{"id":"584471ea-ae78-4236-a7fb-d0709793e34e","arxiv_id":"2412.03018","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"A Hamiltonian neural network with three parallel networks can recover the Hamiltonian, constraint matrix, and Lagrange multipliers of nonholonomically constrained systems from clean or noisy phase-space data.","lead":"This paper builds a three-network neural architecture that learns the Hamiltonian, the constraint equations, and the constraint forces of mechanical systems with nonholonomic constraints directly from trajectory data. A generalist reader might care because this is a step toward data-driven modeling and control of robots and vehicles whose motion is governed by rolling or no-slip constraints.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The loss cannot separate H from the constraint force; the supplement's uniqueness proof covers only one fixed 3D example and fails for holonomic constraints.","rationale":"The reader identified the same weakest point, so I agree with the CONDITIONAL verdict. My check sharpens it: the problem is not merely A/λ gauge (Eqs. 5–6), which the paper acknowledges and which leaves A^Tλ invariant; it is an unacknowledged H↔λ gauge. On holonomic constraints the invariance is exact and elementary, so any unconditional claim of simultaneous recovery of H and λ is unsupported. The method may still be valuable as a trajectory model or as a way to learn the invariant constraint force A^Tλ; the experiments show low errors on those quantities. The appropriate remedy is to weaken the conclusion to the invariant quantities, or to add a verification of supp. Eq. 9 (and a statement about admissible F(g) for holonomic constraints) for each benchmark. Because the paper already receives a conditional verdict, I would not change the verdict; the condition should explicitly require this identifiability check.","tokens_in":15959,"tokens_out":12789,"duration_ms":139125,"concrete_test":"Train the architecture on a particle constrained to a circle, g=x^2+y^2−1, using Eq. 7. Run two runs: (i) H=T+V, λ=λ_true; (ii) H'=H+εg, λ'=λ+ε (A=∇g), same seeds/data. Report terminal loss and predicted trajectories: if they coincide while reported H and A^Tλ differ, the decomposition is non-identifiable. Then, for the rolling-disk and spinning-table A(q), solve the PDE ∇×(A^Tλ*)=0 over the training domain; if a nonzero λ* exists, construct H*=∫A^Tλ*·dq and repeat the same two-loss check.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is that H, A, b and λ are simultaneously recovered. A load-bearing gap is the identifiability of H from the combination ∂H/∂q − A^Tλ. The supplementary proof fixes A and checks supp. Eq. 9 only for the 3D point mass; it does not establish uniqueness for the rolling disk or spinning-table matrices, which are never subjected to Eq. 9. Moreover, the proof misses an exact symmetry of the loss: for a holonomic constraint g(q)=0 with A=∂g/∂q, any smooth F gives H'=H+F(g), λ'=λ+F'(g), and the loss (Eq. 7) is unchanged exactly — ∂(H+F)/∂q − A^T(λ+F') = ∂H/∂q − A^Tλ, ∂(H+F)/∂p = ∂H/∂p, and A qdot − b = dg/dt = 0. Thus infinitely many (H,λ) pairs with different reported energies and constraint forces fit the training loss identically, so the slogan 'simultaneously learns the Hamiltonian and Lagrange multipliers' is false for the holonomic part of the stated scope. For the nonholonomic examples the analogous mixing occurs if the annihilator span contains an exact one-form; the paper gives no proof that it does not. The experiments therefore demonstrate trajectory/conservative-force fitting, not identified physical recovery.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a three-network Hamiltonian-based architecture for learning the dynamics of constrained mechanical systems from phase-space data. One network learns the Hamiltonian H(q,p), a second learns the constraint matrix A(q) and vector b(q), and a third learns the Lagrange multiplier vector λ(q,p). The networks are trained with the single loss in Eq. (7), which enforces the constrained Hamilton equations and the linear constraint A qdot − b = 0. The method is tested on a rolling disk and on solid and hollow balls on a spinning table, with and without 2% multiplicative noise, and a supplementary experiment on a 3D nonholonomically constrained particle is included. Reported results include trajectory predictions, constraint residuals, mean-shifted energy MAE, normalized constraint-force error, and Lyapunov divergence times. The supplementary material also contains a proof intended to establish uniqueness of H and λ for a