Tiny Recursive Models for Solving the J2-Perturbed Lambert Problem

Minduli Wijayatunga; Roberto Armellin

arxiv: 2606.00895 · v1 · pith:DTGEUYK3new · submitted 2026-05-30 · 🧮 math.OC · cs.LG

Tiny Recursive Models for Solving the J2-Perturbed Lambert Problem

Minduli Wijayatunga , Roberto Armellin This is my paper

Pith reviewed 2026-06-28 18:06 UTC · model grok-4.3

classification 🧮 math.OC cs.LG

keywords J2-perturbed Lambert problemtiny recursive modelsneural solverrecursive refinementorbit transferastrodynamicsembedded deployment

0 comments

The pith

A 2.3-million-parameter recursive neural model refines departure velocity to solve the J2-perturbed Lambert problem with median terminal errors below one kilometer.

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

The paper introduces TRM-PL, a weight-shared recursive architecture that applies the same small module repeatedly to simulate J2 trajectories and correct an initial departure velocity guess from the resulting position error. This learned refinement loop replaces classical homotopy or continuation methods with an end-to-end differentiable process that unifies initial-guess generation and iterative correction. Training with target-position supervision alone outperforms joint learning of the Lambert solution and J2 correction across single-revolution LEO, multi-revolution LEO, and multi-revolution Jovian transfers. The same compact model produces the reported error reductions while remaining small enough for potential embedded use.

Core claim

TRM-PL applies a compact reasoning module repeatedly within a two-level latent hierarchy to refine a candidate departure velocity by simulating the J2 trajectory and correcting from the resulting tracking error. The recursive refinement loop serves as a learned alternative to hand-designed continuation schemes, unifying initial-guess generation and iterative correction in a single trainable network. On the evaluated transfers, the position-supervised variant reduces median terminal-position error from 21.7 km to 0.027 km for single-revolution LEO and from 340.9 km to 0.31 km for multi-revolution LEO; one Newton corrector iteration then tightens the Jovian median to 0.063 km.

What carries the argument

The Tiny Recursive Model (TRM) refinement loop that repeatedly simulates the J2 trajectory and corrects departure velocity from position tracking error.

If this is right

Refinement-only training with target-position supervision is more reliable than jointly learning the unperturbed Lambert solution and the J2 correction.
The same 2.3M-parameter architecture handles both single-revolution and multi-revolution cases in LEO and Jovian regimes.
One additional Newton corrector iteration on the TRM-PL output further tightens accuracy on the hardest transfers.
The resulting accuracy and model size support potential use in embedded flight software.

Where Pith is reading between the lines

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

The same recursive refinement structure could be applied to other gravitational or non-gravitational perturbations in astrodynamics.
Explicit documentation of dataset generation, diversity, and train-test splits would be needed to confirm generalization claims.
The architecture might replace classical iterative solvers inside real-time trajectory optimizers.
Integration with existing Lambert solvers could provide fast initial guesses for more expensive numerical methods.

Load-bearing premise

The recursive corrections learned on the training transfers will produce accurate results for arbitrary unseen initial and target conditions.

What would settle it

Testing the trained TRM-PL on a collection of J2-perturbed Lambert transfers whose orbital elements or revolution counts lie outside the training distribution and checking whether median terminal position error stays below one kilometer.

Figures

Figures reproduced from arXiv: 2606.00895 by Minduli Wijayatunga, Roberto Armellin.

**Figure 1.** Figure 1: Illustration of the J2-perturbed Lambert problem. The Keplerian Lambert arc (left, dashed green) connects r1 to the target position r2 exactly under two-body dynamics; propagation under J2 from the same initial condition (r1, v1,Lambert) instead reaches the propagated terminal position rf , producing the terminal miss vector r2 − rf (left, dashed blue). The corrected initial velocity v (K) 1 produced by th… view at source ↗

**Figure 2.** Figure 2: Two-head TRM architecture for the J2-perturbed Lambert problem. The two configurations differ only in the source of v (0) 1 : a learned Initial Decoder in (a), the classical Lambert solver in (b). 5 [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** Figure 3: Performance of TRM on the single-revolution LEO dataset [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗

**Figure 4.** Figure 4: Performance of TRM on the multi-revolution LEO dataset [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗

**Figure 5.** Figure 5: Performance of TRM on the multi-revolution Jovian dataset. Results also include the outcomes after a single [PITH_FULL_IMAGE:figures/full_fig_p012_5.png] view at source ↗

read the original abstract

This paper presents a fast, recursive neural solver for the J2-perturbed Lambert problem based on Tiny Recursive Models (TRM), termed the TRM-Perturbed Lambert (TRM-PL) model. TRM is a weight-shared architecture whose effective capacity emerges from iteration depth rather than parameter count: a compact reasoning module is applied repeatedly within a two-level latent hierarchy, refining a candidate departure velocity by simulating the J2 trajectory and correcting it from the resulting tracking error. This unifies initial-guess generation and iterative correction in a single, end-to-end differentiable architecture. The recursive refinement loop is a learned alternative to the homotopy and continuation schemes of classical perturbed-Lambert solvers: rather than following a hand-designed path from the Keplerian to the perturbed solution, the network learns its own sequence of corrections. We evaluate TRM-PL on three test cases of increasing difficulty: single-revolution low-Earth-orbit (LEO) transfers, multi-revolution LEO transfers, and multi-revolution Jovian transfers. Three training paradigms are compared: jointly learning the Lambert solution and the J2 correction; refining the Lambert initial velocity with target-position and J2-corrected velocity supervision; and refining it with target-position supervision alone. Across all cases, the refinement-only approaches are the most reliable. The position-supervised variant reduces the median terminal-position error from 21.7 km to 0.027 km on single-revolution LEO, from 340.9 km to 0.31 km on multi-revolution LEO, all with the same 2.3M-parameter architecture. A single Newton corrector iteration on the TRM-PL output tightens the Jovian median to 0.063 km, yielding compact models accurate enough for embedded deployment.

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.

Desk Editor's Note private letter to a colleague

TRM-PL cuts terminal errors sharply on J2 Lambert cases with a small recursive net, but missing dataset details make generalization hard to verify.

read the letter

The main thing to know is that the position-supervised TRM-PL drops median terminal-position error from 21.7 km to 0.027 km on single-revolution LEO and from 340.9 km to 0.31 km on multi-revolution LEO using the same 2.3 M parameter model, with one Newton step getting Jovian cases to 0.063 km. That is a concrete improvement over the baselines they compare against.

What is new is the direct application of the weight-shared recursive TRM architecture to unify initial-guess generation and J2 correction inside one differentiable loop, plus the head-to-head of three training paradigms on LEO and Jovian multi-revolution transfers. The recursive refinement replaces hand-designed homotopy paths with a learned correction sequence, which is a clean idea for this class of problems.

The work does well at keeping the model compact while showing that refinement-only supervision is more reliable than joint learning, and at demonstrating that the output is accurate enough to be useful after minimal polishing. The numbers are reported consistently across the three test suites.

The soft spot is exactly the one flagged in the stress-test note. The abstract gives no orbital-element ranges, no sampling distribution, no mention of train/test separation, and no description of how multi-revolution cases were generated. Without those, it is impossible to tell whether the error reductions reflect generalization or simply performance inside the training distribution. That gap is load-bearing for the central claim.

This paper is for people building or evaluating learned solvers for classical astrodynamics problems. A reader already working on Lambert or continuation methods would get value from the training-paradigm comparison and the architecture details. It deserves a serious referee because the core approach is coherent and the reported gains are large enough to matter if the experimental setup holds up.

Referee Report

2 major / 1 minor

Summary. The paper presents Tiny Recursive Models (TRM-PL) as a compact, weight-shared recursive neural architecture for solving the J2-perturbed Lambert problem. It unifies initial-guess generation and iterative correction via repeated application of a reasoning module that simulates J2 trajectories and refines departure velocities from tracking errors. Three training paradigms are compared on single-revolution LEO, multi-revolution LEO, and multi-revolution Jovian transfers; the position-supervised variant is reported to reduce median terminal-position error from 21.7 km to 0.027 km (single-revolution LEO) and 340.9 km to 0.31 km (multi-revolution LEO) with a fixed 2.3 M-parameter model, with one Newton iteration further tightening Jovian results to 0.063 km median.

