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A structure-preserving neural network, LLPNN, learns latent Lie–Poisson dynamics from observable data and reconstructs unobservable momentum trajectories via a conserved Noether invariant.

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2026-08-03 16:50 UTC pith:PYTX54BZ

load-bearing objection Clever Noether-based latent reconstruction and clean experiments, but the central identifiability claim is unproven and false for some linear Hamiltonians—overbroad as stated. the 2 major comments →

arxiv 2607.28939 v1 pith:PYTX54BZ submitted 2026-07-31 cs.LG math.DSmath.OC

Latent Lie-Poisson Neural Networks (LLPNNs): Discovering the motion of Lie-Poisson systems through observable data and latent dynamics

Vakhtang Putkaradze This is my paper
classification cs.LG math.DSmath.OC
keywords Lie–Poisson systemsstructure-preserving neural networksNoether's theoremlatent dynamicsdegenerate Hamiltonianscoadjoint reconstructionsymmetry reductionoptimal control
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper tries to establish that for Hamiltonian systems with a known Lie-group symmetry, it is possible to learn the full latent momentum dynamics from observations of only configurations and velocities, even when the Hamiltonian is degenerate and no Lagrangian or Euler–Poincaré formulation exists. The key bridge is Noether's theorem: for each trajectory the spatial momentum p0 is constant, so the latent momentum at every time is exactly a coadjoint action of the observed group element on p0. Because all latent updates are coadjoint actions, Casimirs are preserved to machine precision and the flow stays on the correct coadjoint orbit. The method is demonstrated on rigid-body motion, an underwater vehicle, and a multi-vehicle optimal control problem, where non-geometric baselines fail because observable velocities do not form a closed dynamical system.

Core claim

Central claim: if the observed configuration g(t) evolves on a known Lie group G and the reduced Hamiltonian h(μ) is time-independent, then the unobservable momentum μ(t) is exactly Ad*_{g(t)} p0 for a constant, per-trajectory spatial momentum p0. This identity turns the learning problem into one of fitting h(μ) (or a pseudo-Lagrangian on active velocities) and a constant vector p0 per trajectory from data; once p0 is found from a short fitting window, both observable and latent trajectories are reconstructed by coadjoint updates and Magnus-based group dynamics. The framework therefore works for degenerate Hamiltonians, where the observable variables need not admit an autonomous evolution eq

What carries the argument

The central object is the coadjoint reconstruction identity μ(t)=Ad*_{g(t)} p0, a direct consequence of Noether's theorem for time-independent, left-invariant Lie–Poisson systems. It provides an exact bridge from the observed configuration g(t) to the unobservable momentum μ(t). Combined with a Hamiltonian decoder h(μ) or a pseudo-Lagrangian encoder ℓ(ξ_act), it yields residual equations for the next step that are solved with a second-order Magnus expansion and Newton–Raphson root-finding. Because each latent update is a coadjoint action, the flow remains on coadjoint orbits and preserves all Casimirs to machine precision.

Load-bearing premise

The entire latent reconstruction rests on the assumption that the system has a known, time-independent Lie-group symmetry, so Noether's theorem yields a constant spatial momentum p0 for each trajectory; if the symmetry is broken or the group is unknown, the mapping μ(t)=Ad*_{g(t)} p0 collapses and the method cannot be formulated.

What would settle it

Train the method on a Hamiltonian system with a known Lie-group symmetry but with a small symmetry-breaking potential added. If reconstruction error grows systematically with the symmetry-breaking magnitude and p0 drifts, the Noether-based identity is falsified. More directly, construct an observation operator that hides the active momentum component: if p0 cannot be recovered from the short fitting window even with clean data, the latent reconstruction fails in a way the paper leaves open.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Long-term prediction of symmetry-reduced Hamiltonian systems becomes possible from observable data alone, with machine-precision Casimir conservation built into the architecture.
  • Degenerate Hamiltonian systems, including optimal control problems with unobservable co-states, become learnable even when no Euler–Poincaré or Lagrangian formulation exists.
  • The learned Hamiltonian closely matches the true Hamiltonian (absolute errors around 1e-4 in the tested region), suggesting the method recovers the physical or control Hamiltonian, not merely trajectory fits.
  • The framework handles both regular systems and systems with passive coordinates, with active and passive variables split from data by identifying constant velocity components.
  • Reconstruction is robust to moderate observational noise (η=0.005), with degraded but still accurate long-term predictions in the tested examples.

