Structure-Preserving Generalized Manifold Galerkin Reduction for Port-Hamiltonian Systems

Hongliang Mu; Silke Glas

REVIEW 2 major objections 5 minor 1 cited by

Structure-Preserving Generalized Manifold Galerkin Reduction for Port-Hamiltonian Systems

T0 review · 2 major / 5 minor · reviewed 2026-08-04 · deepseek-v4-flash

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Pith's one-line read 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}.

desk verdict 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. read the letter →

arxiv 2603.08656 v2 pith:3MISP6SX submitted 2026-03-09 math.NA cs.NA

Silke Glas , Hongliang Mu This is my paper · ORCID

classification math.NAcs.NA

keywords structure-preservingmodelreductionport-HamiltoniansystemsgeneralizedmanifoldGalerkinquadraticallyembeddedmanifoldspassivitynonlinearenergybalance

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

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.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the 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.

The 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.

Minor 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.

My 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.

Recommendation: accept for peer review; ask for a proof or counterexample for the genericity claim and out-of-sample or hold-out validation.

Referee Report

2 major / 5 minor

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.

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 (2)

[Sec. 4.1, Thm. 3.1, Eqs. (12), (17), (18)-(22)] 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.
[Sec. 5, Eqs. (26)-(27)] 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.

minor comments (5)

[Table 1, GMG-QM-ROM row] 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.
[Sec. 5.1] The sentence introducing SP2-POD-ROM contains a missing citation: 'the structure-preserving MOR method presented in ,'.
[Eq. (28)] 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.
[Sec. 2.1 and Thm. 3.1] 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.
[Sec. 2.1, abstract] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the structure-preservation theorem is a self-contained algebraic derivation; the main caveats are an unproved 'generic' claim and in-sample numerical validation.

full rationale

The central result, Theorem 3.1, is derived directly from the GMG reduction definition (9) and the definitions of the reduced quantities W, J, R, H, and B. The proof uses only W^T D_ x phi = I and span(B) subset span(D_ x phi) to rewrite the projected vector field as (W^T J W - W^T R W) grad_ x H(phi(x)) plus W^T B u, and to match the output equation. This is an algebraic identity, not an equivalence with its own conclusion. Condition (ii), D_ x phi in S_{(J-R)^{-1}}, is exactly the invertibility required for W in (9) to exist, so the theorem is a conditional statement rather than a tautology. The cited works [3] and [4] are prior work by a co-author, but they only supply the GMG framework and an approximation lower bound; neither is used to justify pH preservation, and the proof above is self-contained. Two caveats are correctness/validation concerns rather than circularity: the abstract claims the non-degeneracy conditions are 'generically satisfied' without a proof in Sections 3-4, and the numerical basis V, V1, V2 is fitted to the same snapshot matrix X (13) used to evaluate errors (26)-(29), so the reported accuracy is in-sample rather than a predictive test on unseen inputs. Neither caveat makes the derivation equivalent to its inputs by construction.

Assumptions & free parameters 5 free parameters · 6 assumptions · 0 invented entities

The central structure-preservation theorem relies on standard GMG machinery plus an invertibility/non-degeneracy assumption on (J−R)^{-1}. No new physical entities are introduced. The numerical accuracy claims depend on data-driven bases and tuned hyperparameters listed as free parameters.

free parameters (5)

quadratic lifting dimension r_n = 8 (linear example); input to Algorithm 3
Chosen by hand; controls the number of quadratic lift columns in V2 and affects the accuracy achieved by GMG-QM-ROM.
regularization parameter λ_reg = 10^{-3} (linear example); max(0.2*e_proj(r), 10^{-2.5}) (nonlinear example)
Tuned to the data/projection error; determines the least-squares fit of M in (22) and therefore the accuracy of the quadratic embedding.
DEIM tolerance ε_DEIM = 10^{-8}
Algorithmic threshold that selects the DEIM basis size d; affects the reduced Hamiltonian approximation but not the structure-preservation theorem.
reduced order r = 5-20 (linear), 6-20 (nonlinear)
Swept in experiments; larger r gives richer approximation manifold; not a physical constant.
POD bases V1/V2 and DEIM basis U = POD of snapshot matrices X−BB†X and q(x_i)
Data-driven bases from the FOM trajectory; the numerical accuracy claims depend directly on these fitted subspaces.

