{"id":"dd5f368c-5640-4035-9b05-5909ee96b37c","arxiv_id":"2606.31137","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"A Bayesian filtering approach learns Lagrangian dynamics parameterized by neural networks from noisy measurements by forming a stochastic state-space model and jointly estimating parameters and states via maximum likelihood.","lead":"The paper introduces a Bayesian filtering method to learn neural-network-parameterized Lagrangian dynamics from partial noisy measurements by modeling unknown forces as Gaussian noise and jointly estimating states and parameters. This could help build more accurate physical models when real sensor data is imperfect, as in robotics or control systems.","discovery_kind":"unclear","skeptic_critique":{"model":"grok-4.3","headline":"White-Gaussian modeling of unknown external forces may not capture structured disturbances, risking biased Lagrangian recovery.","rationale":"The reader's weakest_assumption directly identifies the same modeling step; because the full manuscript was not supplied to the initial review, the quantitative evidence needed to bound the mismatch risk is still missing, so the UNVERDICTED status is retained but the concern is now explicit and testable.","tokens_in":1626,"tokens_out":321,"duration_ms":23180,"concrete_test":"Re-run the pendulum experiment with an added deterministic sinusoidal torque (known amplitude/frequency) instead of white noise; after training, compare the recovered potential-energy network against the analytic mgl(1-cosθ) on a 100-point grid—if the L2 error exceeds 10% of the true range while the filter likelihood remains high, the noise-model assumption is the dominant error source.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central construction begins from the Euler-Lagrange equations with additive white Gaussian process noise on the generalized forces, producing the continuous-time SSM on which the Gaussian-approximation filters (and the subsequent joint MLE) operate. If the true disturbances are colored, state-dependent, or deterministic, the resulting stochastic dynamics deviate from the physical system; the neural-network kinetic/potential energies can then absorb the mismatch during optimization, yielding parameters that fit the assumed SSM but not the underlying Lagrangian. The pendulum and Duffing demonstrations do not stress this assumption, so the reported gains over standard LNNs may not generalize.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","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.","tokens_in":1765,"tokens_out":350,"duration_ms":33591,"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":[{"comment":"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.","section":"Abstract / modeling description"}],"minor_comments":[{"comment":"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.","section":null}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for the constructive feedback. We address the major comment below and will incorporate clarifications into the revised manuscript.","responses":[{"response":"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_made":"partial","referee_comment":"[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."}],"tokens_in":1219,"tokens_out":382,"duration_ms":33481,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The main takeaway is a probabilistic extension of Lagrangian neural networks that treats unknown external forces as white Gaussian noise, builds the resulting continuous-time stochastic state-space model, and then uses Bayesian filters to do maximum-likelihood joint estimation of the neural network weights and the latent states from noisy partial observations.\n\nWhat is new is the specific combination of the LNN parameterization with these filters rather than separate training or known-model filters. The setup is coherent: the Euler-Lagrange equations with additive white noise give the SSM, the filters approximate the likelihood, and the examples on the pendulum and Duffing oscillator compare against plain LNNs.\n\nThe modeling steps and the filtering machinery look formally grounded. The white-noise choice is explicit and standard for producing a tractable SSM.\n\nThe soft spot is exactly the white-Gaussian assumption on the forces. If real disturbances are colored, state-dependent, or deterministic, the neural networks can absorb the mismatch into the learned kinetic and potential energies, so the recovered Lagrangian fits the assumed model but not the underlying physics. The reported examples do not stress this case, which limits how far the gains over standard LNNs can be trusted. The abstract also gives no numbers or error bars, so the size of any improvement is still unclear.\n\nThis is for people working on learning dynamics from noisy sensor data who already know LNNs and want a filtering-based alternative. A reader focused on probabilistic system identification would get value from the construction.\n\nIt deserves peer review because the framework is well-defined and the filtering step is a natural move, even though the noise-model robustness needs checking.","headline":"The paper integrates Lagrangian neural networks with continuous-time Gaussian-approximation Bayesian filters for joint state and parameter learning under a white-Gaussian force noise model, but that noise assumption is the main practical limitation.","tokens_in":2237,"tokens_out":410,"would_cite":false,"duration_ms":23404,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"Bayesian filters jointly learn neural-network parameters and hidden states inside a Lagrangian dynamics model from partial noisy measurements.","keywords":["Bayesian filtering","Lagrangian neural networks","stochastic state-space models","dynamics learning","noisy measurements","maximum likelihood","Gaussian approximation"],"falsifier":"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.","tokens_in":2527,"feed_emoji":"","tokens_out":650,"duration_ms":28820,"temperature":0.7,"pith_summary":"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.","feed_headline":"Bayesian filters learn Lagrangian dynamics from noisy measurements","feed_subtitle":"Joint maximum-likelihood estimation of neural parameters and states works on pendulum and Duffing examples where standard LNNs do not model","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"fun_headline_variants":["Bayesian filters jointly learn Lagrangian NN dynamics from noise","Stochastic SSM enables Bayesian learning of neural Lagrangians","Joint estimation of states and NN parameters via Bayesian filters","Gaussian filters learn Lagrangian from partial noisy measurements"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"Unknown external forces can be represented adequately as additive white Gaussian noise in the continuous-time equations of motion.","fun_headline_variants_meta":{"raw":{"variants":["Bayesian filters jointly learn Lagrangian NN dynamics from noise","Stochastic SSM enables Bayesian learning of neural Lagrangians","Joint estimation of states and NN parameters via Bayesian filters","Gaussian filters learn Lagrangian from partial noisy measurements"]},"model":"grok-4.3","cost_usd":0.005157,"raw_usage":{"total_tokens":2455,"prompt_tokens":569,"num_sources_used":0,"completion_tokens":60,"cost_in_usd_ticks":51574500,"prompt_tokens_details":{"text_tokens":569,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":1826,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":569,"tokens_out":60,"duration_ms":18030,"temperature":1.0,"reasoning_tokens":1826,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-01T06:55:56.126442+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"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.","supporting_citations":[],"review_version":1}