pith. sign in

arxiv: 2606.30936 · v1 · pith:YDMTUMTJnew · submitted 2026-06-29 · 💻 cs.LG · astro-ph.EP· astro-ph.IM· cs.AI

Physics-informed Conditional Normalizing Flows for Angles-only Cislunar Orbit Determination

Walther Litteri , Massimiliano Vasile This is my paper

Pith reviewed 2026-07-01 06:08 UTC · model grok-4.3

classification 💻 cs.LG astro-ph.EPastro-ph.IMcs.AI
keywords normalizing flowsorbit determinationcislunarangles-onlyconditional density estimationnear rectilinear halo orbitsgenerative modeling
0
0 comments X

The pith

A conditional normalizing flow trained on perturbed NRHO observations learns flexible posteriors over initial states from short-arc angles-only measurements.

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

The paper frames cislunar orbit determination as a conditional density estimation task solved with normalizing flows. The model is trained on sequences of topocentric angle observations generated from Near Rectilinear Halo Orbits under realistic perturbations. Once trained, the flow produces samples from the posterior distribution over possible initial spacecraft states given new measurements. These samples serve as statistically consistent starting points that are then polished by classical nonlinear least-squares. The result is a method that can represent uncertainty and multimodality while remaining anchored in the underlying orbital dynamics.

Core claim

A conditional normalizing flow trained on perturbed topocentric observations from Near Rectilinear Halo Orbits learns a flexible and potentially multimodal posterior over initial states conditioned on angles-only measurements; sampling from this density yields physics-informed state hypotheses that are refined by nonlinear least-squares to give competitive warm starts for classical orbit determination algorithms.

What carries the argument

Conditional normalizing flow performing density estimation of initial orbital states given short sequences of angles-only topocentric observations.

If this is right

  • The learned density can represent multimodal posteriors when short-arc angles-only data leave multiple orbits consistent with the measurements.
  • State hypotheses sampled from the flow are statistically consistent with the training dynamics and measurement noise model.
  • The sampled hypotheses supply warm starts that improve convergence of nonlinear least-squares for angles-only problems.
  • Generative modelling is extended from simulation to a practical cislunar orbit determination pipeline.

Where Pith is reading between the lines

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

  • The same training strategy could be repeated for other cislunar orbit families or different sensor combinations to broaden coverage.
  • If the posterior quality holds under distribution shift, the approach may reduce reliance on long observation arcs or additional range data.
  • Embedding the flow in an onboard filter could support autonomous navigation with limited ground contact.

Load-bearing premise

Training exclusively on perturbed topocentric observations from Near Rectilinear Halo Orbits produces a posterior that remains accurate and physics-consistent for real or out-of-distribution angles-only measurements in the cislunar regime.

What would settle it

Test the trained flow on real angles-only observations from a cislunar spacecraft on a non-NRHO trajectory and measure whether the refined state estimates recover the true orbit within the uncertainty reported by the posterior.

read the original abstract

Generative Astrodynamics is advanced in this work by extending generative modelling to an orbit determination problem in the cislunar environment. The task is formulated as conditional density estimation, aiming to infer the probability distribution of the initial state from angles-only measurements over short observation arcs. A normalising flow is trained on perturbed topocentric observations from Near Rectilinear Halo Orbits, enabling a flexible and potentially multimodal posterior representation. Given new measurements, the learned density is sampled to generate statistically consistent and physics-informed state hypotheses. These estimates are refined via nonlinear least-squares minimisation, providing a competitive warm start for classical algorithms.

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 proposes training a conditional normalizing flow on perturbed topocentric angles from simulated Near Rectilinear Halo Orbit (NRHO) trajectories to perform angles-only orbit determination in the cislunar regime. Given new short-arc measurements, the flow models the posterior over initial states, which is sampled to produce candidate state vectors that are then refined by nonlinear least-squares to supply warm starts for classical estimators.

Significance. If the central claims hold, the work would demonstrate a practical use of conditional normalizing flows to represent potentially multimodal posteriors in a physics-constrained setting, offering an alternative initialization strategy for angles-only cislunar OD where traditional batch estimators can be sensitive to poor initial guesses. The explicit coupling of generative density estimation with subsequent NLS refinement is a methodological strength that could be extended to other under-observed regimes.

major comments (2)
  1. [Abstract] Abstract: the claim that the sampled states are 'statistically consistent and physics-informed' and provide 'competitive warm starts' cannot be evaluated because the abstract (and the description supplied) contains no quantitative results, error metrics, baseline comparisons, or validation details on either in-distribution or out-of-distribution cases.
  2. [Abstract] Abstract / training description: the method is trained exclusively on perturbed topocentric observations from NRHO trajectories, yet the central claim of applicability to 'general angles-only cislunar OD' is not supported by any reported hold-out tests on other orbit families, different dynamical regimes, or realistic sensor noise models; if the conditional density overfits to NRHO-specific geometry and observability, the generated hypotheses will not remain accurate or reliable for NLS initialization outside the training distribution.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments on our manuscript. We agree that the abstract requires quantitative support and that the scope of validation should be clarified. We have revised the abstract accordingly and added explicit discussion of limitations. Point-by-point responses follow.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the claim that the sampled states are 'statistically consistent and physics-informed' and provide 'competitive warm starts' cannot be evaluated because the abstract (and the description supplied) contains no quantitative results, error metrics, baseline comparisons, or validation details on either in-distribution or out-of-distribution cases.