fixed constraint matrix A.","tokens_in":16206,"tokens_out":10591,"duration_ms":107123,"significance":"If the central identifiability claim were established, the paper would make a useful contribution to data-driven nonholonomic mechanics: recovering H, A, b, and λ simultaneously from unlabelled phase-space data would enable energy and constraint-force estimation in robotics and control. The paper is clearly written, the architecture and training loss are fully specified, and the trajectory predictions in Figs. 3, 5, and 6 are accurate with small constraint residuals, both clean and noisy. The supplementary uniqueness attempt is a genuinely constructive step beyond purely empirical benchmarking. However, as detailed below, the identifiability proof covers only one toy example, an exact symmetry invalidates the claim for the stated holonomic scope, and the ball-on-table experiments omit a necessary holonomic constraint on the Euler-Rodrigues parameters. The paper therefore currently demonstrates trajectory and constraint-residual fitting rather than the certified recovery of H and λ.","major_comments":[{"comment":"The uniqueness proof is not sufficient to support the claim that H, A, b, and λ are simultaneously learned. The proof is carried out only for the point-mass constraint matrix A = [−y, 0, 1], and the rolling-disk and spinning-table matrices (Eqs. 10 and 15) are never checked against condition Eq. (9). Moreover, the proof fixes A, whereas in training A is a learned quantity. Because the loss Eq. (7) constrains only the combined term ∂H/∂q − A^T λ and the residual A qdot − b, any alternative decomposition that leaves these two combinations unchanged is invisible to the training objective. Please either prove that no nonzero λ* satisfies ∇ × (A^T λ*) = 0 for each experimental constraint matrix, including the gauge transformations of Eqs. (5)–(6), or explicitly introduce an identifiability assumption/regularization and state the resulting scope of the claim.","section":"Supplementary §2; main §2.2, Eq. (7)"},{"comment":"The stated scope includes holonomic constraints, and there the central claim is exactly false. For a holonomic constraint g(q) = 0 with A = ∂g/∂q in the time-differentiated form, any smooth F gives H' = H + F(g) and λ' = λ + F'(g), and one verifies that ∂H'/∂p = ∂H/∂p, ∂H'/∂q − A^T λ' = ∂H/∂q − A^T λ, and A qdot − b = dg/dt = 0, so the loss Eq. (7) is unchanged exactly. Thus infinitely many (H, λ) pairs with different reported energies and constraint forces fit the training objective identically, and the mean-shifted energy MAE in Tables 5–6 cannot distinguish them. The authors should either narrow the claim to the nonholonomic setting or add a mechanism that breaks this symmetry, such as an additional energy-conservation residual or auxiliary information about H.","section":"§2.1, §2.2, Eq. (7)"},{"comment":"The ball-on-spinning-table experiments omit the holonomic unit-norm constraint on the Euler-Rodrigues parameters. The Hamiltonian in Eq. (13) and the momenta in Eq. (14) are only valid on the unit sphere ||e||^2 = 1, and with e_dot = p_e/(4I) the unconstrained flow leaves that sphere unless the additional velocity constraint e · e_dot = 0 is imposed. The learned A network outputs only two constraint rows, those in Eq. (15), so the architecture has no explicit mechanism to preserve ||e|| = 1. Please include the quaternion normalization as an additional constraint row (or justify that it is automatically satisfied), and report the e-norm error along the learned trajectories, particularly in the noisy experiments.","section":"§3.2, Eqs. (13)–(15)"}],"minor_comments":[{"comment":"The learning-rate entry for the λ and A networks is marked with an asterisk, but the table has no footnote; the text mentions reductions at 1.5e5 and 3e5 epochs, so the table should state the schedule explicitly.","section":"§3.1, Table 1"},{"comment":"The word 'asses' is used repeatedly where 'assess' is intended; please run a spelling pass over the manuscript.","section":"Throughout"},{"comment":"The in-text citation appears as 'Alltawaitan et al.' while the reference list spells the first author 'Altawaitan'; please unify the spelling.","section":"Reference [11]"},{"comment":"The Lyapunov-exponent calculation is described only briefly: initial 30% of the error vector is removed and the remainder is normalized, but the fitting window and the number of trajectories used are not stated. Please specify these details for reproducibility.","section":"§3.4"},{"comment":"The expression for phi_dot is ambiguous because the parentheses are unbalanced