assumptions (6)

domain assumption J=-J^T, R=R^T⪰0, and det(J−R)≠0 for the pH system
Introduced in §2.1; the invertibility of J−R is needed to define G=(J−R)^{-1} and is not part of the standard pH definition, excluding pH systems with singular dissipation/DAEs.
ad hoc to paper The non-degeneracy condition Dˇxϕ(ˇx) ∈ S_{(J−R)^{−1}} holds for all ˇx
Assumed in Theorem 3.1 and used throughout; the abstract claims it is generically satisfied, but no proof appears in §3 or §4. For quadratic maps this must hold at every point of the manifold and is not guaranteed by construction.
standard math GMG reduction framework from [4] (point projection property and tangent reduction map)
Recalled in §2.2; the reduction map W is built on this framework and its properties (W^T V=I) are used in Theorem 3.1.
domain assumption Structure-preserving DEIM approximation from [6] yields a valid reduced Hamiltonian
Used in Algorithms 2-3 and in the nonlinear example; the ROM is pH with respect to the approximate Hamiltonian H_DEIM, not the original H.
domain assumption Gauss-Legendre time integration preserves pH structure
Invoked in §5.1 by reference [17]; needed for the numerical energy-balance comparisons to be meaningful.
domain assumption Snapshot matrix X is representative of the dynamics being reduced
The POD bases V1,V2 and DEIM basis are built from a single trajectory of the FOM; the numerical error claims assume this training trajectory adequately represents the ROM's operating regime.

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Cite this review

Pith. "Pith review of Structure-Preserving Generalized Manifold Galerkin Reduction for Port-Hamiltonian Systems." pith.science (2026). https://pith.science/paper/3MISP6SX

@misc{pith2026260308656,
  author       = {Pith},
  title        = {Pith review of: Structure-Preserving Generalized Manifold Galerkin Reduction for Port-Hamiltonian Systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3MISP6SX}},
  note         = {Machine review of arXiv:2603.08656}
}

read the original abstract

This paper considers structure-preserving model order reduction (MOR) techniques for port-Hamiltonian (pH) systems, which are typically derived from energy-based modeling. To keep favorable properties of \pH systems such as passivity in a reduced order model (ROM), we use structure-preserving methods in the reduction process. Although projection-based structure-preserving MOR methods for nonlinear pH systems based on nonlinear approximation ansatzes have recently been proposed, existing approaches typically rely on specific structures of the approximation map and the underlying pH system. To address this limitation, we propose a \MOR framework based on generalized manifold Galerkin (GMG) reduction. The resulting framework can employ general nonlinear approximation maps while preserving the pH structure. We establish sufficient conditions for structure preservation, show that the associated non-degeneracy conditions are generically satisfied. We further present linear and quadratic approximation maps within the proposed framework. Numerical examples for a linear and a nonlinear mass-spring-damper system show that the proposed \MOR methods have lower relative reduction error compared to existing methods.

Figures

Figures reproduced from arXiv: 2603.08656 by the authors.

**Figure 1.** Linear mass-spring-damper system A realization of this system for order n = 6 is B =        0 1 0 0 0 0        , J =   J2 0 0 0 J2 0 0 0 J2  , R = diag(0,c1,0,c2,0,c3), H: x 7→ 1 2 x ⊤Qx, with Q =          k1 0 −k1 0 0 0 0 1 m1 0 0 0 0 −k1 0 k1 +k2 0 −k2 0 0 0 0 1 m2 0 0 0 0 −k2 0 k2 +k3 0 0 0 0 0 0 1 m3          , where J2 = · 0 1 −1 0¸ is the two-dimensional canonical Poisson ma… view at source ↗

**Figure 2.** Reduction and output error obtained by SP1-POD-ROM, GMG-POD-ROM, and GMG-QM-ROM for the linear mass-spring-damper system with the inputs u(t) = 0.1 and u(t) = 0.1sin(t). Linear-app represents the projection error of corresponding linear embedding maps. The lower bound for the ROMs resulting from the quadratic embeddings maps is computed by (28). modelling of this system given by Σ(J,R,H,B,u,y,x0) where B =   0n 1 … view at source ↗

**Figure 3.** Reduction and output error obtained by SP2-POD-ROM, GMG-POD-ROM, and GMG-QM-ROM for the nonlinear mass-spring-damper system with the inputs u(t) = 0.1 and u(t) = 0.1sin(t). Linear-app represents the projection error of corresponding linear embedding maps. The lower bound for the ROMs resulting from the quadratic embeddings maps is computed by (28) [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: Error in the energy balance equation (30) of the FOM and ROMs with a reduced order of 16 obtained by the SP2-POD-ROM, GMG-POD-ROM, and GMG-QM-ROM for the nonlinear mass-spring-damper system with two different inputs. 6 Conclusion and outlook We propose a new structure-…

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Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

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Pith tools

Reviewed August 4, 2026 · model on record in the stance chip above.

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