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Learning mechanical systems from real-world data using discrete forced Lagrangian dynamics

T0 review · 3 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read A discrete forced Lagrangian neural network learns damped mechanical systems from position measurements alone.

desk verdict A useful discrete variational extension with a real identifiability gap in the conservative/forced split, especially for the real-data task. read the letter →

arxiv 2505.20370 v1 pith:F67EFKOY submitted 2025-05-26 eess.SY cs.LGcs.SY

Martine Dyring Hansen , Elena Celledoni , Benjamin Kwanen Tapley This is my paper
classification eess.SYcs.LGcs.SY MSC 70H0365P1068T07
keywords discreteLagrange-d'AlembertforcedEuler-Lagrangeequationsposition-onlysystemidentificationstructure-preservingneuralnetworksdissipativedynamicslatent-spacehumanmotioncapturesymplecticintegration
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper proposes a discrete forced Lagrangian neural network (DFLNN) that learns the equations of motion of a mechanical system from position measurements only, without velocities or momenta. The model is built on the discrete Lagrange-d'Alembert principle: the physics loss is the squared norm of the forced discrete Euler-Lagrange residual (Eq. 9) evaluated on position triplets $(q_{n-1}, q_n, q_{n+1})$, with separate neural networks $L_\theta$ and $F_\theta$ for the Lagrangian and the external force. Because the two components are learned separately, the method can separate conservative dynamics from dissipation and other non-conservative forces, and can simulate the corresponding conservative system by switching the force term off. The authors validate this on a damped double pendulum, a dissipative charged particle in a magnetic field, pixel frames compressed by an autoencoder, and real human motion-capture data. If the claim holds, physically interpretable models of damped mechanical systems can be learned from data sources where only positions are available, such as motion capture, video, or low-resolution tracking.

What carries the argument

The central object is the forced discrete Euler-Lagrange equation (5), which says a discrete trajectory is physical when the variation of the discrete action plus the virtual work of the discrete force vanishes at every interior step. The paper uses a midpoint discretization, so the training residual (9) involves three consecutive positions through midpoint position and velocity. Minimizing the squared norm of this residual is the physics loss; a log-determinant barrier (12) enforces regularity of the Lagrangian, and the architecture choices — a positive-definite mass matrix $M_\theta$, a Rayleigh dissipation form for $F_\theta$, or dropout on the free-force network — keep the conservative and non-conservative components from collapsing into one another.

What would settle it

Train two DFLNN models from different random seeds on the same damped-oscillator position data, with the free-force architecture and no Rayleigh prior. If both achieve small residuals but return different Lagrangian/force splits — for example, one model moves part of the potential into the force term — the claimed separation of conservative from non-conservative dynamics is not uniquely determined by position data alone.

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Extended reading notes

Core claim

The central claim is that the inverse problem of identifying a mechanical system from position data can be posed as a regression on the discrete forced Euler-Lagrange equations. For each observed triple $(q_{n-1}, q_n, q_{n+1})$, the paper forms midpoint approximations of position and velocity, computes the residual of Eq. (9), and minimizes its squared norm over the dataset; $L_\theta$ and $F_\theta$ are separate feed-forward networks, so the learned model carries a distinct Lagrangian and a distinct external force. A log-determinant barrier keeps the Lagrangian regular, and when dissipation is expected the force is restricted to a Rayleigh form $F_\theta = -K_\theta \dot q$; when the force is unknown, dropout slows the free-force network so that it does not absorb conservative dynamics. Rollouts are generated by solving the residual equation for the next position, and setting $F_\theta = 0$ gives the corresponding conservative system. The experiments show accurate long-time rollouts on damped double-pendulum, charged-particle, pixel-pendulum, and human motion-capture data, and the recovered conservative/non-conservative split is demonstrated by extrapolating to the undamped regime.

Load-bearing premise

The load-bearing premise is that the observed position traces come from a single mechanical system of the standard kind — kinetic energy that is invertible in velocity and a force depending only on position and velocity — so that the residual can pick out one true energy/force pair; the paper does not prove this uniqueness.