Significance. If the generalization claims hold, the work would demonstrate a parameter-efficient, end-to-end differentiable alternative to classical homotopy/continuation methods for perturbed Lambert solvers, with potential utility for onboard astrodynamics where model size and speed matter. The recursive refinement as a learned substitute for hand-designed paths is conceptually interesting, but the absence of supporting experimental details prevents assessing whether this constitutes a substantive advance.

major comments (2)

[Abstract] Abstract: The headline performance claims (median terminal-position error reductions to 0.027 km and 0.31 km) depend on the assertion that the same 2.3 M-parameter TRM-PL generalizes to unseen transfers. No description is supplied of how the three test suites were generated, including orbital-element ranges, sampling distributions, multi-revolution handling, train/test splits, or any hold-out protocol. Without this information the reported error reductions cannot be distinguished from interpolation on trajectories the model has seen during training.
[Abstract] Abstract (training paradigms section): The manuscript states that three paradigms were compared (joint learning, target-position + J2-velocity supervision, and position supervision alone) but supplies no information on the loss functions, the precise form of the differentiable J2 simulation inside the recursion, convergence criteria, or statistical validation (e.g., number of test cases, variance, or outlier handling). These omissions are load-bearing for the claim that refinement-only approaches are “most reliable.”

minor comments (1)

[Abstract] Abstract: The phrase “all with the same 2.3M-parameter architecture” is repeated without clarifying whether the architecture depth or latent dimension is held fixed across the three test regimes.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and valuable suggestions. We will revise the manuscript to provide the missing experimental details on data generation, training, and validation to strengthen the generalization claims.

read point-by-point responses

Referee: [Abstract] Abstract: The headline performance claims (median terminal-position error reductions to 0.027 km and 0.31 km) depend on the assertion that the same 2.3 M-parameter TRM-PL generalizes to unseen transfers. No description is supplied of how the three test suites were generated, including orbital-element ranges, sampling distributions, multi-revolution handling, train/test splits, or any hold-out protocol. Without this information the reported error reductions cannot be distinguished from interpolation on trajectories the model has seen during training.

Authors: We agree that these details are essential for validating the generalization claims. In the revised manuscript, we will include a comprehensive description of the test suite generation process, specifying the orbital-element ranges, sampling distributions, multi-revolution handling, train/test splits, and hold-out protocols used. revision: yes
Referee: [Abstract] Abstract (training paradigms section): The manuscript states that three paradigms were compared (joint learning, target-position + J2-velocity supervision, and position supervision alone) but supplies no information on the loss functions, the precise form of the differentiable J2 simulation inside the recursion, convergence criteria, or statistical validation (e.g., number of test cases, variance, or outlier handling). These omissions are load-bearing for the claim that refinement-only approaches are “most reliable.”

Authors: We acknowledge that the current manuscript omits these critical implementation and validation details. The revised version will provide explicit information on the loss functions employed for each training paradigm, the structure of the differentiable J2 simulation within the recursive module, the convergence criteria, and statistical validation including the number of test cases, variance measures, and outlier handling methods. revision: yes

Circularity Check

0 steps flagged

No circularity; empirical ML results on simulated trajectories with no self-referential derivation

full rationale

The manuscript presents a trained neural architecture (TRM-PL) whose outputs are obtained by supervised learning on generated trajectory data. Performance claims (error reductions from 21.7 km to 0.027 km, etc.) are empirical evaluations of a fitted model rather than any first-principles derivation or prediction that reduces by construction to fitted parameters or self-citations. No equations, uniqueness theorems, or ansatzes are invoked that collapse to the inputs; the recursive refinement is explicitly learned. The paper is therefore self-contained as a standard supervised-learning result against external simulation benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

Performance claims rest on the trained neural weights and the assumption that the J2 model plus recursive refinement suffices for the tested regimes; no independent evidence for generalization is supplied.

free parameters (1)

TRM model weights = 2.3 million
2.3 million parameters whose values are fitted during training to achieve the reported position errors.

