{"id":"9e389f4b-98a7-47ab-8668-def2c04681c8","arxiv_id":"2603.08656","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":5,"one_line_summary":"A generalized manifold Galerkin projection with G=(J−R)^{-1} produces reduced port-Hamiltonian models that preserve pH structure for linear and quadratic embeddings.","lead":"This paper shows how to shrink large port-Hamiltonian models—physics models that track energy flow—into smaller ones that keep the same energy structure. The method allows curved, nonlinear approximations and reports lower errors than two existing structure-preserving methods on spring-mass-damper tests.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Non-degeneracy of Dϕ in S_{(J−R)^{−1}} is asserted 'generic' but never proved; for the quadratic embedding (17) the singular set may intersect the ROM trajectory, making the GMG map W undefined.","rationale":"The central theoretical result, Theorem 3.1, is a correct sufficient condition: provided both hypotheses hold pointwise, the ROM is a pH system. The reader correctly identified the weakest point as the unproven 'generic' non-degeneracy assertion. This is load-bearing because the whole method depends on W being well-defined along the trajectory; without a proof, the paper's claim that the method can employ general nonlinear approximation maps is not fully substantiated. The quadratic map (17) is the most at risk since Dϕ varies with ˇx, and no argument is given to keep V(ˇx) inside S_{(J−R)^{−1}}. A concrete check on the paper's own numerical examples can show whether the condition actually fails there or whether the examples are safe; either way, the theoretical gap remains. The reader's CONDITIONAL verdict appropriately reflects that the central claim is plausible but needs an explicit verification of condition (ii) or a proof of genericity. I therefore see no reason to change the verdict.","tokens_in":15657,"tokens_out":10738,"duration_ms":93765,"concrete_test":"For the GMG-QM-ROM in the two examples of §5, compute s(ˇx(t_i)) = σ_min(V(ˇx(t_i))^T (J−R)^{−1} V(ˇx(t_i))) at every reduced-state iterate along the ROM trajectories. If min_i s is below, say, 10^{-6} times max_i s, the non-degeneracy condition is effectively violated and the ROM is ill-conditioned; if it stays bounded away from zero, the numerical claims survive for these examples, but the 'generic' assertion still lacks proof.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 3.1 requires D_ˇxϕ(ˇx) ∈ S_{(J−R)^{−1}} for every ˇx along the ROM trajectory. For the quadratic embedding (17), D_ˇxϕ is affine in ˇx (second block is V_1 plus a term linear in ˇx_2), so det(V(ˇx)^T (J−R)^{−1} V(ˇx)) is a polynomial in ˇx of degree 2r. The abstract claims this non-degeneracy is 'generically satisfied,' but no proof appears in §3–§4. The construction (12) only enforces span(B) ⊆ span(Dϕ), not condition (ii). For a fixed data-driven ϕ, the singular set is a proper algebraic hypersurface; nothing in the algorithm prevents the reduced trajectory from crossing it. If it does, W (9) diverges and the ROM (10) is not a well-defined pH system. The numerical examples in §5 may avoid this by luck, but the generic guarantee is unsupported. The method also inherits det(J−R)≠0 from §2.1, excluding pH systems with singular J−R.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript proposes a structure-preserving model-order-reduction framework for port-Hamiltonian (pH) systems based on the generalized manifold Galerkin (GMG) reduction. For a (possibly nonlinear) embedding \\phi:\\mathbb{R}^r\\to\\mathbb{R}^N, the reduced model is defined by enforcing the residual to be orthogonal with respect to the GMG projector W(\\hat x)=F_{(J-R)^{-1}}(D_{\\hat x}\\phi)^T. Theorem 3.1 gives sufficient conditions—span(B)⊆span(D_{\\hat x}\\phi) and D_{\\hat x}\\phi(\\hat x)\\in S_{(J-R)^{-1}} for every \\hat x—under which the resulting ROM is again a pH system, with \\hat J=W^TJW, \\hat R=W^TRW, \\hat H=H\\circ\\phi, and \\hat B=W^TB. The authors instantiate the framework with a linear embedding (GMG-POD-ROM) and a quadratic embedding (GMG-QM-ROM), both built from snapshot data, and use structure-preserving DEIM for nonlinear Hamiltonians. Numerical comparisons on a linear and a nonlinear mass-spring-damper system report lower relative state and output errors than the SP1-POD-ROM and SP2-POD-ROM baselines.","tokens_in":16073,"tokens_out":11955,"duration_ms":126208,"significance":"If the key non-degeneracy hypothesis can be guaranteed or checked for the constructed embeddings, the theorem provides an elegant and quite general structure-preserving reduction framework. The algebraic derivation is clean and self-contained, the