Editorial extensions

If this is right

  • Mechanical systems can be identified from data streams without velocity sensors, making motion capture, video, and low-resolution tracking usable training sources.
  • A trained model can switch off the learned force term and simulate the underlying conservative dynamics, so a dissipative dataset can generate predictions for the equivalent undamped system.
  • Because the force-free discretization is variational, rollouts preserve the symplectic structure of the Hamiltonian system and remain accurate over longer time spans than black-box models.
  • The architecture applies in latent space: an autoencoder can compress pixel frames, and the dynamics can be learned and simulated in the low-dimensional latent coordinates.
  • The method extends to real biomechanics data, suggesting that interpretable equations of motion can be extracted from human motion capture.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Beyond the paper's explicit claims, the residual's gauge freedom means a gradient-like term can be moved from the Lagrangian into the force without changing the fit; the learned conservative/non-conservative split is therefore fixed by the chosen priors (Rayleigh form, dropout, regularity) rather than guaranteed by the data alone.
  • A testable extension would train the same architecture on data with a known external input and check whether the learned force matches the input, turning the method into a position-only estimator of actuation or control.
  • If the data contain non-mechanical modes or multiple interacting agents, the learned 'physics' will be a spurious fit, so the model should be applied only after verifying that a single forced mechanical system generated the observations.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper introduces the Discrete Forced Lagrangian Neural Network (DFLNN), a method for learning mechanical systems from position data only. The model is trained by minimizing the residual of the discrete forced Euler-Lagrange equations (Eq. 9), with separate neural-network representations of the Lagrangian Lθ and the non-conservative force Fθ. The authors evaluate DFLNN on synthetic damped double-pendulum, dissipative charged-particle, and pixel-pendulum tasks, as well as on real CMU motion-capture data of a person swinging from a bar. They report favorable extrapolation performance against a continuous Generalized Lagrangian Neural Network (GLNN) and a Neural ODE baseline, and they claim that switching off the learned force term yields physically interpretable conservative extrapolations, thereby separating conservative from non-conservative dynamics.

Significance. If the central separation claim were firmly established, DFLNN would be a useful contribution: it avoids finite-difference velocity estimates, respects the discrete Lagrange-d'Alembert structure, and shows strong predictive rollouts on several synthetic tasks. The paper also demonstrates a sensible integration with autoencoder latent representations for pixel observations and reports modest computational cost. However, the core interpretability claim—that the learned pair (Lθ, Fθ) isolates conservative from non-conservative dynamics—is not justified by the training objective, because the loss is invariant under a gauge transformation that transfers gradients between L and F. The use of dropout or a Rayleigh ansatz is presented as an informal remedy rather than a structural guarantee. For these reasons, the significance of the contribution is currently conditional on the authors providing an identifiability analysis or a reframed, more modest claim.

major comments (3)
  1. [Section 3.1, Eq. (9); Section 1.2(c); Section 3.3] The loss in Eq. (9) is invariant under the transformation (Lθ, Fθ) -> (Lθ + S(q), Fθ − ∇S(q)) for any smooth scalar function S(q). Because all terms in the residual are evaluated at the same midpoints (qbar, vbar), the contributions ∇S cancel identically, so the learned decomposition of conservative versus non-conservative dynamics is not identifiable from the training loss alone. The paper provides no identifiability condition, and the Rayleigh form (Eq. 13) and dropout in Section 3.3 are informal priors rather than proven constraints. This directly affects the headline claim that the method "allows us to separate conservative from non-conservative dynamics," and it undermines the practice of turning off Fθ for conservative extrapolation: the resulting conservative system depends on which representative of the equivalence class was learned. Please provide either a structural identifiability proof, an explicit constraint that removes the gauge freedom, or an experiment with known L and F that verifies the recovered split.
  2. [Section 3.3] Dropout is the only mechanism offered to prevent Fθ from absorbing conservative dynamics when Fθ is a free network. Dropout is a stochastic regularizer and does not constrain the function space in a way that removes the gauge freedom described above, nor does it provide any convergence or uniqueness guarantee. The claim that it "encourages the Lagrangian term to fit the data when possible" is qualitative and requires empirical support, for example an ablation where the same task is trained with and without dropout, or a comparison of the learned L and F against ground truth on a system where the true split is known.
  3. [Section 4.4] The real-data experiment assumes that a person swinging from a bar is well described by a multi-pendulum model with a generalized potential U(q, qdot) and a force of the form −Kθ(q) qdot + F_free. This assumption is not verified, and the reported extrapolation error (11.75 ± 10.89 at k=35) plus visual rollouts do not establish that the learned conservative/non-conservative split is physically correct. Because no ground-truth L and F are available for human motion, the separation claim on real data remains anecdotal. Please report held-out quantitative metrics, compare against a purely black-box baseline with comparable capacity, or validate the same architecture on a synthetic system with known mechanical ground truth and similar trajectory statistics.
minor comments (4)
  1. [Section 4, Eq. (6)] The normalization factor NT(N+1) in Eq. (6) appears inconsistent with the sum over n=1,...,N−1; Eq. (15) uses NT(N−1) for the physics term. Please correct the denominator.
  2. [Table 2] The entry "Tabel 2" in Section 4.3 is a typo for "Table 2," and the NaN result for GLNN on Task 4 is not explained; please state whether it is a numerical instability or an implementation issue.
  3. [Appendix G] The phrase "Newtons second low" in the charged-particle section should be "Newton's second law." There is also a typo "resoures" in Appendix E.
  4. [Section 4.4] The latent dimension is written as "l=6" but the notation for the latent space is introduced as z∈R^l; please keep the notation uniform and clarify whether l is a scalar dimension.