in the printed form; please rewrite it with unambiguous notation.","section":"§3.1, Eq. (11)"},{"comment":"No statement of code or data availability is included; providing the trained-model code would improve reproducibility and allow readers to test the identifiability issue directly.","section":"General"}],"recommendation":"major_revision","confidential_remarks":"The trajectory-modeling results appear sound and the architecture is clearly described, but the load-bearing identifiability claim is not supported for the reported systems, the holonomic case admits an exact counterexample to uniqueness, and the ball-on-table model appears to omit the quaternion normalization constraint. These issues are substantial but fixable by narrowing claims, adding identifiability tests for the actual constraint matrices, correcting the ball model, and reporting additional diagnostics. I therefore recommend major revision rather than rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Plainly: the paper is a worthwhile architecture paper, but its central claim—that it simultaneously recovers the Hamiltonian, constraints, and Lagrange multipliers—outruns the evidence. The identifiability problem is real, and the supplement does not close it.\n\nWhat's new: unlike Altawaitan et al., which assumes the constraints are known, this three-network setup learns A, b, and λ from data. The loss is a direct encoding of the pseudo-Hamiltonian equations plus the constraint residual, and the architecture is straightforward. The experiments are internally consistent: rolling disk, solid and hollow sphere on a spinning table, and a point mass in the supplement, each with and without noise. The authors even ask the right question about whether the networks learn the intended quantities and provide a supplementary proof attempt. That is honest and useful.\n\nThe soft spot is the identifiability of the decomposition. The loss only constrains ∂H/∂q − A^T λ and A q̇ − b. The gauge freedoms in Eqs. 5–6 are acknowledged, but the supplement fixes A and then checks the curl condition (Eq. 9) only for the 3D point mass. The rolling disk and spinning table matrices are never subjected to Eq. 9. Worse, there is an exact symmetry the paper misses: for a holonomic constraint g(q)=0 with A=∂g/∂q, any smooth F gives H'=H+F(g), λ'=λ+F'(g), and the loss is unchanged identically. Since the paper explicitly includes holonomic constraints, the slogan about simultaneously learning H and λ is false in that case. For nonholonomic systems the analogous mixing can occur if the constraint annihilator contains an exact one-form; the paper gives no reason it does not. So the experiments show trajectory and constraint fitting, not uniquely identified physics.\n\nMinor but worth noting: no baselines, no repeated-seed error bars, no released code or data. Those could be fixed in revision. The 229-hour training time on one disk trajectory is also worth scrutiny, though not a flaw by itself.\n\nWho should read it: people building HNN variants and anyone looking for a cautionary tale about identifiability in physics-informed learning. The architecture is a reasonable contribution, but the claims need to be reined in or proven. I'd send it to peer review—a serious referee can sort out the identifiability question—but I would not cite it as evidence of simultaneous physical recovery until that is resolved.","headline":"A clean three-network architecture for learning constrained dynamics from data, but the simultaneous-recovery claim overreaches; the identifiability gap is real and the supplement only closes it for one low-dimensional example.","tokens_in":16733,"tokens_out":2768,"would_cite":false,"duration_ms":25485,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"A single physics-informed loss trains three networks to learn the Hamiltonian, constraints, and constraint forces of nonholonomic systems from data alone.","keywords":["Hamiltonian neural networks","nonholonomic constraints","Lagrange multipliers","pseudo-Hamiltonian systems","physics-informed machine learning","constraint discovery","acatastatic constraints"],"falsifier":"Train the three-network architecture on a nonholonomic system whose constraint matrix admits a nonzero perturbation $\\lambda^*$ satisfying $\\nabla \\times (A^T \\lambda^*)=0$, the condition the paper itself identifies as the only obstruction to uniqueness. If the trajectory predictions remain accurate but the learned constraint force $\\hat A^T \\hat\\lambda$ differs from the analytical force $A^T \\lambda$ along the same trajectory, then the decomposition is not