construction of the reduced Hamiltonian and the constant port matrix in the special cases is natural, and the numerical study includes state errors, output errors, and energy-balance errors. The main weakness is that the central sufficient condition of Theorem 3.1 is never verified for the data-driven embeddings, and the abstract's claim that the condition is 'generically satisfied' is not substantiated anywhere in the paper. Because the GMG map W in Eq. (9) is undefined if the condition fails, this gap is load-bearing for the claimed generality.","major_comments":[{"comment":"The non-degeneracy condition D_{\\hat x}\\phi(\\hat x)\\in S_{(J-R)^{-1}} is asserted but never proved for the proposed embeddings. After Eq. (12), Section 4.1 verifies only condition (i), namely span(B)⊆span(D_{\\hat x}\\phi); condition (ii) is not addressed. For the quadratic map (17), D_{\\hat x}\\phi is affine in \\hat x_2, so det((D_{\\hat x}\\phi)^T(J-R)^{-1}D_{\\hat x}\\phi) is a polynomial in the reduced state. The constructions (18)-(22) impose no constraint on its zeros, and Algorithm 2 gives no check even for the linear map, where V=[B,\\bar V] is built only from a POD of X-BB^\\dagger X. If the determinant vanishes on the ROM trajectory, W in (9) is undefined and the ROM (10) is not a pH system. The abstract's 'generically satisfied' claim therefore needs either a rigorous proof or an algorithmic safeguard/check.","section":"Sec. 4.1, Thm. 3.1, Eqs. (12), (17), (18)-(22)"},{"comment":"The numerical errors are computed on the same trajectories used to construct the approximation maps: the snapshot matrix X in (13) is used to build the POD bases in (18)-(19) and the quadratic lifting M in (22), and the same X is then used to evaluate e_{x,red} and e_y. Thus the reported 'lower relative reduction error' is a training/interpolation error, not a predictive error on unseen inputs or initial conditions. The comparison is still informative, but the claim should be qualified as an interpolation result and, ideally, supplemented by a hold-out test or cross-validation to support the advertised accuracy advantage.","section":"Sec. 5, Eqs. (26)-(27)"}],"minor_comments":[{"comment":"The displayed formula reads 'B\\hat x_1 + V_1\\hat x_1 + V_2M(\\hat x_2\\otimes\\hat x_2)'; the second term should presumably be V_1\\hat x_2.","section":"Table 1, GMG-QM-ROM row"},{"comment":"The sentence introducing SP2-POD-ROM contains a missing citation: 'the structure-preserving MOR method presented in ,'.","section":"Sec. 5.1"},{"comment":"The formula for the lower bound appears to be missing the relative-error denominator; as printed, it is a product of two norms without a division by \\sum_i\\|x_i\\|_2^2.","section":"Eq. (28)"},{"comment":"The definition (1) assumes constant J,R,B, but the reduced system in Theorem 3.1 has state-dependent \\hat J,\\hat R,\\hat B. A short remark reconciling this with the definition (or extending the definition to state-dependent matrices) would improve readability.","section":"Sec. 2.1 and Thm. 3.1"},{"comment":"The assumption det(J-R)\\neq 0 is not part of the standard definition of pH systems and is imposed for the entire paper. The abstract and introduction currently claim applicability to 'pH systems' without this caveat; the scope restriction should be stated explicitly in the abstract and revisited in the conclusion.","section":"Sec. 2.1, abstract"}],"recommendation":"major_revision","confidential_remarks":"The main theorem is sound under its hypotheses, and the framework is a genuine contribution. The critical issue is the unproved 'generic' non-degeneracy: the algorithms as written can produce an undefined GMG map. This seems fixable by adding a rigorous genericity argument for the specific embedding classes, or by adding an explicit invertibility check/regularization, and by qualifying the numerical claims as training errors. I therefore recommend major revision rather than rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"First, what is new and good. The paper takes the generalized manifold Galerkin reduction from Buchfink et al. and applies it to pH systems by setting the structure matrix G = (J-R)^{-1}. The main theorem, Theorem 3.1, gives sufficient conditions—span(B) inside the tangent of the embedding and the tangent in the non-degenerate set S_{(J-R)^{-1}}—under which the reduced model is again pH. The proof is short and the algebra is correct: with W from the GMG map, the reduced matrices are W^T J W and W^T R W, and the port condition makes the output equation work. The paper also gives linear and quadratic embeddings where the port-matrix condition is built in by construction, which is a thoughtful design. This is not a re-coordinatization of existing work; it extends [4] in a way that is useful for practice.