{"id":"499edc1e-17f5-4ba6-9ded-b7c09ee09831","arxiv_id":"2506.17868","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":6,"one_line_summary":"Geometric Contact Flows learn dissipative dynamical systems by mapping a simple contact-Hamiltonian baseline through learned contact-preserving transformations, with ensemble uncertainty and geodesic reshaping for robust generalization.","lead":"This paper introduces a machine learning method that builds 'contact geometry' into models of moving physical systems, letting them naturally capture energy loss and gain. It adds uncertainty detection and safe steering, and is tested on physical simulations and real robot manipulation tasks.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The empirical claims hinge on an underdetermined canonical-coordinate lift: App. D.3 leaves s0 unspecified, so s may be an invented progress signal that baselines never receive.","rationale":"The reader's weakest assumption identifies the most load-bearing juncture: the canonical-coordinate lift is required by every experiment, and the unspecified s0 plus the absence of baselines that receive s make the main quantitative comparisons uncontrolled. I did not find a more fundamental objection to the framework itself; the learned contactomorphisms, the latent integration, and the ensemble uncertainty mechanism are coherent as described. The sign inconsistency in Eq. (8) is real, but the implemented vector field (31) gives the intended dissipative behavior, so it is a correctable presentation error rather than a reason to reject the framework. Because these concerns are concrete and fixable and do not, by themselves, disprove the algorithmic proposal, the existing CONDITIONAL verdict remains appropriate; I would neither accept the paper as is nor escalate to REJECT.","tokens_in":32713,"tokens_out":9210,"duration_ms":103626,"concrete_test":"Fix the lift and rerun the key experiments with documented s0 choices: (a) s0=0, (b) s0=1, and (c) the s0 implied by imposing sddot0=0 in Eq. (43) at t=0. Then rerun the handwriting and spring-mesh comparisons while feeding the same constructed s coordinate to DHNN and MLP as an additional input. If the GCF DTWD advantage and convergence ratios persist under all three s0 choices and with s-augmented baselines, the concern is refuted; if results shift materially or the baseline gap narrows, the headline generalization claims are not attributable to the contact structure.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is that contact Hamiltonian structure, not auxiliary inputs, explains GCF's accuracy and generalization. Every experiment depends on converting observed (q, qdot) trajectories into canonical coordinates (q, p, s). Appendix D.3 defines s by integrating Eq. (43), a second-order ODE whose right-hand side involves s itself and the observed q, qdot, qddot. The initial conditions in Eq. (44) state sddot0=0 and sdot0=qdot0^T qdot0, but leave s0 unspecified. For a second-order ODE, s0 is part of the initial data: one cannot impose sddot0=0 without solving for s0, or if s0 is chosen freely then sddot0 is determined by Eq. (43), not by fiat. Because s0 is unspecified, the entire s trajectory is underdetermined. For the handwriting data, beta(q)=T-t further makes s a proxy for time-to-go along each demonstration. Baselines (EF, NCDS, DHNN) never receive this coordinate, so the headline DTWD reductions (57%, 60%) and convergence metrics may reflect an extra 'progress clock' rather than contact geometry. The contactomorphism-versus-diffeomorphism ablation does not isolate this issue, since both variants receive s. Separately, Eq. (8) has a sign error: with the contact vector field (31), dH/dt = -H * dH/ds, so the exponent should be negative; Eq. (40) implicitly uses the negative sign. This is secondary but indicates the theoretical presentation is not yet fully settled.