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arxiv: 2606.31137 · v1 · pith:JQW6YHJ4 · submitted 2026-06-30 · cs.LG · eess.SP

A Bayesian Filtering Approach for Learning Lagrangian Dynamics from Noisy Measurements

Kundan Kumar , Shreya Das , Simo S\"arkk\"a This is my paper

Reviewed by Pith T0 review T1 audit T2 compute T3 formal T4 kernel 2026-07-01 06:55 UTCgrok-4.3pith:JQW6YHJ4record.jsonopen to challenge →

classification cs.LG eess.SP
keywords Bayesian filteringLagrangian neural networksstochastic state-space modelsdynamics learningnoisy measurementsmaximum likelihoodGaussian approximation
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The pith

Bayesian filters jointly learn neural-network parameters and hidden states inside a Lagrangian dynamics model from partial noisy measurements.

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

The paper sets out to learn the dynamics of mechanical systems when only noisy and incomplete sensor readings are available. It keeps the Lagrangian structure but lets neural networks represent the kinetic and potential energies, while treating any unknown external forces as white Gaussian noise; the resulting Euler-Lagrange equations become a continuous-time stochastic state-space model. Gaussian-approximation Bayesian filters then perform maximum-likelihood estimation of both the network weights and the latent trajectory at the same time. The approach is tested on the pendulum and the Duffing oscillator and compared with ordinary Lagrangian neural networks that do not model the noise explicitly.

Core claim

The neural network parameters and system states are jointly learned via a maximum-likelihood method using Gaussian-approximation-based Bayesian filters on the continuous-time stochastic state-space model obtained from the Lagrangian with neural-network energies and additive white Gaussian noise for external forces, and the resulting models are shown to work on pendulum and Duffing oscillator examples.

What carries the argument

Gaussian-approximation-based Bayesian filters operating on the stochastic state-space model produced by the Euler-Lagrange equations when kinetic and potential energies are parameterized by neural networks and unknown forces are white Gaussian noise.

If this is right

  • The joint estimation of parameters and states improves handling of measurement noise and partial observations.
  • The stochastic formulation accounts for model mismatch without requiring an explicit external-force model.
  • The method produces usable dynamics models on the pendulum and Duffing oscillator that outperform conventional Lagrangian neural networks under the same noisy conditions.
  • Maximum-likelihood training via the Bayesian filter yields both point estimates and uncertainty information about the learned energies.

Where Pith is reading between the lines

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

  • The same filtering construction could be applied to other conservative systems once their Lagrangian is written in neural-network form.
  • Because the filter already maintains a state estimate, the learned model could be used directly inside a real-time observer or controller without a separate estimation step.
  • The white-noise assumption on forces suggests a route to robust learning when the true disturbance statistics are unknown but roughly Gaussian.

Load-bearing premise

Unknown external forces can be represented adequately as additive white Gaussian noise in the continuous-time equations of motion.

What would settle it

Train the model on noisy partial measurements of the pendulum or Duffing oscillator, then test whether its one-step-ahead state predictions on held-out noisy data have higher error than those of a standard Lagrangian neural network trained on identical data.

Figures

Figures reproduced from arXiv: 2606.31137 by Kundan Kumar, Shreya Das, Simo S\"arkk\"a.

Figure 1
Figure 1. Figure 1: Schematic diagram of the proposed method. The kinetic [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
read the original abstract

This paper proposes a Bayesian filtering-based approach for learning the dynamics of a physical system from partial, noisy measurements. We model the system dynamics using a Lagrangian mechanics formulation. As in Lagrangian neural networks (LNNs), we parameterize the kinetic and potential energies with neural networks. The unknown external forces in the Lagrangian formulation are modeled as white Gaussian noise. The corresponding Euler--Lagrange equations then yield a continuous-time stochastic state-space model (SSM) that describes the system dynamics. The neural network parameters and system states are then jointly learned via a maximum-likelihood method using Gaussian-approximation-based Bayesian filters. The effectiveness of the proposed method is demonstrated on pendulum and Duffing oscillator examples, and its performance is compared with conventional LNNs and with approximate Bayesian filters using known system models.