    Authors: We agree that the original abstract was too concise and omitted supporting metrics. The revised abstract now includes summary quantitative results from the experiments, such as median position and velocity errors on the test set, the fraction of samples that converge under NLS refinement, and a direct comparison against a random-initialization baseline. These additions allow the claims to be evaluated without requiring the reader to consult the full text. revision: yes

  2. Referee: [Abstract] Abstract / training description: the method is trained exclusively on perturbed topocentric observations from NRHO trajectories, yet the central claim of applicability to 'general angles-only cislunar OD' is not supported by any reported hold-out tests on other orbit families, different dynamical regimes, or realistic sensor noise models; if the conditional density overfits to NRHO-specific geometry and observability, the generated hypotheses will not remain accurate or reliable for NLS initialization outside the training distribution.

    Authors: The work deliberately targets NRHOs because they represent a demanding cislunar regime with limited observability and strong nonlinearity. The conditional flow architecture itself is not orbit-family-specific, but all reported training and testing data are drawn from NRHO trajectories. We have revised the abstract and added a dedicated limitations paragraph stating that generalization beyond NRHOs has not been demonstrated and would require retraining and validation on additional orbit classes. The core methodological contribution on coupling conditional density estimation with NLS refinement therefore remains scoped to the demonstrated regime. revision: yes

Circularity Check

0 steps flagged

No circularity: standard conditional density estimation on simulated data

full rationale

The paper trains a conditional normalizing flow on perturbed topocentric angles from NRHO trajectories to represent the posterior over initial states, then samples and refines via NLS. This is ordinary supervised generative modeling with no equation that equates a claimed prediction to a fitted input by construction, no load-bearing self-citation chain, and no uniqueness theorem imported from the authors' prior work. The derivation chain remains self-contained against external simulation benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract supplies insufficient technical detail to enumerate free parameters, axioms, or invented entities.

pith-pipeline@v0.9.1-grok · 5634 in / 1030 out tokens · 28926 ms · 2026-07-01T06:08:25.409838+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

18 extracted references · 9 canonical work pages · 1 internal anchor

  1. [1]

    Vallado,Fundamentals of Astrodynamics and Applications

    D. Vallado,Fundamentals of Astrodynamics and Applications. Mar. 1997

    1997

  2. [2]

    Lunar Water: A Brief Review,

    M. Anand, “Lunar Water: A Brief Review,”Earth, Moon, and Planets, V ol. 107, Dec. 2010, pp. 65–73, 10.1007/s11038-010-9377-9

    work page doi:10.1007/s11038-010-9377-9 2010

  3. [3]

    Artemis: An Overview of NASA’s Activities to Re- turn Humans to the Moon,

    S. Creech, J. Guidi, and D. Elburn, “Artemis: An Overview of NASA’s Activities to Re- turn Humans to the Moon,”2022 IEEE Aerospace Conference (AERO), 2022, pp. 1–7, 10.1109/AERO53065.2022.9843277

    work page doi:10.1109/aero53065.2022.9843277 2022

  4. [4]

    State-Space Inference for Non-Linear Latent Force Models with Application to Satellite Orbit Prediction

    J. Hartikainen, M. Seppanen, and S. Sarkka, “State-Space Inference for Non-Linear Latent Force Models with Application to Satellite Orbit Prediction,” June 2012. arXiv:1206.4670 [cs], 10.48550/arXiv.1206.4670

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.1206.4670 2012

  5. [5]

    Machine Learning in Orbit Estimation: a Survey,

    F. Caldas and C. Soares, “Machine Learning in Orbit Estimation: a Survey,”Acta Astronautica, V ol. 220, July 2024, pp. 97–107. arXiv:2207.08993 [astro-ph], 10.1016/j.actaastro.2024.03.072

    work page doi:10.1016/j.actaastro.2024.03.072 2024

  6. [6]

    An Expert's Guide to Training Physics- informed Neural Networks,

    S. Wang, S. Sankaran, H. Wang, and P. Perdikaris, “An Expert’s Guide to Training Physics-informed Neural Networks,” Aug. 2023. arXiv:2308.08468 [cs], 10.48550/arXiv.2308.08468

    work page doi:10.48550/arxiv.2308.08468 2023

  7. [7]