Where Pith is reading between the lines

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

  • A direct consequence of the paper's identity is that, for any time-independent Lie–Poisson system with known symmetry, the latent momentum is identifiable up to an overall scale from configuration data alone—if p0 is recoverable from a short window. This suggests a general observability criterion could be developed for Lie–Poisson systems.
  • The framework could be extended to inverse optimal control: by learning the Hamiltonian from observed agent trajectories, one might recover the underlying cost or coordination structure without ever measuring co-states.
  • The paper itself flags that the known-symmetry assumption is restrictive (Limitation 1, Section 5) and that p0 identifiability under general observation operators remains open (Section 6). A natural next step is to characterize when the short fitting window is sufficient.
  • Since the reconstruction identity depends on g(t) but not directly on ξ(t), replacing velocity observations with finite differences of g might make the method applicable to position-only data; this is an editorial inference, not a claim of the paper.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 5 minor

Summary. The paper introduces Latent Lie–Poisson Neural Networks (LLPNNs), a geometric deep-learning method for systems whose state is a Lie–Poisson momentum μ(t) that is not directly observed. Available data are Lie-group configurations g(t) and reduced velocities ξ(t). The central identity is Noether's theorem: for a left-invariant Hamiltonian, the spatial momentum p0 is constant per trajectory and μ(t)=Ad^*_{g(t)} p0. The paper develops two training variants: a Hamiltonian decoder that learns h(μ)≈h_NN(μ) and fits p0 per trajectory via the nonlinear loss (23); and a pseudo-Lagrangian encoder that learns ℓ(ξ_act) and fits p0 from a linear system (32). Reconstruction propagates μ by coadjoint actions and g by a second-order Magnus update, so Casimirs are preserved exactly. The method is demonstrated on a non-quadratic rigid body (SO(3)), Kirchhoff's underwater vehicle (SE(3)), and a degenerate three-vehicle optimal-control Hamiltonian (SE(2)^3), reporting long-horizon MAE and conservation errors below baselines (Neural ODE, DeepONet, LPNet).

Significance. If the identifiability issues are resolved, the paper makes a useful contribution: it provides a principled way to infer latent momentum variables from observable data by leveraging the Noether constant, and it handles degenerate Hamiltonians for which an Euler–Poincaré formulation does not exist. The construction is elegant and the numerical evidence is encouraging; Casimir conservation is exact by construction rather than by penalty. The code is publicly available, which supports reproducibility. However, the central claim of latent reconstruction is not universally valid without an observability condition on p0, and the paper's own Section 6 acknowledges the gap. The recommendations below therefore ask for a sharpened statement of scope and either a proof or explicit assumption of identifiability.