identifiable and the simultaneous recovery claim fails.","tokens_in":15719,"feed_emoji":"⚙️","tokens_out":8638,"duration_ms":78246,"temperature":0.7,"pith_summary":"This paper claims that a single physics-informed loss can train three neural networks to jointly learn everything needed to describe a mechanical system with constraints: the Hamiltonian (energy function), the constraint equations, and the Lagrange multipliers that represent the forces enforcing the constraints. The key move is to write nonholonomically constrained systems, such as a disk rolling without slipping, in pseudo-Hamiltonian form, where the constraint forces appear explicitly through the term $A^T \\lambda$. Working only from data on positions, momenta, and their time derivatives, the proposed architecture recovers trajectories, energy, and constraint forces that match analytical solutions for a rolling disk and for a ball on a spinning table. This matters because constraints are usually assumed known in physics-informed models; here they are learned from data, which could make such models useful for robotics and control when exact constraint equations are unavailable.","feed_headline":"One loss teaches three networks the physics of no-slip rolling","feed_subtitle":"It recovers motion, constraints, and constraint forces from phase-space data alone, with no labels for H, A, or λ.","key_machinery":"The central object is the pseudo-Hamiltonian formulation of nonholonomic mechanics: $\\dot q = \\partial H/\\partial p$, $\\dot p = -\\partial H/\\partial q + A^T(q) \\lambda$, subject to the linear constraint $A(q) \\dot q = b(q)$. The architecture factors this identity into three networks, one for each learned quantity, and uses a single loss whose three terms are the squared residuals of the two Hamilton equations and of the constraint equation. A supplementary uniqueness argument shows that for a fixed constraint matrix $A$, the split between $H$ and $\\lambda$ is identifiable whenever $\\nabla \\times (A^T \\lambda^*) = 0$ admits no nonzero solution $\\lambda^*$; the paper verifies this condition for its point-mass example.","core_discovery":"The paper's central claim is that the constrained Hamilton equations $\\dot q = \\partial H/\\partial p$ and $\\dot p = -\\partial H/\\partial q + A^T \\lambda$, together with the constraint equation $A \\dot q = b$, can serve as a single training objective. Three parallel multilayer perceptrons represent the Hamiltonian $H(q,p)$, the constraint matrix $A(q)$ with offset $b(q)$, and the multiplier vector $\\lambda(q,p)$; automatic differentiation yields the required derivatives of $H$, and the loss sums squared residuals of both Hamilton equations and of the constraint equation. In clean data, the learned energy and constraint forces match ground truth with mean normalized force errors around $4 \\times 10^{-4}$; under 2% multiplicative noise, force errors stay near $10^{-2}$ although long-time trajectory predictions diverge. The paper concludes that this extends Hamiltonian-based learning from conservative systems to systems with holonomic and nonholonomic constraints, producing models of the constraints and their forces without any labels for $H$, $A$, $b$, or $\\lambda$.","pith_inferences":["Because the loss only pins down the combined term $\\partial H/\\partial q - A^T \\lambda$, any practical deployment should validate learned constraint forces against an analytical force law before relying on them for control; the paper's uniqueness analysis covers only a fixed constraint matrix $A$.","The same architecture could serve as a data-driven way to discover nonholonomic constraint equations from raw robot trajectories, since the constraint network directly outputs a matrix $A(q)$ and vector $b(q)$ that can be read off after training.","For systems where measurements provide positions and velocities but not momenta, the method could be chained to a learned momentum map, though the paper lists such extensions as future work.","Because gauge transformations such as rescaling or linearly combining constraints leave the dynamics invariant, learned representations of $A$ and $\\lambda$ may differ between runs; quantifying this gauge freedom would tell users how much of the learned structure is physical."],"forward_implications":["A single training run produces a model of the unconstrained dynamics through $H$, of the constraints through $A$ and $b$, and of the constraint forces through $A^T \\lambda$, using only phase-space data.","The method handles acatastatic constraints, where $b$ is nonzero and energy is not conserved, as