\n\nThe soft spots are two. First, the abstract asserts that the non-degeneracy condition Dϕ(ˇx) ∈ S_{(J-R)^{-1}} is generically satisfied, but no proof is offered. For the quadratic embedding, Dϕ is affine in ˇx2, so the determinant is a polynomial in ˇx; the construction does not prevent the ROM trajectory from hitting the singular set. If it does, W diverges and the reduced model stops being well-defined. This is a real gap, though it looks patchable—either with a careful genericity argument or by checking the condition online. The stress-test note is accurate here. Second, the numerical evaluation is in-sample: the POD/quadratic bases are built from the same trajectories used to measure error, and the hyperparameters λ_reg and r_n are tuned on the test data. So the reported lower error is partly a fitting result. This is common in MOR, but it deserves an explicit disclaimer. I also would have liked a comparison to data-driven pH methods like pH-OpInf or SOBMOR, not just to the two POD baselines.\n\nMinor note: the assumption det(J-R)≠0 is stated clearly in Section 2.1, so that is not a hidden flaw. The paper does not ship code or data, so reproducibility is limited.\n\nMy overall take: the central theorem is solid and the class of general nonlinear embeddings genuinely extends what was available. The genericity claim is the main thing I would push on in review—it is asserted rather than proved, and it is load-bearing for the quadratic case. But the method itself is reasonable and the examples make a plausible case that it works. I would send this to a serious referee. It is not a major breakthrough, but it is a clean, useful contribution that the MOR community should see.\n\nRecommendation: accept for peer review; ask for a proof or counterexample for the genericity claim and out-of-sample or hold-out validation.","headline":"Clean pH-preservation theorem built on GMG reduction; the generic non-degeneracy claim is unproved and the numerics are in-sample, but the core extension is real and worth refereeing.","tokens_in":16447,"tokens_out":3023,"would_cite":true,"duration_ms":31758,"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":"A generalized manifold Galerkin reduction preserves the port-Hamiltonian form for any approximation map whose tangent space contains the port matrix and avoids the singular set of (J−R)^{-1}.","keywords":["structure-preserving model reduction","port-Hamiltonian systems","generalized manifold Galerkin reduction","quadratically embedded manifolds","passivity","nonlinear model reduction","energy balance"],"falsifier":"Compute det(V(ˇx(t))^T (J−R)^{−1} V(ˇx(t))) along the reduced trajectory for GMG-QM-ROM on a mass-spring-damper with a strong input; if it crosses zero at any time, the ROM is no longer a pH system (or the projection breaks), disproving the genericity claim.","tokens_in":15610,"feed_emoji":"⚙️","tokens_out":12454,"duration_ms":102347,"temperature":0.7,"pith_summary":"This paper proposes a model order reduction framework for port-Hamiltonian systems that works with general nonlinear approximation maps, not just linear or specially structured ones. The central claim is that if the embedding's Jacobian contains the port matrix and stays in the non-degeneracy set of (J−R)^{-1}, the reduced model obtained by generalized manifold Galerkin projection is again port-Hamiltonian, so passivity and the energy balance are preserved. The paper proves this theorem and gives explicit linear and quadratic embeddings that satisfy the conditions. Numerical examples on linear and nonlinear mass-spring-damper systems show that the reduced models have lower relative state and output errors than existing structure-preserving methods. If correct, this lifts a major restriction on nonlinear structure-preserving MOR for energy-based systems.","feed_headline":"Two conditions preserve port-Hamiltonian structure in reduced models","feed_subtitle":"Any nonlinear embedding that satisfies two tangent-space conditions yields a reduced port-Hamiltonian model, with lower errors in tests.","key_machinery":"The load-bearing piece is the generalized manifold Galerkin (GMG) reduction map defined with structure matrix G = (J−R)^{−1}. For a given embedding Jacobian V = D_x ϕ, it produces the tangent projection W = (F_G(V))^T, where F_G(V) = (V^T G V)^{−1} V^T G. This projection becomes the 'reduction map' that turns the full-order residual into the reduced dynamics. The non-degeneracy condition V^T G V invertible and the port-inclusion condition span(B) ⊆ span(V) together force the reduced system