major comments (2)
  1. [Section 3.1.1 / 6 (Eqs. (10), (21)–(23))] The reconstruction of μ(t) is predicated on identifiability of p0 from a short window of (g,ξ), but no sufficient condition is given. This is not a technicality: there are Hamiltonians satisfying all stated assumptions for which p0 is completely unobservable. For SE(2)^N with Ψ=0 in Eq. (65), h(μ)=Σ_k μ_{k1}. Then ξ_k=(1,0,0) for every μ, so g(t)=g0 exp(t ξ) and the entire observed trajectory is independent of μ. The LLPNN loss (23) vanishes identically for any p0, and the learned latent trajectory is arbitrary. This is a linear, time-independent Hamiltonian inside the paper's scope. The paper should either formulate and prove observability conditions on (h, observation map), or explicitly restrict the central claims to systems for which p0 is identifiable. As written, the abstract and Section 2.3 overstate the generality.
  2. [Remark 2.1 / Section 4] The momentum scale ambiguity in Remark 2.1 means that μ is only reconstructed up to a positive constant c; the reported latent trajectories are rescaled post hoc using an initial test window or Hessian trace. Consequently, the method does not identify the physical scale of the momentum/co-state from observable data. This is a second fundamental identifiability gap that should be stated in the abstract and conclusions. The quantitative latent momentum errors reported in Figures 4, 9, and 13 are scale-adjusted, and the text should make this explicit rather than presenting them as direct estimates of μ.
minor comments (5)
  1. [Section 3.6] The LPNet baseline is trained on latent trajectories produced by the learned LLPNN Hamiltonian, as the paper states. This comparison should be presented as a sanity check, not as evidence that LPNet would fail if given true latent data. Please add a clear caveat in the results section and avoid drawing strong conclusions from this baseline.
  2. [Equations (26)–(27)] Notation is inconsistent: h is used both for the Hamiltonian and for the time step in (27). Use Δt or τ for the time step and make the Magnus residual equation dimensionally explicit.
  3. [Figure captions (Figures 3, 4, 12, 13)] The 'Fit Window' is not quantitatively specified in the captions. Please state N_fit and the noise level for each figure, and define the legend entries consistently across panels.
  4. [Section 4.2] Training and test initial momenta are drawn from different distributions (N(0,I3) vs. N(0,0.25I3)). The paper does not discuss why this distribution shift was chosen or its effect on generalization. Please comment on this choice in the text.
  5. [References] Reference [23] contains a typo in the author name ('Vakhtangee'); the name should be corrected. Please also check reference formatting throughout.

Circularity Check

1 steps flagged

Core derivation is independent supervised Lie–Poisson learning; only the post-hoc momentum-scale calibration is a fitted (not predicted) quantity, and the paper's own observability limitation is a correctness caveat, not circularity.

specific steps
  1. fitted input called prediction [Remark 2.1 (Momentum Gauge Ambiguity and Scaling); cf. Figures 4/9/13 'Scaled Momentum Components']
    "Because ch(µ/c) generates identical velocity fields ξ, the absolute scale of µ cannot be uniquely determined from velocity observations alone. To evaluate learned latent states against physical ground-truth trajectories, predicted momenta are rescaled post-hoc using the Hessian trace ratio at the origin or a scale factor estimated from an initial test window."

    The latent-momentum reconstruction μ(t)=Ad*_{g(t)} p0 inherits the gauge freedom of Eq. (10): any rescaling of p0 (equivalently of h) yields exactly the same observable velocity data. The paper therefore cannot identify the absolute magnitude of μ from observable data alone, and it supplies the missing scale post-hoc using ground-truth information or the test window. The reported 'Scaled Momentum Components' thus compare ground truth against a curve whose absolute scale was fitted/calibrated to the target, so the magnitude part of the latent 'reconstruction' is fitted rather than predicted. This is a disclosed, partial circularity; the future velocity predictions and the shape of μ(t) remain genuinely out-of-sample.

full rationale

I walked the derivation chain: Eq. (10) (Noether coadjoint reconstruction) is standard textbook mechanics; the passive/active split and ansatz (20) are stated modeling assumptions, not definitions of the target quantities. Training minimizes (23) jointly over h_NN and per-trajectory p0_j, which is supervised fitting, and test-time p0 is fitted to a short initial window (Eqs. 24, 33) before forecasting the remaining trajectory — ordinary initial-condition fitting, not a circular prediction. The self-citations [27]–[29] are used for baseline LPNets and for the SE(2)^N optimal-control model, but the load-bearing reconstruction formula is not imported from those papers as an unverified uniqueness result. The only concrete circular element is Remark 2.1: because the absolute scale of μ is unidentifiable from ξ, the paper rescales predicted momenta post-hoc using the Hessian trace ratio or a test-window scale factor, so the reported momentum magnitude is fitted, not predicted. Section 6 explicitly says observability of p0 remains an open question; that is a genuine identifiability limitation and a correctness risk, but it is not a circular derivation. Overall the central velocity-prediction claim is self-contained and out-of-sample, so the circularity score is low.