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces Geometric Contact Flows (GCF), a framework for learning dissipative second-order dynamical systems from data. GCF couples a latent contact Hamiltonian model, chosen to encode properties such as stability or energy depletion, to the observed dynamics through an ensemble of learned contactomorphisms. The ensemble provides predictive uncertainty, which is used to reshape a Riemannian metric and guide trajectories toward the data manifold. Experiments are reported on a 60-dimensional spring mesh, a single-mode bosonic quantum system, two handwriting datasets (LASA and DigiLeTs), and two robotic manipulation tasks, with comparisons against EF, NCDS, HNN, DHNN, and MLP baselines. The paper claims a 57% DTWD reduction on the spring mesh, a 60% reduction on the quantum system, superior convergence to the data manifold on handwriting generalization, and successful energy-aware robot task execution.","tokens_in":33028,"tokens_out":8309,"duration_ms":78974,"significance":"The core idea is attractive and potentially significant for the learning-dynamics community: it brings the odd-dimensional contact Hamiltonian formalism, with its physically meaningful action variable, into a neural-network framework that supports uncertainty quantification and control via metric reshaping. The paper is also honest in its reporting: error bars, ablations (contactomorphism vs diffeomorphism, latent loss weight, ensemble vs single map), and architectural details are provided, and the flow-based contactomorphisms are analytically invertible. If the empirical claims survive scrutiny, the framework would be a meaningful advance over first-order diffeomorphism-based methods and over Hamiltonian networks lacking a dissipation coordinate. However, the current validation is undermined by the underdetermined construction of the canonical-coordinate lift, by an internal sign inconsistency in the theoretical energy law, and by at least one headline claim that is contradicted by the paper's own tables. These issues are fixable within the scope of the manuscript, but they must be addressed before the central claims can be accepted.","major_comments":[{"comment":"The canonical-coordinate lift used in every experiment is underdetermined and internally inconsistent. Equation (43) is a second-order ODE for s, so a well-posed initial value problem requires both s0 and sdot0. The paper lists sdot0 = qdot0^T qdot0 and imposes sddot0 = 0, but does not specify s0; conversely, if s0 is chosen freely, then sddot0 is determined by Eq. (43) and cannot be imposed independently. Moreover, with the stated sdot0 the first term in Eq. (43) vanishes identically, so the condition sddot0 = 0 reduces to 2 qdot0^T qddot0 = 0, which is not satisfied by generic trajectories. In addition, the parametrization beta(q) = T - t introduces explicit time dependence into the 'Hamiltonian' (38) and makes s a time-to-go proxy, which conflicts with the autonomy assumption stated in App. D.1. Because the baselines EF, NCDS, and DHNN never receive s, the headline 57% and 60% DTWD reductions may reflect the presence of this extra input rather than contact geometry. The contactomorphism-versus-diffeomorphism ablation does not resolve this, since both variants receive s. This issue is load-bearing for the central empirical claim of the paper.","section":"App. D.3, Eqs. (43)–(44)"},{"comment":"Equation (8) has a sign error. For the contact Hamiltonian vector field written in Eq. (31), a direct computation gives dH/dt = -H * dH/ds, so the correct energy law is H(t) = H(0) * exp(-∫ dH/ds dτ). Equation (8) displays the opposite sign, and it is also contradicted by the paper's own Eq. (40) in App. D.3, which uses the negative sign. The text immediately after Eq. (8) uses the displayed expression to argue that the '+s' term in Eq. (7b) depletes the system energy; with the sign as written, the opposite behavior would follow. This internal inconsistency affects the theoretical motivation for the latent Hamiltonians HgB and HgC and should be corrected.","section":"Sec. 3, Eq. (8)"},{"comment":"The claim that GCF achieves 'the highest average convergence ratio and lowest variance' on the handwriting generalization tests is not supported by the reported numbers. In Table 4, on LEAF 2, NCDS achieves 0.94 ± 0.19 while GCF achieves 0.69 ± 0.19; on ELLE, GCF achieves 0.66 ± 0.12, which is within one standard deviation of EF's 0.61 ± 0.30. The conclusion in the text that 'GCF shows greater reliability, ensuring all the dynamics predictions for the two characters converge to the data manifold' is too strong given these overlapping values. The authors should either restrict the claim to the cases where the advantage is statistically meaningful or add the appropriate significance tests.","section":"Sec. 6, Table 4 and Fig. 6"},{"comment":"The uncertainty-aware control mechanism is not operationalized. The text states that trajectories are steered by a control input u obtained from the optimal control problem (18), but no solver, discretization