to inherit the pH structure. The paper constructs embeddings of the form ϕ(ˇx) = B ˇx_1 + V η(ˇx_2) so that the port condition holds automatically, and instantiates η as linear and quadratic maps.","core_discovery":"The core discovery is that the pH structure survives the GMG reduction under two explicitly checkable conditions on the approximation map's Jacobian V: span(B) ⊆ span(V) and V ∈ S_{(J−R)^{−1}}, i.e., V^T (J−R)^{−1} V invertible. Under these conditions the reduced system is exactly of port-Hamiltonian form with ˇJ = W^T J W, ˇR = W^T R W, ˇH = H∘ϕ, and ˇB = W^T B, where W = (F_{(J−R)^{−1}}(V))^T. The proof works by rewriting the GMG projection so that it factors into the desired pH structure, and the output equation is matched through the port-inclusion condition. This is the first such result allowing a completely general nonlinear embedding map.","pith_inferences":["The paper asserts that the non-degeneracy condition is 'generically satisfied' but provides no proof; a measure-theoretic or probabilistic statement about the set of initial conditions/trajectories would turn this into a rigorous guarantee.","The assumption det(J−R)≠0 excludes significant classes of pH systems, such as lossless systems with singular J or systems with dependent ports; extending the framework to differential-algebraic pH systems would broaden the reach.","The quadratic embedding could be replaced by higher-degree polynomial or neural-network maps without any change to the structural argument, provided the tangent-space condition is enforced during training—this suggests a direct route to learning-based structure-preserving MOR.","The numerical experiments are limited to mass-spring-damper systems; testing on fluid or electrical networks would check whether the claimed accuracy gains persist outside mechanical examples."],"forward_implications":["With the two conditions satisfied, reduced-order models of both linear and nonlinear pH systems remain provably port-Hamiltonian, preserving passivity and the dissipation inequality.","The framework removes the need to restrict to linear embeddings or special separable ansatzes, opening the door to data-driven nonlinear embeddings (e.g., neural-network maps) as long as the tangent condition is enforced.","The explicit reduced matrices (^J, ^R, ^B) and reduced Hamiltonian (^H) are given in closed form, so the resulting ROM can be simulated with standard port-Hamiltonian solvers.","Combining the method with structure-preserving DEIM keeps the ROM cheap while retaining the pH structure, as the numerical examples illustrate.","The quadratic embedding realization (GMG-QM-ROM) achieves lower state and output errors than the linear one and is close to the theoretical lower bound, indicating the approach can exploit curvature of the solution manifold."],"fun_headline_variants":["Two conditions keep port-Hamiltonian form in reduction","General nonlinear embeddings preserve pH structure under two checks","Structure-preserving reduction: any embedding works if two conditions hold","New reduction method lowers error while keeping passivity","Two tangent-space conditions guarantee port-Hamiltonian ROMs"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The construction requires that det(J−R) ≠ 0 and that, for every reduced state along the trajectory, the Jacobian of the nonlinear embedding satisfies det((D_xϕ)^T (J−R)^{−1} D_xϕ) ≠ 0; the paper assumes this non-degeneracy for all reduced states without giving a constructive guarantee that it holds away from the training data.","fun_headline_variants_meta":{"raw":{"variants":["Two conditions keep port-Hamiltonian form in reduction","General nonlinear embeddings preserve pH structure under two checks","Structure-preserving reduction: any embedding works if two conditions hold","New reduction method lowers error while keeping passivity","Two tangent-space conditions guarantee port-Hamiltonian ROMs"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00016,"raw_usage":{"total_tokens":1075,"prompt_tokens":758,"completion_tokens":317,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":502,"completion_tokens_details":{"reasoning_tokens":240}},"tokens_in":502,"tokens_out":317,"duration_ms":3725,"temperature":1.0,"reasoning_tokens":240,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-04T05:50:21.624970+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute det(V(ˇx(t))^T (J−R)^{−1} V(ˇx(t))) along the reduced trajectory for GMG-QM-ROM on a mass-spring-damper with a strong input; if it crosses zero at any time, the ROM is no longer a pH system (or the projection breaks), disproving the genericity claim.","supporting_citations":[],"review_version":1}