axioms (1)

domain assumption The standard J2 gravitational perturbation model accurately captures the dominant dynamics for the LEO and Jovian test cases considered.
Invoked when the network simulates trajectories and computes tracking error.

pith-pipeline@v0.9.1-grok · 5851 in / 1392 out tokens · 28162 ms · 2026-06-28T18:06:27.511936+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

17 extracted references · 11 canonical work pages

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Zhen Yang, Ya-Zhong Luo, Jin Zhang, and Guo-Jin Tang. Homotopic perturbed Lambert algorithm for long- duration rendezvous optimization.Journal of Guidance, Control, and Dynamics, 38(11):2215–2223, 2015. doi: 10.2514/1.G001198

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Mohammad Alhulayil, Ahmad Bani Younes, and James D. Turner. Higher order algorithm for solving Lambert’s problem.The Journal of the Astronautical Sciences, 65(4):400–422, 2018. doi: 10.1007/s40295-018-0137-9. 13 Tiny Recursive Models for theJ 2-Perturbed Lambert Problem Table 5: Sample statistics for the three training datasets, including mean, standard d...

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Tiny recursive control: Iterative reasoning for efficient optimal control

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work page doi:10.2514/6.2026-2380 2026
[15]

Wijayatunga, Roberto Armellin, and Laura Pirovano

Minduli C. Wijayatunga, Roberto Armellin, and Laura Pirovano. Exploiting scaling constants to facilitate the convergence of indirect trajectory optimization methods.Journal of Guidance, Control, and Dynamics, 46(5), 2023

2023
[16]

Transformer- based model predictive control: Trajectory optimization via sequence modeling.IEEE Robotics and Automation Letters, 9(11):9820–9827, 2024

Davide Celestini, Daniele Gammelli, Tommaso Guffanti, Simone D’Amico, and Marco Pavone. Transformer- based model predictive control: Trajectory optimization via sequence modeling.IEEE Robotics and Automation Letters, 9(11):9820–9827, 2024

2024
[17]

RT-2: Vision-language-action models transfer web knowledge to robotic control, 2023

Anthony Brohan et al. RT-2: Vision-language-action models transfer web knowledge to robotic control, 2023. Appendix: Training and Validation Dataset Statistics 14 Tiny Recursive Models for theJ 2-Perturbed Lambert Problem Table 6: Sample statistics for the validation dataset, including mean, standard deviation, 10th percentile p10, and 90th percentilep 90...

2023

[1] [1]

Battin.An Introduction to the Mathematics and Methods of Astrodynamics

Richard H. Battin.An Introduction to the Mathematics and Methods of Astrodynamics. AIAA Education Series, revised edition, 1999. doi: 10.2514/4.861543

work page doi:10.2514/4.861543 1999

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Bate, Donald D

Roger R. Bate, Donald D. Mueller, and Jerry E. White.Fundamentals of Astrodynamics. Dover Publications, 1971

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Wijayatunga, Roberto Armellin, Harry Holt, Laura Pirovano, and Aleksander A

Minduli C. Wijayatunga, Roberto Armellin, Harry Holt, Laura Pirovano, and Aleksander A. Lidtke. Design and guidance of a multi-active debris removal mission.Astrodynamics, 2023. doi: 10.1007/s42064-023-0159-3

work page doi:10.1007/s42064-023-0159-3 2023

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Izzo, Revisiting Lambert’s problem, Celestial Mechanics and Dynamical Astronomy 121 (1) (2015) 1–15

Dario Izzo. Revisiting Lambert’s problem.Celestial Mechanics and Dynamical Astronomy, 121(1):1–15, 2015. doi: 10.1007/s10569-014-9587-y

work page doi:10.1007/s10569-014-9587-y 2015

[5] [5]

San Juan

Roberto Armellin, David Gondelach, and Juan F. San Juan. Multiple revolution perturbed Lambert problem solvers.Journal of Guidance, Control, and Dynamics, 41(9):2019–2032, 2018. doi: 10.2514/1.G003531

work page doi:10.2514/1.g003531 2019

[6] [6]

Potential of Diffractive Sail’s Sun-Earth Equilibria for Continuous Polar Observation,