scheme, cost weight for sigma_z, or stopping criteria are provided, and no algorithm is given in the appendix. The generalization experiments in App. E.3 and the ensemble ablation in Table 16 depend on this mechanism, since the ensemble's uncertainty is claimed to reshape the latent metric. Without a precise specification, the convergence-to-data-manifold results are not reproducible, and the contribution of the uncertainty-aware geodesics cannot be separated from the behavior of the latent dynamics alone.","section":"Sec. 5, Eq. (18)"}],"minor_comments":[{"comment":"The acronym is inconsistent: the abstract uses 'GFC' while the main text uses 'GCF'; please choose one spelling and use it consistently.","section":"Throughout"},{"comment":"The two rows in Table 3 are not labeled with the character names; the corresponding rows in Table 14 of App. E.3 should be reproduced or referenced so the reader knows which characters are being compared.","section":"Sec. 6, Table 3"},{"comment":"The notation beta(q) = T - t is misleading because T - t is a function of time, not of q; this should be clarified and reconciled with the statement in App. D.1 that the model treats the dynamical system as autonomous.","section":"App. D.3"},{"comment":"The derivation of Eq. (41) by substituting Eq. (39) into Eq. (40) is hard to follow and appears circular as written; please spell out the substitution and the role of H0, or add a reference for this step.","section":"App. D.3, Eqs. (39)–(41)"},{"comment":"The baselines enumeration lists EF, NCDS, HNN, and DHNN but Tables 1 and 2 also report MLP; MLP should be included in the enumeration for completeness.","section":"Sec. 6"},{"comment":"The caption says 'From top to bottom rows, EF, NCDS, DHNN and GCF methods' but the figure is arranged as a 4-row-by-2-column grid; please clarify what the two columns represent.","section":"Fig. 5"}],"recommendation":"major_revision","confidential_remarks":"The paper has a strong core idea and a substantial experimental effort, including real robot demonstrations. The most serious issue is the underdetermined lift in App. D.3, which affects every experimental claim; a revision should specify s0 (or a well-posed initial condition), test sensitivity to that choice, and add baseline variants that also receive the s coordinate. The Eq. (8) sign error is fixable but signals that the theoretical section needs a careful pass. The overclaim in Table 4/Fig. 6 can be corrected with more precise language. I would not reject, as the core methodology is defensible and the problems appear addressable within the scope of the manuscript."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know. First, the core construction is real: a latent contact Hamiltonian dynamics, transferred to data through learned contactomorphisms (flows of contact Hamiltonian vector fields), plus an ensemble for uncertainty and metric reshaping. That combination is new, and the paper gives a plausible recipe for learning dissipative second-order dynamics with stability and energy priors. Second, the empirical claims are weaker than they look. The main comparisons give GCF an extra input, the Lagrangian action s, that the baselines never receive, and the way s is constructed is underdetermined. Until that is controlled, the 57-60% DTWD numbers cannot be read as evidence for contact geometry.\n\nWhat is good. The paper is honest: error bars, ablations, and a diffeomorphism-vs-contactomorphism ablation that shows a large degradation. The real-robot wrap-and-pull and dishwasher-loading experiments are concrete, and the implementation is public. B.3 checks the contactomorphism conditions. I do not see a circularity problem; the model is fitted to observed trajectories.\n\nWhere it is soft. Eq. (8) has a sign error. With the contact vector field (31), dH/dt = -H ∂H/∂s, so the exponent should be negative; Eq. (40) later uses the negative sign. Likely a typo, but in a theory paper the main energy law should be stated correctly.