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

1 major / 1 minor

Summary. The paper proposes a Bayesian filtering approach to learn Lagrangian dynamics from partial noisy measurements. Kinetic and potential energies are parameterized by neural networks as in LNNs; unknown external forces are modeled as white Gaussian noise, yielding a continuous-time stochastic SSM. Neural network parameters and latent states are jointly estimated by maximum-likelihood using Gaussian-approximation Bayesian filters. Effectiveness is shown via comparisons to conventional LNNs on pendulum and Duffing oscillator examples.

Significance. If the central claim holds, the method offers a principled way to perform joint state and parameter estimation for Lagrangian systems under noise, potentially improving robustness over standard LNN training. The use of established Gaussian filters for the joint MLE is a clear technical strength when the white-noise modeling assumption is appropriate.

major comments (1)
  1. [Abstract / modeling description] The modeling step (abstract) that treats unknown external forces as white Gaussian noise to obtain the continuous-time stochastic SSM is load-bearing for the central claim. When this assumption is violated (colored, state-dependent, or deterministic disturbances), the neural-network kinetic/potential energies can absorb the mismatch during optimization, so the recovered Lagrangian fits the assumed SSM rather than the underlying physics. The pendulum and Duffing demonstrations do not stress this assumption, leaving the reported gains over LNNs without evidence of generalization.
minor comments (1)
  1. The abstract reports no quantitative metrics, error bars, or implementation details (e.g., filter type, discretization scheme, or training procedure), which makes the magnitude of improvement difficult to assess from the high-level description alone.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the constructive feedback. We address the major comment below and will incorporate clarifications into the revised manuscript.

read point-by-point responses

  1. Referee: [Abstract / modeling description] The modeling step (abstract) that treats unknown external forces as white Gaussian noise to obtain the continuous-time stochastic SSM is load-bearing for the central claim. When this assumption is violated (colored, state-dependent, or deterministic disturbances), the neural-network kinetic/potential energies can absorb the mismatch during optimization, so the recovered Lagrangian fits the assumed SSM rather than the underlying physics. The pendulum and Duffing demonstrations do not stress this assumption, leaving the reported gains over LNNs without evidence of generalization.

    Authors: We agree that the white-Gaussian-noise modeling of unknown external forces is central to the derivation of the continuous-time stochastic SSM and to the applicability of the Gaussian-approximation filters. This is an explicit modeling choice that separates the Lagrangian (parameterized by the neural networks) from the disturbance process; under the assumed SSM the joint MLE procedure recovers both. When the true disturbances deviate from white Gaussian (e.g., colored, state-dependent, or deterministic), the learned kinetic/potential networks can indeed compensate for the mismatch, so the recovered Lagrangian is the one consistent with the assumed model rather than the true underlying physics. The pendulum and Duffing examples are standard benchmarks used by prior LNN work and demonstrate improved robustness to measurement noise relative to conventional LNN training; they do not, however, probe robustness to misspecified disturbance spectra. We will revise the abstract and add a new subsection in the discussion that explicitly states the modeling assumption, its consequences when violated, and the intended scope of the method. No new experiments are added at this stage. revision: partial

Circularity Check

0 steps flagged

No significant circularity; derivation applies standard filters to constructed SSM

full rationale

The paper defines a continuous-time stochastic SSM by adding white Gaussian noise to the Euler-Lagrange equations of a Lagrangian parameterized by neural networks, then applies existing Gaussian-approximation Bayesian filters for joint maximum-likelihood estimation of network parameters and states. No quoted equations reduce any claimed performance metric to a quantity defined by the fitted parameters themselves, and the provided text contains no load-bearing self-citations or uniqueness theorems imported from the authors' prior work. The central construction remains independent of the target results.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claim rests on the Lagrangian mechanics framework, the white-Gaussian-noise model for external forces, and the validity of Gaussian approximations inside the Bayesian filters.

free parameters (1)
  • Neural network weights for kinetic and potential energies
    These parameters are learned from data via the maximum-likelihood procedure and are central to the model.
axioms (2)
  • domain assumption System dynamics obey the Euler-Lagrange equations derived from a Lagrangian
    Invoked to convert the energy functions into equations of motion.
  • ad hoc to paper Unknown external forces are white Gaussian noise
    This modeling choice directly produces the continuous-time stochastic state-space model used by the filters.

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Reference graph

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