Axiom & Free-Parameter Ledger

4 free parameters · 4 axioms · 0 invented entities

The method introduces per-trajectory latent constants p0 and neural-network parameters as fitted degrees of freedom. The main domain assumptions are known symmetry and observability of p0 from a short window; no new physical entities are postulated.

free parameters (4)
  • Noether constant p0^j per training trajectory = M trajectories × dim(g*) (e.g., 3 for SO(3), 6 for SE(3), 9 for SE(2)^3)
    In Step 1 (Eq. 21), each training trajectory is assigned a constant spatial momentum p0^j, optimized jointly with network weights in loss (23). This adds M*dim(g*) parameters that store initial latent states.
  • Neural network weights W for h_NN or ℓ_NN = ~2K parameters per network (3 hidden layers × 32 units)
    Optimized via loss (23) or (32).
  • Warm-start matrices B and K = K × K symmetric matrices
    Trainable symmetric matrices in (37)-(38), initialized via KKT system with tr(B)=1; part of the model.
  • Post-hoc momentum scale factor = estimated per trajectory from initial test window or Hessian trace ratio
    Remark 2.1: velocity data determine µ only up to scaling c>0; predicted momenta are rescaled using a scale estimated from an initial test window before computing MAE in Figures 4, 9, 13. This uses test data to set the scale, weakening the latent-reconstruction measurement.
axioms (4)
  • domain assumption The system has a known Lie group symmetry G and its Lie–Poisson bracket Λ(µ) is known a priori; the Hamiltonian h(µ) is unknown but time-independent.
    Problem formulation (Section 2.3) assumes the Lie–Poisson matrix is known; Remark 3.2 assumes time-independent h. The method's core reconstruction (10) relies on left-invariance and Noether conservation.
  • standard math Conserved spatial momentum p0 exists and is constant per trajectory (Noether's theorem)
    Eq. (10) and Appendix B; valid for left-invariant systems. This is a theorem, so standard_math.
  • domain assumption Observed data comprise both group elements g(t) and velocities ξ(t), with enough information to determine p0 from a short fitting window
    Section 2.3 and Section 6: observability of p0 is assumed but not proven; the paper states it remains an open question.
  • domain assumption Active/passive decomposition of coordinates is identifiable from data (Definition 3.1) and passive velocities are constant; active velocity–momentum map is locally invertible for the Lagrangian LLPNN (Eq. 29)
    Required for the pseudo-Lagrangian encoder; if the active Hessian is singular, the Lagrangian formulation fails and the Hamiltonian formulation's p0 identification may also fail.

pith-pipeline@v1.3.0-daily-deepseek · 35751 in / 14483 out tokens · 122383 ms · 2026-08-03T16:50:24.372967+00:00 · methodology