Circularity Check

0 steps flagged · score 1.0 of 10

No circular derivation: the DFLNN residual is a discretization of the forced Euler-Lagrange equations, and conservative extrapolation is checked against ground truth in synthetic tasks; the (L,F) gauge ambiguity is an identifiability caveat, not a circular fit.

full rationale

The paper's derivation chain is not circular. The physics loss (Eq. 9) is obtained by discretizing the discrete Lagrange-d'Alembert principle (Eqs. 4-5), with L_theta and F_theta as free network parameters; there is no fitted parameter that is defined in terms of a target quantity and then renamed as a prediction. Training minimizes the squared residual of the discrete forced Euler-Lagrange equations, and prediction solves the same equations forward, which is a standard inverse-problem setup rather than a tautology. The claimed conservative extrapolation is genuinely external in the synthetic tasks: the model is trained on damped trajectories and then evaluated by turning off F_theta against ground-truth conservative rollouts (Section 4, Figures 3-4), so the conservative behavior is not an input to the fit. The self-citations ([29,30]) appear only in related-work context and are not load-bearing. A real limitation is that, for a general free force F_theta, the residual (Eq. 9) is invariant under (L,F) -> (L+S(q), F - grad S(q)), so the conservative/non-conservative split is not identifiable from data alone; however, this is a uniqueness/identifiability gap, not a circularity, and in Tasks 1-2 the Rayleigh form supplies additional structure that can break the gauge. The paper would be stronger with an explicit identifiability discussion, especially for Task 4, but the core derivation and its synthetic validation remain self-contained and non-circular.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The central claim rests on standard discrete variational mechanics plus several informal modeling choices. The core derivation is self-contained given Marsden-West theory, but the unique separation of conservative and dissipative dynamics is assumed rather than proved, showing up as hand-set weights, dropout, and latent-space assumptions.

