arxiv: 2605.06246 · v1 · submitted 2026-05-07 · 💻 cs.LG · cs.RO

Recognition: unknown

Structure-Preserving Gaussian Processes Via Discrete Euler-Lagrange Equations

Jan-Hendrik Ewering, Kathrin Fla{\ss}kamp, Niklas Wahlstr\"om, Thomas B. Sch\"on, Thomas Seel

Authors on Pith no claims yet

Pith reviewed 2026-05-08 13:23 UTC · model grok-4.3

classification 💻 cs.LG cs.RO

keywords Lagrangian Gaussian processesdiscrete Euler-Lagrangestructure preservationdynamics learningGaussian processesvariational methodsrobotics

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The pith

Gaussian processes conditioned on discrete Euler-Lagrange equations preserve dynamical structure for stable predictions

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

This paper develops Lagrangian Gaussian Processes for learning dynamical systems from data. It conditions the Gaussian process on linear operators obtained from discrete forced Euler-Lagrange equations and variational discretizations. As a result, the models respect the geometric structure of the Lagrange-d'Alembert principle without external forces. This built-in preservation prevents unphysical energy drift and supports accurate long-term forecasts. The approach works from position measurements only, which is useful when velocities are unavailable.

Core claim

Lagrangian Gaussian Processes are constructed such that their conditioning operators derive directly from the discrete forced Euler-Lagrange equations. In the absence of external forces, this ensures that the discrete trajectories satisfy the geometric structure of the underlying continuous dynamics by construction, yielding physically consistent models that support stable long-term predictions from sparse positional observations.

What carries the argument

Linear operators for Gaussian process conditioning, constructed from discrete forced Euler-Lagrange equations via variational discretization schemes; these operators embed the structure preservation into the learning process.

If this is right

Dynamics can be learned solely from discrete position snapshots without velocity data
Learned models exhibit no artificial energy drift during long-term integration
The method provides probabilistic predictions with uncertainty estimates
It demonstrates effectiveness on real-world systems such as soft robots with hysteresis

Where Pith is reading between the lines

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

The same discretization-based conditioning could be applied to other physical principles beyond Lagrangian mechanics
In robotic control, these models might enable more reliable planning over extended horizons
Further analysis could quantify the approximation error introduced by the discretization for different sampling rates

Load-bearing premise

The chosen variational discretization schemes applied to sequences of position measurements faithfully capture the continuous-time dynamics without errors that would undermine the structure preservation.

What would settle it

A long-term simulation of a learned force-free system showing substantial deviation from conserved energy would indicate that the structure preservation does not hold in practice.

Figures

Figures reproduced from arXiv: 2605.06246 by Jan-Hendrik Ewering, Kathrin Fla{\ss}kamp, Niklas Wahlstr\"om, Thomas B. Sch\"on, Thomas Seel.

**Figure 1.** Figure 1: We propose Lagrangian Gaussian Processes (LGPs) for learning probabilistic, non-conservative dynamics models using only position measurements, without access to velocity or momentum data. Incorporating Lagrange-d’Alembert principle into Gaussian Processes (GPs), the methods enable physically consistent long-term predictions. structural preservation of the underlying energy conservation law, (ii) uncertaint… view at source ↗

**Figure 2.** Figure 2: The proposed LGPs enable learning probabilistic models of a simulated pendulum’s Lagrangian L and external forces F only from position data. Incorporating further physics knowledge into the kernel improves the accuracy of L and F as well as the predictive performance. See Appendices A.5 and C for further details. -3 q 3 -4 q˙ 4 H = const. True Hamiltonian t = 0 t = 0.5 t = 1 t = 1.5 -3 q 3 Learned Hamilton… view at source ↗

**Figure 3.** Figure 3: The proposed LGPs yield structure-preserving and accurate long-term forward simulations of a controlled pendulum. Continuous LGPs generalize to prediction time step sizes not seen during training ∆tpred ̸= ∆ttrain. Figure 2a illustrates how additional physics information in the kernel choice can improve the accuracy of the learned Lagrangian and force GPs in terms of their posterior mean and covariance. We… view at source ↗