\n\nMore serious is App. D.3. The action s is defined by integrating a second-order ODE with s0 left free and sddot0 set to zero, which is not the right initial data: sddot0 is determined by the ODE and by q, qdot, qddot. Vary s0 and the s trajectory changes. For handwriting, beta(q)=T-t makes s roughly a time-to-go clock. Since EF, NCDS, and DHNN never see s, the reported gains may come from that clock rather than from contact structure. The contactomorphism ablation does not fix this, because both variants receive s. I would want (a) s0 specified or a proof that results are invariant to it; (b) a baseline that gets the same extra coordinate (e.g., DHNN augmented with s) or an ablation with s removed; and (c) a comparison to the closest contact-Hamiltonian baseline, Zadra [2023], which is cited but absent. The single-vs-ensemble ablation also changes both the ensemble and the uncertainty-aware geodesics, so the contribution of the geodesic controller is not isolated. That is a smaller point.\n\nBottom line: this is a serious paper for the geometric-deep-learning and robot-learning crowds, and it deserves a real referee. I would not desk-reject; I would send it out with the request to fix the s-lift and the sign, and to add the missing control. If the s-lift survives scrutiny, the framework stands.","headline":"Genuinely new contact-geometric learning framework with real robot demos, but the headline gains are not yet controlled because the latent action coordinate s is underdetermined and withheld from baselines; send to review with a request to fix the s-lift and the Eq. (8) sign.","tokens_in":33560,"tokens_out":4308,"would_cite":true,"duration_ms":42662,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"Geometric Contact Flows claims that contactomorphisms, by preserving the contact form, let a latent contact Hamiltonian dynamics reconstruct dissipative second-order systems and steer generalization toward the data manifold.","keywords":["contact Hamiltonian dynamics","contactomorphisms","dissipative dynamical systems","Riemannian metric learning","uncertainty-aware geodesics","physics-informed learning","robot manipulation","motion generation"],"falsifier":"Run GCF on a damped mechanical system with a known non-unit-mass Hamiltonian, compute the $s$ series from Eq. (43), and compare it to the true mechanical action; if the reconstructed $s$ diverges from the true action and reconstruction error nonetheless stays low, the method is fitting the invented coordinate rather than the physics. A cleaner test: give the baselines the same estimated $s$ as input and see whether the reported 57% and 60% dynamic-time-warping margins survive.","tokens_in":32468,"feed_emoji":"🌀","tokens_out":6566,"duration_ms":63117,"temperature":0.7,"pith_summary":"Geometric Contact Flows (GCF) claims that a learned, second-order dynamical system can be made accurate and physically interpretable by anchoring it to a simple contact Hamiltonian dynamics in a latent space and connecting that latent space to the observed state space through contactomorphisms, which are diffeomorphisms preserving the contact form up to a scaling factor. The paper's thesis is that this geometric prior fixes what black-box models and existing physics-inspired baselines miss: dissipative systems with energy exchange need an extra odd-dimensional variable, the Lagrangian action, that symplectic and Hamiltonian methods do not carry. Concretely, the paper reports that GCF reduces dynamic-time-warping reconstruction error by 57% on a spring-mesh benchmark and by 60% on stochastic quantum dynamics, and that trajectories initialized from off-manifold grid points reliably converge back to the data manifold on handwriting tasks. Why it would matter: dynamics models built this way inherit stability or energy behavior from the latent system, come with a principled uncertainty signal from an ensemble of maps, and can be steered away from unsafe or data-poor regions by reshaping the metric. The same machinery is demonstrated on real robot tasks, including an energy-aware safety stop when an unexpected load drives the action variable to zero.","feed_headline":"Contact maps make learned dissipative dynamics reliable","feed_subtitle":"Latent contact Hamiltonians plus structure-preserving maps cut reconstruction error by ~60% and steer motion to data.","key_machinery":"The load-bearing object is the contact Hamiltonian vector field on the canonically coordinated contact manifold $T^*\\mathbb{R}^d\\times\\mathbb{R}$ with state $\\{q,p,s\\}$, whose local equations are $\\dot q=\\partial H/\\partial p$, $\\dot p=-\\partial H/\\partial q-p\\,\\partial H/\\partial s$, and $\\dot s=p^\\top\\partial H/\\partial p-H$; here $s$ is the Lagrangian action and $H$'s dependence on $s$ controls energy dissipation through Eq. (8). A contactomorphism is a diffeomorphism preserving the contact form up to a positive factor, and the paper builds each one as the