    Machine Learning with Physics Knowledge for Prediction: A Survey,

    J. Watson, C. Song, O. Weeger, T. Gruner, A. T. Le, K. Pompetzki, A. Hendawy, O. Arenz, W. Tro- jak, M. Cranmer, C. D’Eramo, F. B ¨ulow, T. Goyal, J. Peters, and M. W. Hoffman, “Machine Learning with Physics Knowledge for Prediction: A Survey,” May 2025. arXiv:2408.09840 [cs], 10.48550/arXiv.2408.09840

    work page doi:10.48550/arxiv.2408.09840 2025

  8. [8]

    Physics-Informed Orbit Deter- mination for Cislunar Space Applications,

    A. Scorsoglio, A. D’Ambrosio, L. Ghilardi, R. Furfaro, and V . Reddy, “Physics-Informed Orbit Deter- mination for Cislunar Space Applications,” 2023

    2023

  9. [9]

    a ubener, Sophie Fellenz, Asja Fischer, Thomas G \

    L. Manduchi, K. Pandey, C. Meister, R. Bamler, R. Cotterell, S. D ¨aubener, S. Fellenz, A. Fischer, T. G¨artner, M. Kirchler, M. Kloft, Y . Li, C. Lippert, G. d. Melo, E. Nalisnick, B. Ommer, R. Ranganath, M. Rudolph, K. Ullrich, G. V . d. Broeck, J. E. V ogt, Y . Wang, F. Wenzel, F. Wood, S. Mandt, and V . Fortuin, “On the Challenges and Opportunities in...

    work page doi:10.48550/arxiv.2403.00025 2025

  10. [10]

    Litteri, A

    W. Litteri, A. Francisco Gil, M. Vasile, V . Rodriguez-Fernandez, and D. Camacho,Generative Astrody- namics: Trajectory Analysis and Design in the Restricted Three-Body Problem. Jan. 2025

    2025

  11. [11]

    Generation of Periodic Orbits in the Restricted Three-Body Problem with a Variational Autoencoder,

    W. Litteri, A. F. Gil, M. Vasile, V . Rodriguez-Fernandez, and D. Camacho, “Generation of Periodic Orbits in the Restricted Three-Body Problem with a Variational Autoencoder,” Jan. 2026. ISSN: 2693- 5015, 10.21203/rs.3.rs-8651179/v1

    work page doi:10.21203/rs.3.rs-8651179/v1 2026

  12. [12]

    Generating stable and metastable critical points in uncertain systems via flow-based models,

    C. Wilson and M. Vasile, “Generating stable and metastable critical points in uncertain systems via flow-based models,”Expert Systems, V ol. 43, Jan. 2026, 10.1111/exsy.70196

    work page doi:10.1111/exsy.70196 2026

  13. [13]

    S. Ross, W. Koon, M. Lo, and J. Marsden,Dynamical Systems, the Three-Body Problem, and Space Mission Design. Oct. 2022

    2022

  14. [14]

    Normalizing Flows: An Introduction and Review of Current Methods,

    I. Kobyzev, S. J. D. Prince, and M. A. Brubaker, “Normalizing Flows: An Introduction and Review of Current Methods,”IEEE Transactions on Pattern Analysis and Machine Intelligence, V ol. 43, Nov. 2021, pp. 3964–3979

    2021

  15. [15]

    NICE: Non-linear Independent Components Estimation,

    L. Dinh, D. Krueger, and Y . Bengio, “NICE: Non-linear Independent Components Estimation,”Pro- ceedings of the International Conference on Learning Representations (ICLR) Workshop, San Diego, CA, USA, 2015

    2015

  16. [16]

    Normaliz- ing Flows for Probabilistic Modeling and Inference,

    G. Papamakarios, E. Nalisnick, D. J. Rezende, S. Mohamed, and B. Lakshminarayanan, “Normaliz- ing Flows for Probabilistic Modeling and Inference,”Journal of Machine Learning Research, V ol. 22, No. 57, 2021, pp. 1–64

    2021

  17. [17]

    Density Estimation Using Real NVP,

    L. Dinh, J. Sohl-Dickstein, and S. Bengio, “Density Estimation Using Real NVP,”Proceedings of the International Conference on Learning Representations (ICLR), Toulon, France, Apr. 2017

    2017

  18. [18]

    Attention Is All You Need,

    A. Vaswani, N. Shazeer, N. Parmar, J. Uszkoreit, L. Jones, A. N. Gomez, Ł. Kaiser, and I. Polosukhin, “Attention Is All You Need,”Advances in Neural Information Processing Systems, V ol. 30, 2017. 16

    2017