free parameters (3)
  • loss weights and dropout rate = Task-dependent: e.g. Task 1 omega_physics=0.5, omega_reg=0.5, dropout=0.5; Task 3 omega_physics=0.9, omega_reg=0.1…
    These hyperparameters control the balance between fitting the physics residual, enforcing nondegeneracy of Ltheta, reconstruction, and preventing F_free from absorbing conservative dynamics. The claimed separation depends on these choices, and no sensitivity analysis is given.
  • mass matrix regularization constant epsilon = small positive, value not reported
    Introduced in Eq. (10) to guarantee Mtheta is symmetric positive definite.
  • latent dimension l = l=1 (Task 3), l=6 (Task 4)
    Hand-chosen per task; the mechanical structure is imposed in this latent space, so the central claims about separating dynamics depend on it.
assumptions (5)
  • standard math The discrete Lagrange-d'Alembert principle and the forced discrete Euler-Lagrange equations (Eq. 5) correctly describe the observed dynamics.
    Central modeling assumption inherited from Marsden-West [11]; accepted as background, not proved in this paper.
  • domain assumption The midpoint quadrature approximations in Eqs. (7)-(8) accurately represent the continuous Lagrangian and force integrals over each step.
    Used to construct the residual (9); for coarse h or strongly nonlinear forces the residual is only an approximation, and the paper does not study the resulting bias.
  • ad hoc to paper There exists a regular Lagrangian L and fiber-preserving force F such that the observed position sequences satisfy the discrete forced Euler-Lagrange equations.
    The inverse problem assumes data are generated by a single mechanical system of this form; real human motion (Task 4) is asserted to behave like a multi-pendulum, but no test or identifiability check is provided.
  • ad hoc to paper Latent coordinates from the autoencoder support a mechanical Lagrangian structure.
    For pixel and human tasks, Ltheta and Ftheta act on autoencoder latents; the paper does not show that the latent dynamics are Lagrangian or Hamiltonian.
  • ad hoc to paper Dropout prevents Ftheta from absorbing conservative dynamics.
    Section 3.3 states dropout 'encourages slower learning of the force term'; this informal bias is load-bearing for the claimed separation.

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Cite this review

Pith. "Pith review of Learning mechanical systems from real-world data using discrete forced Lagrangian dynamics." pith.science (2026). https://pith.science/paper/F67EFKOY

@misc{pith2026250520370,
  author       = {Pith},
  title        = {Pith review of: Learning mechanical systems from real-world data using discrete forced Lagrangian dynamics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/F67EFKOY}},
  note         = {Machine review of arXiv:2505.20370}
}
read the original abstract

We introduce a data-driven method for learning the equations of motion of mechanical systems directly from position measurements, without requiring access to velocity data. This is particularly relevant in system identification tasks where only positional information is available, such as motion capture, pixel data or low-resolution tracking. Our approach takes advantage of the discrete Lagrange-d'Alembert principle and the forced discrete Euler-Lagrange equations to construct a physically grounded model of the system's dynamics. We decompose the dynamics into conservative and non-conservative components, which are learned separately using feed-forward neural networks. In the absence of external forces, our method reduces to a variational discretization of the action principle naturally preserving the symplectic structure of the underlying Hamiltonian system. We validate our approach on a variety of synthetic and real-world datasets, demonstrating its effectiveness compared to baseline methods. In particular, we apply our model to (1) measured human motion data and (2) latent embeddings obtained via an autoencoder trained on image sequences. We demonstrate that we can faithfully reconstruct and separate both the conservative and forced dynamics, yielding interpretable and physically consistent predictions.

Figures

Figures reproduced from arXiv: 2505.20370 by the authors.

Figure 1
Figure 1. In this paper, we propose a structure-preserving approach for learning non-conservative [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. A schematic illustration of the data flow through the proposed model. We use segments of [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Combined results for Task 1 (left) and Task 2 (right). Solid Green lines are DFLNN [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figures from the paper (6 more)
Figure 4
Figure 4. Figure 4: Results for a simple pendulum represented through pixel images. Green pendulums are [PITH_FULL_IMAGE:figures/full_fig_p009_4.png]
Figure 5
Figure 5. Figure 5: Results and reconstruction (trial 2) of human motion. Green notation is DFLNN (proposed), yellow is the GLNN model, and pink lines a Neural ODE. The ground truth is indicated in black. The left panel depicts the full movement represented as a skeletal sketch. The right…
Figure 6
Figure 6. Figure 6: An autoencoder is applied to reduce the high-dimensional input dimension to a lower [PITH_FULL_IMAGE:figures/full_fig_p015_6.png]
Figure 7
Figure 7. Figure 7: Results for Task 1 with different time resolution over the dataset. Solid Green lines are [PITH_FULL_IMAGE:figures/full_fig_p017_7.png]
Figure 8
Figure 8. Figure 8: Results for Task 2 with different time resolution over the dataset. Solid Green lines are [PITH_FULL_IMAGE:figures/full_fig_p017_8.png]
Figure 9
Figure 9. Figure 9: Synthetic datasets. From the left-panel: a damped double pendulum, a charged particle moving through a magnetic field, and a damped single pendulum represented with pixel frames. force proportional to its velocity, the motion is governed by the following equations: ¨θ1…

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.