Bin Yang, Shuang Li, Jinglang Feng, and Massimiliano Vasile. Fast solver for J2-perturbed Lambert problem using deep neural network.Journal of Guidance, Control, and Dynamics, 45(5):875–884, 2022. doi: 10.2514/1. G006091

work page doi:10.2514/1 2022

[7] [7]

L. G. Kraige, John L. Junkins, and L. D. Ziems. Regularized integration of gravity-perturbed trajectories: A numerical efficiency study.Journal of Spacecraft and Rockets, 19(4):291–293, 1982. doi: 10.2514/3.62255

work page doi:10.2514/3.62255 1982

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Russell, Nathan Strange, and David Ottesen

Nitin Arora, Ryan P. Russell, Nathan Strange, and David Ottesen. Partial derivatives of the solution to the Lambert boundary value problem.Journal of Guidance, Control, and Dynamics, 38(9):1563–1572, 2015. doi: 10.2514/1.G001030

work page doi:10.2514/1.g001030 2015

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Woollands, Ahmad Bani Younes, and John L

Robyn M. Woollands, Ahmad Bani Younes, and John L. Junkins. New solutions for the perturbed Lambert problem using regularization and Picard iteration.Journal of Guidance, Control, and Dynamics, 38(9):1548–1562,

[10] [10]

doi: 10.2514/1.G001028

work page doi:10.2514/1.g001028

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Homotopic perturbed Lambert algorithm for long- duration rendezvous optimization.Journal of Guidance, Control, and Dynamics, 38(11):2215–2223, 2015

Zhen Yang, Ya-Zhong Luo, Jin Zhang, and Guo-Jin Tang. Homotopic perturbed Lambert algorithm for long- duration rendezvous optimization.Journal of Guidance, Control, and Dynamics, 38(11):2215–2223, 2015. doi: 10.2514/1.G001198

work page doi:10.2514/1.g001198 2015

[12] [12]

Mohammad Alhulayil, Ahmad Bani Younes, and James D. Turner. Higher order algorithm for solving Lambert’s problem.The Journal of the Astronautical Sciences, 65(4):400–422, 2018. doi: 10.1007/s40295-018-0137-9. 13 Tiny Recursive Models for theJ 2-Perturbed Lambert Problem Table 5: Sample statistics for the three training datasets, including mean, standard d...

work page doi:10.1007/s40295-018-0137-9 2018

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Less is more: Recursive reasoning with tiny networks, 2025

Alexia Jolicoeur-Martineau. Less is more: Recursive reasoning with tiny networks, 2025. URL https://arxiv. org/abs/2510.04871

Pith/arXiv arXiv 2025

[14] [14]

Tiny recursive control: Iterative reasoning for efficient optimal control

Amit Jain and Richard Linares. Tiny recursive control: Iterative reasoning for efficient optimal control. InAIAA SciTech F orum, Orlando, FL, 2026. doi: 10.2514/6.2026-2380

work page doi:10.2514/6.2026-2380 2026

[15] [15]

Wijayatunga, Roberto Armellin, and Laura Pirovano

Minduli C. Wijayatunga, Roberto Armellin, and Laura Pirovano. Exploiting scaling constants to facilitate the convergence of indirect trajectory optimization methods.Journal of Guidance, Control, and Dynamics, 46(5), 2023

2023

[16] [16]

Transformer- based model predictive control: Trajectory optimization via sequence modeling.IEEE Robotics and Automation Letters, 9(11):9820–9827, 2024

Davide Celestini, Daniele Gammelli, Tommaso Guffanti, Simone D’Amico, and Marco Pavone. Transformer- based model predictive control: Trajectory optimization via sequence modeling.IEEE Robotics and Automation Letters, 9(11):9820–9827, 2024

2024

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RT-2: Vision-language-action models transfer web knowledge to robotic control, 2023

Anthony Brohan et al. RT-2: Vision-language-action models transfer web knowledge to robotic control, 2023. Appendix: Training and Validation Dataset Statistics 14 Tiny Recursive Models for theJ 2-Perturbed Lambert Problem Table 6: Sample statistics for the validation dataset, including mean, standard deviation, 10th percentile p10, and 90th percentilep 90...

2023