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read the original abstract

Structure-preserving neural networks are essential for the long-term prediction of Hamiltonian systems from data. Many important Hamiltonian systems in mechanics and control admit symmetry reduction to Lie--Poisson systems, including rigid bodies, underwater vehicles, fluids, plasmas, and optimal control problems. A fundamental challenge in learning such systems is that their dynamics evolve in momentum variables that are typically unobservable, while available data consist only of observable quantities such as configurations and velocities. In optimal control applications, the situation is further complicated because the latent variables contain unobservable co-states and the Hamiltonian may be degenerate, preventing the existence of a corresponding Lagrangian and rendering the encoder-decoder approaches inapplicable. We introduce Latent Lie--Poisson Neural Networks (LLPNNs), a structure-preserving framework for learning Lie--Poisson dynamics directly from observable data. The proposed approach exploits three geometric ingredients: (i) learning either a Hamiltonian decoder or a pseudo-Lagrangian encoder on the active variables, (ii) constructing latent trajectories through a universal Noether invariant arising from Lie--Poisson symmetry reduction, and (iii) reconstructing observable and latent dynamics through Lie--Poisson flows combined with Magnus-based Lie-group updates. The resulting method preserves the geometric structure and is applicable to both regular and degenerate Hamiltonian systems. We demonstrate the method on three examples: a generalized rigid body on SO(3), Kirchhoff's underwater vehicle on SE(3), and an optimal-control problem for interacting vehicles on $SE(2)^N$. Numerical experiments show excellent long-term predictive accuracy, strong robustness to noise, and competitive performance using only modest datasets and lightweight neural-network architectures.

Figures

Figures reproduced from arXiv: 2607.28939 by Vakhtang Putkaradze.

Figure 1
Figure 1. Figure 1: Illustration of the structural relationship between observable configura [PITH_FULL_IMAGE:figures/full_fig_p012_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Schematic overview of the proposed LLPNN framework in both Lagrangian and [PITH_FULL_IMAGE:figures/full_fig_p024_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Predicted versus ground-truth velocity components [PITH_FULL_IMAGE:figures/full_fig_p034_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Reconstructed scaled latent momentum trajectories [PITH_FULL_IMAGE:figures/full_fig_p034_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: 3D phase-space trajectories in latent momentum space [PITH_FULL_IMAGE:figures/full_fig_p035_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Time-evolution of Mean Absolute Error (MAE) for momentum [PITH_FULL_IMAGE:figures/full_fig_p036_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Conservation properties for non-quadratic [PITH_FULL_IMAGE:figures/full_fig_p036_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Predicted versus ground-truth body velocity components [PITH_FULL_IMAGE:figures/full_fig_p040_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Reconstructed latent body momentum trajectories [PITH_FULL_IMAGE:figures/full_fig_p041_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Time evolution of Mean Absolute Error (MAE) for body momentum [PITH_FULL_IMAGE:figures/full_fig_p042_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Conservation error metrics for non-quadratic [PITH_FULL_IMAGE:figures/full_fig_p042_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Time evolution of body velocity components [PITH_FULL_IMAGE:figures/full_fig_p048_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Time evolution of scaled body momentum components [PITH_FULL_IMAGE:figures/full_fig_p049_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Reconstructed 2D spatial trajectories (xk(t), yk(t)) for N = 3 vehicles on the SE(2) plane corresponding to the predictions in [PITH_FULL_IMAGE:figures/full_fig_p050_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Mean Absolute Error (MAE) for scaled momentum [PITH_FULL_IMAGE:figures/full_fig_p050_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Casimir invariant conservation error |Ck(t)−Ck(0)| evaluated across individual vehicle momentum components, confirming machine-precision preservation for Lie–Poisson formulations [PITH_FULL_IMAGE:figures/full_fig_p051_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: Relative Hamiltonian conservation drift |H(t) − H(0)|/|H(0)| for the Hamilto￾nian LLPNN under clean (η = 0) and noisy (η = 0.005) conditions. 1.00 0.75 0.50 0.25 0.00 0.25 0.50 0.75 1.00 Physical active coordinate 2 1.00 0.75 0.50 0.25 0.00 0.25 0.50 0.75 1.00 P h y sic al a c tiv e c o o r din a t e 3 True vs. learned H 1 = 0.5 True Hamiltonian Learned Hamiltonian 1.00 0.75 0.50 0.25 0.00 0.25 0.50 0.75 … view at source ↗
Figure 18
Figure 18. Figure 18: Hamiltonian level-set comparison in active momentum space ( [PITH_FULL_IMAGE:figures/full_fig_p052_18.png] view at source ↗

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