**Figure 4.** Figure 4: Prediction tasks in a controlled real-world double pendulum. The LGPs yield accurate forward simulations despite learning only from noisy real-world position data (∆ttrain = 61 ms, nq = 2, N = 300 data points). In the absence of inputs, the learning-based system energy (Hamiltonian H) along the trajectory—consistent with the underlying physics—decays due to dissipation. Task 2.1 is visualized in Figure A10… view at source ↗

**Figure 5.** Figure 5: Prediction task in a controlled pneumatic real-world soft robot. Although the soft robot exhibits complex nonlinear kinematics and dynamics with hysteresis effects, the LGPs enable accurate forward simulations of the shape-describing parameters q⊤ = [∆x, ∆y, δℓ] (∆ttrain = 20 ms, nq = 3, N = 200 data points). Photo and data adopted from [30]. 4.2 Task 2: Controlled Real-World Double Pendulum Second, we eva… view at source ↗

read the original abstract

In this paper, we propose Lagrangian Gaussian Processes (LGPs) for probabilistic and data-efficient learning of dynamics via discrete forced Euler-Lagrange equations. Importantly, the geometric structure of the Lagrange-d'Alembert principle, which governs the motion of dynamical systems, is preserved by construction in the absence of external forces. This allows learning physically consistent models that overcome erroneous drift in the system's energy, thereby providing stable long-term predictions. At the core of our approach lie linear operators for Gaussian process conditioning, constructed from discrete forced Euler-Lagrange equations and variational discretization schemes. Thereby and unlike prior work, the method enables learning dynamics from discrete position snapshots, i.e., without access to a system's velocities or momenta. This is particularly relevant for a large class of practical scenarios where only position measurements are available, for instance, in motion capture or visual servoing applications. We demonstrate the data-efficiency and generalization capabilities of the LGPs in various synthetic and real-world case studies, including a real-world soft robot with hysteresis. The experimental results underscore that the LGPs learn physically consistent dynamics with uncertainty quantification solely from sparse positional data and enable stable long-term predictions.

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

LGPs turn discrete EL equations into GP operators for structure-preserving learning from positions only, with good practical tests but a question mark on continuous rollout stability.

read the letter

LGPs turn discrete EL equations into GP operators for structure-preserving learning from positions only. This is the main contribution, and it directly helps with the common case of having only position measurements in robotics. They do a good job laying out how the variational discretization leads to linear operators that enforce the structure by construction. The real-world soft robot example with hysteresis is a nice test case because it includes effects that standard models might struggle with. The claims about data efficiency and stable predictions are backed by those case studies. One area that could use more scrutiny is the prediction phase. The discrete scheme preserves the structure at the training level, but the paper needs to show clearly that the GP posterior samples or mean, when rolled out with numerical integration, maintain the no-drift property over long times. Small kernel-induced variations could accumulate otherwise. Readers working on hybrid physics-ML models for robotics will get the most from this. It sits at the intersection of variational mechanics and probabilistic learning, so specialists in either area can follow the contribution. The paper shows clear thinking on how to adapt existing principles to the GP setting without obvious internal contradictions. I think it should go to peer review.

Referee Report

2 major / 1 minor

Summary. The manuscript proposes Lagrangian Gaussian Processes (LGPs) that combine Gaussian process regression with linear operators derived from discrete forced Euler-Lagrange equations and variational discretization schemes. This construction is intended to preserve the geometric structure of the Lagrange-d'Alembert principle by construction in the absence of external forces, enabling learning of physically consistent dynamics solely from discrete position snapshots (without velocities or momenta) and yielding energy-stable long-term predictions with uncertainty quantification. The approach is demonstrated on synthetic benchmarks and a real-world soft robot exhibiting hysteresis.

Significance. If the central claim of exact structure preservation holds through the GP conditioning and rollout, the work would offer a useful advance in physics-informed probabilistic modeling of dynamics. It addresses a practical gap by operating from position-only data while enforcing variational principles, potentially improving long-term stability and data efficiency in robotics and control applications. The emphasis on uncertainty-aware, structure-preserving models from sparse observations is a positive contribution, though its impact hinges on rigorous confirmation that the GP posterior and continuous predictions retain the claimed conservation properties.

major comments (2)