time-$T$ composition of flows of learned contact Hamiltonians $H_{r_{\\theta_k}}=\\tfrac12 p^\\top M_{\\theta_k}(p)p+V_{\\theta_k}(q)+F_{\\theta_k}(q)s$, integrated with an analytically invertible contact splitting integrator. This machinery does two jobs: it transfers the latent system's stability or safety properties to the observed dynamics, and it supplies an ensemble whose disagreement defines an uncertainty metric that reshapes the latent geodesic problem, steering trajectories away from high-uncertainty, data-poor regions.","core_discovery":"The core claim is that composing a latent contact Hamiltonian flow with an ensemble of learned contactomorphisms yields a model of non-conservative, second-order dynamics that is both accurate on the data support and stable outside it. In the paper's construction, the latent dynamics $\\dot{z}=Z_{H_g}$ is chosen to encode desired behavior, such as periodic, stable, or safe motion, and the learned maps $\\phi_{r_n}$ carry that behavior to the observed coordinates through $x(t)=\\phi_r^{-1}\\circ \\phi_g(t)\\circ \\phi_r(x_0)$, so a long-horizon prediction needs only one forward and one inverse mapping. The paper argues that preserving the contact form is what keeps the ambient dynamics physically interpretable: the damping coefficient $\\partial H_m/\\partial s$ and the action $s$ retain their mechanical meaning, and the ensemble's variance, injected into the latent metric as an extra traversal cost, makes off-manifold predictions bend toward the data. On the evidence offered, this yields the reported error reductions and convergence rates, and on the robot tasks it produces a stable energy-consumption profile and a safety stop when the Lagrangian action reaches zero under unexpected loading.","pith_inferences":["Because the action variable $s$ is estimated from $(\\mathbf{q},\\dot{\\mathbf{q}})$ rather than measured, a fair stress test would feed the same estimated $s$ to symplectic-style baselines or train GCF without it; the paper does not run this control, so part of the reported margin could reflect the extra input channel.","The same uncertainty-as-cost construction could be repurposed for costs other than ensemble variance, such as predicted collision risk, actuator limits, or time-to-contact, without changing the optimization scheme in Eq. (18).","If the contactomorphism ansatz behaves like a normal form, skills learned on one robot platform might transfer to another by re-estimating only the inertia-scaled momentum map, a conjecture the paper does not test.","The sensitivity of the learned dynamics to the unspecified initial value $s_0$ in Eq. (44) is a natural ablation: on datasets with non-unit mass, if reconstruction error depends strongly on this constant, the latent action should be treated as an additional learned variable rather than a fixed preprocessing outcome."],"forward_implications":["If the central claim is right, geometry alone, rather than hand-tuned losses, can bias a learned dynamics toward dissipative and even odd-dimensional phase spaces, since the action variable $s$ is part of the state.","A model trained once with stable latent dynamics can reproduce a demonstrated robot skill under loading never seen during training at roughly the same energy expenditure, because the contact structure keeps the energy bookkeeping physically consistent.","Ensemble uncertainty is not just a diagnostic but an active control signal, so predictions started outside the data support can be expected to converge back to the manifold rather than diverge.","The reported reductions of 57% and 60% in dynamic-time-warping distance imply that on spring-mesh and quantum-dynamics benchmarks, the structure-preserving map extracts more usable signal from the same trajectories than the best baselines compared in the paper."],"supporting_citations":[{"why":"Supplies the contact Hamiltonian mechanics and the energy-variation law $H(t)=H(0)e^{\\int \\partial H/\\partial s\\,d\\tau}$ used for the latent dynamics and damping.","marker":"Bravetti et al., 2017"},{"why":"Provides the local form of the contact Hamiltonian vector field and the contact splitting integrator used to integrate each learned contactomorphism flow.","marker":"Zadra, 2023"},{"why":"Grounds the identification of contact flows with geodesics on the augmented space-time manifold via Maupertuis' principle, which motivates the metric reshaping.","marker":"Abraham & Marsden, 2008"},{"why":"Defines the Euclideanizing