demonstrated by the ball on a spinning table.","Adding 2% multiplicative noise to the training data increases trajectory divergence over long horizons but keeps normalized constraint-force errors low, around 1-2%, suggesting that the learned constraint structure is relatively robust to imperfect data.","The architecture is not restricted to holonomic constraints; it treats any linear constraint of the form $A \\dot q = b$, integrable or not.","If the learned Hamiltonian and constraints are correct, the network can be used to predict energy and constraint forces for applications such as noise prediction, dissipation detection, and robot control."],"supporting_citations":[{"why":"Supplies the Hamiltonian neural network formulation that this work extends to constrained systems.","marker":"[9]"},{"why":"Establishes the pseudo-Hamiltonian equations for nonholonomic systems, which form the basis of the loss function.","marker":"[18]"},{"why":"Introduces Eden's bracket, an almost-Poisson bracket used to justify the pseudo-Hamiltonian formulation.","marker":"[20]"},{"why":"Previous Hamiltonian neural ODE treatment of a nonholonomic robot with known constraints, the baseline this work generalizes by learning constraints.","marker":"[11]"},{"why":"Modern analysis of nonholonomic brackets, supporting the claim that constrained dynamics admit an almost-Poisson formulation.","marker":"[19]"}],"fun_headline_variants":["Hamiltonian networks learn constraints and forces from raw trajectories","Three networks, one loss: physics-aware ML for nonholonomic systems","No labels needed: neural nets discover nonholonomic constraints","Pseudo-Hamiltonian networks learn rolling disk and spinning ball dynamics","One loss for Hamiltonian, constraints, and multipliers in nonholonomic systems"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The approach assumes that the training data let the networks tell the Hamiltonian apart from the constraint forces, meaning that the learned constraint matrix $A$ converges to the true one and that $H$ does not secretly absorb part of the constraint force, even though the loss only pins down the combined term $\\partial H/\\partial q - A^T \\lambda$.","fun_headline_variants_meta":{"raw":{"variants":["Hamiltonian networks learn constraints and forces from raw trajectories","Three networks, one loss: physics-aware ML for nonholonomic systems","No labels needed: neural nets discover nonholonomic constraints","Pseudo-Hamiltonian networks learn rolling disk and spinning ball dynamics","One loss for Hamiltonian, constraints, and multipliers in nonholonomic systems"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000827,"raw_usage":{"total_tokens":3618,"prompt_tokens":950,"completion_tokens":2668,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":566,"completion_tokens_details":{"reasoning_tokens":2580}},"tokens_in":566,"tokens_out":2668,"duration_ms":19076,"temperature":1.0,"reasoning_tokens":2580,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T22:50:48.205940+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Train the three-network architecture on a nonholonomic system whose constraint matrix admits a nonzero perturbation $\\lambda^*$ satisfying $\\nabla \\times (A^T \\lambda^*)=0$, the condition the paper itself identifies as the only obstruction to uniqueness. If the trajectory predictions remain accurate but the learned constraint force $\\hat A^T \\hat\\lambda$ differs from the analytical force $A^T \\lambda$ along the same trajectory, then the decomposition is not identifiable and the simultaneous recovery claim fails.","supporting_citations":[{"cited_title":"Van der Schaft, B","cited_arxiv_id":null,"evidence_quote":"Establishes the pseudo-Hamiltonian equations for nonholonomic systems, which form the basis of the loss function."},{"cited_title":"Eden, The hamiltonian dynamics of non-holonomic systems, Pro- ceedings of the Royal Society of London","cited_arxiv_id":null,"evidence_quote":"Introduces Eden's bracket, an almost-Poisson bracket used to justify the pseudo-Hamiltonian formulation."},{"cited_title":"Hamiltonian Dynamics Learning from Point Cloud Observations for Nonholonomic Mobile Robot Control","cited_arxiv_id":"2309.09163","evidence_quote":"Previous Hamiltonian neural ODE treatment of a nonholonomic robot with known constraints, the baseline this work generalizes by learning constraints."},{"cited_title":"de Le´ on, M","cited_arxiv_id":null,"evidence_quote":"Modern analysis of nonholonomic brackets, supporting the claim that constrained dynamics admit an almost-Poisson formulation."}],"review_version":1}