five windows fail for both approaches, and pooled over all tested windows success is roughly 87% versus roughly 42%. Main limitations: no multi-start baseline, no ablations of the headline components, validation on a single planet pair under two-body dynamics, and GPU-seed versus CPU-baseline runtime fairness.
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Assessment at a glance
5
risk flags identified
● high 2
● medium 2
● low 1
4 strengths · 0 weaknesses · 5 author questions
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Key claims under review
- A transformer-based diffusion model with one-step-lag integrated Hamiltonian context generates diverse joint state–costate trajectory samples that satisfy boundary conditions and nearly satisfy optimality conditions.
- Diffusion-generated initializations refined by IPOPT achieve 100% convergence in 13 of 15 tested departure-shift windows of an Earth–Mars benchmark, versus 16.8%–62% for IPOPT from a single random costate guess.
- The diffusion pipeline is faster on average (18.5 s vs 68.4 s per trial) than the random-initialization baseline, which uses CPU only while the pipeline uses an RTX 4090.
- A dataset-generation procedure that backward-integrates from λm(tf)=0 with randomized terminal conditions plus gradient refinement yields ~49,152 diverse extremal trajectories exhibiting bang-bang thrust.
- The trained model generalizes beyond the training dataset to held-out Earth–Mars transfers including one-revolution solutions.
Strengths
Clean train/eval separation supports the generalization claim
The DM is trained only on backward-integrated extremals with randomized terminal conditions, never on the Earth–Mars evaluation case (Sec. III.B); hitting 100% convergence on 13 of 15 shifted departure windows (Table II) is genuine out-of-distribution evidence, including one-revolution solutions (Fig. 3).
Dynamics-informed conditioning is mechanistically motivated
Providing forward-integrated state–costate nodes as auxiliary tokens (Eq. 17-18) exposes the multiple-shooting residuals to self-attention without backpropagating through the ODE — a pragmatic, defensible design choice (Sec. II.C).
Honest, verifiable success criterion
Failure windows disclosed
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Scope & Claims
The paper proposes a pipeline that fuses DDPM sampling with multiple-shooting indirect optimal control. A transformer-based diffusion model (five token streams: noisy states, noisy costates, forward-integrated states, forward-integrated costates, and a diffusion-step token) generates joint state–costate trajectory samples with fixed boundary components masked from corruption (Eq. 12); IPOPT (CasADi) then refines samples into solutions of the multiple-shooting HBVP. Claims: (1) first implicit one-step-lag guidance combining DMs with multiple-shooting indirect control, (2) transformer architecture enabling context learning, (3) a dataset-generation procedure producing diverse optimality-satisfying samples. The benchmark is a fixed-time (348.795-day) Earth–Mars fuel-optimal low-thrust transfer (nx=7, nu=4, Tmax=0.5 N, Isp=2000 s, m0=1000 kg).
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Frequently asked questions
What does 'Diffusion-Based Multiple-Shooting Indirect Optimal Control' actually do?
It pairs a diffusion transformer (conditioned on one-step-lag Hamiltonian integrations of the trajectory) with multiple-shooting indirect optimal control. The diffusion model samples joint state–costate trajectories that satisfy boundary conditions, and IPOPT refines them into solutions of the Hamiltonian boundary-value problem, avoiding hand-tuned costate guesses.
How well does the method converge compared to standard indirect methods?
On a fixed-time 348.795-day Earth–Mars low-thrust transfer, it reached 100% convergence in 13 of 15 tested departure windows over 250 random-seed trials each, versus 16.8%–62% for IPOPT started from a single random costate guess. Five windows failed for both methods, and overall success across all tested windows is about 87% versus about 42% for the baseline.
Does this work generalize to real interplanetary missions?
Not automatically. It was trained only on synthetic two-body extremal trajectories (about 49,152 of them) and was tested on a single Earth–Mars benchmark under two-body dynamics. Other planet pairs, multi-revolution transfers, and higher-fidelity dynamics (e.g., ephemerides, perturbations) are not demonstrated.
Is the code available?
The authors state that code and visualizations are available at saeidtafazzol.github.io/Diffusion_Indirect_Control. The paper text does not confirm whether trained model weights are included, so full reproduction may require retraining the diffusion model.
About this review
This is an AI-generated preliminary review by Evolve.Pub, produced from the openly available text of the paper. It is not peer review and does not represent the authors' views. The canonical record of the paper is at the publisher link above.
- Equation rendering is partially garbled in the extracted text; the mathematical formulation was assessed from recoverable context
- Figures (1-4) are reduced to captions/OCR fragments; visual evidence (e.g., MDS coverage in Fig. 2, Fig. 4 pre/post-refinement curves) could not be inspected
- The contents of the linked project page (completeness of released code, trained weights) could not be verified from the text
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