Flows baseline that GCF extends by replacing diffeomorphisms with contactomorphisms.","marker":"Rana et al., 2020"},{"why":"Supplies the contractive NCDS baseline against which GCF's off-manifold convergence is compared.","marker":"Mohammadi et al., 2024"},{"why":"Supplies the Dissipative Hamiltonian Neural Network baseline and the limitation to even-dimensional phase spaces that GCF addresses.","marker":"Sosanya & Greydanus, 2022"},{"why":"Provides the spring-mesh benchmark dataset used for the reported 57% dynamic-time-warping reduction.","marker":"Otness et al., 2021"},{"why":"Provides the LASA handwriting benchmark used for the reconstruction and generalization tests.","marker":"Lemme et al., 2015"},{"why":"Provides the DigiLeTs second-order handwriting dataset with self-crossing trajectories used to test GCF beyond first-order dynamics.","marker":"Fabi et al., 2022"}],"fun_headline_variants":["Contact geometry tames learned dissipative dynamics","Structure-preserving contact flows cut error 60%","Geometric contact flows for dependable robot control","Contact Hamiltonians stabilize learned dissipative dynamics","Contact maps make learned physics stay on track"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument depends on the unmeasured 'Lagrangian action' coordinate $s$ being faithfully reconstructed from the observed positions and velocities; if that reconstructed coordinate does not represent real energy exchange, the comparisons against methods that never receive $s$ are not controlled.","fun_headline_variants_meta":{"raw":{"variants":["Contact geometry tames learned dissipative dynamics","Structure-preserving contact flows cut error 60%","Geometric contact flows for dependable robot control","Contact Hamiltonians stabilize learned dissipative dynamics","Contact maps make learned physics stay on track"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001194,"raw_usage":{"total_tokens":4909,"prompt_tokens":912,"completion_tokens":3997,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":528,"completion_tokens_details":{"reasoning_tokens":3929}},"tokens_in":528,"tokens_out":3997,"duration_ms":25931,"temperature":1.0,"reasoning_tokens":3929,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T18:59:54.422219+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run GCF on a damped mechanical system with a known non-unit-mass Hamiltonian, compute the $s$ series from Eq. (43), and compare it to the true mechanical action; if the reconstructed $s$ diverges from the true action and reconstruction error nonetheless stays low, the method is fitting the invented coordinate rather than the physics. A cleaner test: give the baselines the same estimated $s$ as input and see whether the reported 57% and 60% dynamic-time-warping margins survive.","supporting_citations":[{"cited_title":"Contact H amiltonian mechanics","cited_arxiv_id":null,"evidence_quote":"Supplies the contact Hamiltonian mechanics and the energy-variation law $H(t)=H(0)e^{\\int \\partial H/\\partial s\\,d\\tau}$ used for the latent dynamics and damping."},{"cited_title":"Topics in contact H amiltonian systems: analytical and numerical perspectives","cited_arxiv_id":null,"evidence_quote":"Provides the local form of the contact Hamiltonian vector field and the contact splitting integrator used to integrate each learned contactomorphism flow."},{"cited_title":"and Marsden, J","cited_arxiv_id":null,"evidence_quote":"Grounds the identification of contact flows with geodesics on the augmented space-time manifold via Maupertuis' principle, which motivates the metric reshaping."},{"cited_title":"A., Li, A., Fox, D., Boots, B., Ramos, F., and Ratliff, N","cited_arxiv_id":null,"evidence_quote":"Defines the Euclideanizing Flows baseline that GCF extends by replacing diffeomorphisms with contactomorphisms."},{"cited_title":"B., Hauberg, S., Arvanitidis, G., Figueroa, N., Neumann, G., and Rozo, L","cited_arxiv_id":null,"evidence_quote":"Supplies the contractive NCDS baseline against which GCF's off-manifold convergence is compared."},{"cited_title":"An extensible benchmark suite for learning to simulate physical systems","cited_arxiv_id":null,"evidence_quote":"Provides the spring-mesh benchmark dataset used for the reported 57% dynamic-time-warping reduction."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the DigiLeTs second-order handwriting dataset with self-crossing trajectories used to test GCF beyond first-order dynamics."}],"review_version":2}