Abstract: The claim that the Lagrange-d'Alembert structure 'is preserved by construction' is load-bearing for the central contribution, yet the description leaves open whether the GP posterior mean (or samples) and the numerical integration used for continuous-time rollout exactly satisfy the discrete forced Euler-Lagrange equations at every step. Any deviation introduced by the kernel, conditioning, or integrator could allow energy drift, undermining the stability guarantee.
Core construction of linear operators: It is unclear from the provided description how the variational discretization scheme interacts with the GP covariance function to ensure that the conditioned model remains exactly consistent with the discrete EL equations outside the training discretization points, particularly when external forces are absent.

minor comments (1)

The abstract refers to 'various synthetic and real-world case studies' without quantifying data sparsity, baseline comparisons, or specific metrics for energy drift; adding these details would improve clarity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. The points raised highlight opportunities to strengthen the presentation of our core claims regarding structure preservation. We will revise the manuscript to provide additional mathematical detail and clarifications on how the GP posterior and rollout satisfy the discrete Euler-Lagrange equations.

read point-by-point responses

Referee: Abstract: The claim that the Lagrange-d'Alembert structure 'is preserved by construction' is load-bearing for the central contribution, yet the description leaves open whether the GP posterior mean (or samples) and the numerical integration used for continuous-time rollout exactly satisfy the discrete forced Euler-Lagrange equations at every step. Any deviation introduced by the kernel, conditioning, or integrator could allow energy drift, undermining the stability guarantee.

Authors: We appreciate this observation and agree that precision on this point is essential. In the revised manuscript we will clarify that the linear operators derived from the discrete forced Euler-Lagrange equations are used to condition the GP directly on the position snapshots. Consequently, both the posterior mean and samples satisfy the discrete EL equations exactly at every discrete time step corresponding to the observed positions (and at the same steps during rollout). For continuous-time predictions we employ the identical variational discretization scheme as a structure-preserving integrator; this guarantees that the discrete conservation properties (including energy stability in the absence of external forces) are maintained at every integration step. We will update the abstract for accuracy and add an appendix containing a short proof that the conditioned posterior and the discrete rollout incur no spurious energy drift. revision: yes
Referee: Core construction of linear operators: It is unclear from the provided description how the variational discretization scheme interacts with the GP covariance function to ensure that the conditioned model remains exactly consistent with the discrete EL equations outside the training discretization points, particularly when external forces are absent.

Authors: Thank you for identifying this need for elaboration. The variational discretization yields a set of linear operators (finite-difference approximations to velocities and accelerations consistent with the Lagrange-d'Alembert principle) that are applied to the underlying GP. Conditioning the GP on these operators equaling the (learned) force terms enforces the discrete EL equations at the discrete time instants where the operators are evaluated. Because the operators are linear, the posterior covariance is modified globally; any draw from the posterior therefore satisfies the discrete EL equations exactly at those instants, independent of whether they coincide with training locations. When external forces are absent the same operators reduce to the homogeneous discrete EL equations, yielding exact discrete conservation. For points between the discrete steps the GP provides a smooth interpolation that is consistent with the learned variational dynamics. In the revision we will expand the methods section with the explicit operator-kernel interaction and a short derivation showing that consistency holds at all discrete steps used by the model. revision: yes

Circularity Check

0 steps flagged

Structure preservation is enforced by construction in the discrete scheme; no reduction of central claim to fitted inputs or self-citation chain

full rationale

The derivation constructs linear operators for GP conditioning directly from discrete forced Euler-Lagrange equations and variational discretization schemes (external to the paper). This enforces the Lagrange-d'Alembert structure by design for the discrete case in the absence of forces, rather than deriving it from data or self-referential definitions. The GP learning step remains a standard conditioning on position snapshots, with structure preservation as an added constraint rather than a tautology. No load-bearing self-citation, fitted parameter renamed as prediction, or ansatz smuggled via prior work is present; the approach is self-contained against the cited variational principles.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Review based solely on abstract; full details on parameters and assumptions unavailable. The approach relies on standard variational mechanics but introduces the LGP construction.

axioms (1)

domain assumption The Lagrange-d'Alembert principle governs the motion of dynamical systems and its geometric structure can be preserved via discrete forced Euler-Lagrange equations.
Directly invoked in the abstract as the foundation for structure preservation by construction.

pith-pipeline@v0.9.0 · 5523 in / 1193 out tokens · 46811 ms · 2026-05-08T13:23:07.298972+00:00 · methodology

discussion (0)

Reference graph

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