{"id":"825e4fe0-7b5c-496f-aee1-8c13b6198f8b","arxiv_id":"2506.06471","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Energy-stable port-Hamiltonian systems merge two energy-based formalisms and keep their structure under Galerkin projection and model reduction.","lead":"Mathematicians combine two existing ways to write energy-based physical models, energy-stable systems and port-Hamiltonian systems, into one form called energy-stable port-Hamiltonian (es-pH) systems. The new form is designed so that standard numerical reduction methods automatically preserve the system's energy structure, which could simplify simulations and control of complex physical systems.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The temporal discretization claim in §2.5 is not proven: the discrete EBE requires testing with the input u_N, but only y_N∈W_IO,N is assumed; the missing inclusion u_N∈W_IO,N invalidates the proof as stated.","rationale":"The reader's weakest_assumption bundled two concerns: representability of arbitrary iso-pH systems as es-pH, and the temporal subspace condition. I agree with the temporal concern and sharpen it: the missing admissibility is on the input u_N∈W_IO,N, not merely on y_N∈W_IO,N. This is the most load-bearing issue because it bears directly on the paper's advertised structure-preserving time discretization, even for systems already in the es-pH class. The representability issue is real but less decisive: the paper introduces es-pH as a class of energy-stable systems with ports and only claims that es-pH systems are pH systems, not that every pH system admits an es-pH form; the capacitor example shows the class is not universal, but the mathematical closure arguments for the class itself are sound. The paper honestly states that initial/boundary conditions are not specified and that a numerical study is future work, which makes the missing verification of §2.5 more salient. If the proposed scalar test shows a nonzero residual, the temporal claim should be revised to include u_N∈W_IO,N or an explicit projection-error bound; if it shows zero residual, the condition may be unnecessary. In either case, the conditional verdict remains appropriate, so I do not change the reader's verdict.","tokens_in":10279,"tokens_out":14090,"duration_ms":129002,"concrete_test":"Set H(x)=x^2/2, ω=0, ρ=1, γ=1, π=μ=σ=0 so that (1.2) becomes ẋ=-x+u, y=ẋ. Discretize (2.8) on [0,T] with V_N=continuous piecewise linears, W_N=W_IO,N=piecewise constants, V_IO,N=continuous piecewise linears, V_y,N=piecewise constants, and input u_N the piecewise-linear interpolant of u(t)=t. This satisfies every subspace condition stated in §2.5 except u_N∈W_IO,N, which is not stated. Compute the discrete EBE residual R := H(x_N(T))-H(x_N(0)) + ∫_0^T ⟨φ[ẋ_N,u_N],[ẋ_N,u_N]⟩ dt - ∫_0^T y_N u_N dt. If R is nonzero on uniform refinements, the claimed discrete EBE requires the missing test-space inclusion; if R vanishes, the stated condition can be relaxed or replaced by a projection argument.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section 2.5 asserts that the Petrov–Galerkin time discretization (2.8) inherits an EBE/DI provided d_t x_N∈W_N and y∈W_IO,N. To reproduce the proof of Theorem 2.1, one must insert the test pair (v,v_IO)=(d_t x_N,u_N) into (2.8). The first inclusion is stated, but the second test function u_N is only assumed to lie in V_IO,N; no assumption puts u_N into W_IO,N. The stated condition y∈W_IO,N is neither necessary for that test nor sufficient to make u_N admissible. As written, the discrete EBE therefore does not follow. In particular, the natural choice V_IO,N=continuous piecewise linears and W_IO,N=piecewise constants violates the required inclusion, and projection error in the output equation breaks the discrete energy balance. The paper gives no example of spaces that satisfy the hypotheses, so the headline advantage for time discretization is conditional on an unverified and apparently misstated admissibility condition.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript introduces energy-stable port-Hamiltonian (es-pH) systems as an extension of energy-stable systems by an input–output port. It gives three formulations: a finite-dimensional matrix form (1.2), a Banach-space variational form (2.2)–(2.5), and a finite-dimensional Dirac-structure form (3.6). Theorem 2.1 establishes a power balance equation and a dissipation inequality for classical solutions. The paper then claims that Galerkin projection in space, model-order reduction, and a Petrov–Galerkin time discretization all preserve the es-pH structure, and that the construction is a genuine pH system via a Dirac structure.","tokens_in":10466,"tokens_out":12124,"duration_ms":104101,"significance":"If the main structural claims hold, the es-pH formulation is attractive because structure preservation would follow from a plain (Petrov-)Galerkin projection, without additional correction terms or fitting. The proof of Theorem 2.1 is concise and correct, and the Dirac-structure embedding in Section 3.3 is a faithful mapping rather than a circular restatement of the desired result. The paper contains no fitted parameters and the central algebraic conditions are stated in a checkable way. The main reservation is that the temporal-discretization claim advertised in the abstract and discussed in Section 2.5 is not established as written; until that is repaired, the claimed advantage over existing pH discretization techniques is fully supported only for the spatial-discretization and MOR settings.","major_comments":[{"comment":"The discrete energy balance does not follow from the stated hypotheses. Reproducing the proof of Theorem 2.1 requires inserting the test pair (v,v_IO) = (d_t x_N, u_N) into (2.8). The paper assumes d_t x_N ∈ W_N, but u_N is only assumed to belong to V_IO,N; the stated condition y_N ∈ W_IO,N does not make u_N an admissible test function in the output test space. Consequently the displayed EBE and integrated DI after (2.8) are unproven as stated. The admissibility condition should be corrected (for example, by requiring u_N ∈ W_IO_N, or by proving an appropriate compatibility condition), and a concrete example of temporal spaces satisfying it should be supplied; none of the methods cited from [3] is shown to satisfy the condition in the port setting.","section":"§2.5 (Eq. (2.8))"},{"comment":"The abstract promises structure preservation 'in space and time' and in model reduction. The spatial and MOR parts are rigorously supported by Theorem 2.1 and the Galerkin argument in §2.4. The temporal part, however, depends entirely on the unverified admissibility condition discussed in the previous comment. Because no concrete temporal spaces are constructed or verified, the paper does not currently deliver on the time-discretization portion of the headline claim. The authors should either supply the missing construction or explicitly restrict the claim to spatial discretization and model reduction.","section":"Abstract and §2.5"}],"minor_comments":[{"comment":"The output y is consistently an element of the dual space V'_IO, see Eq. (2.3); the statement after (2.5) that the Galerkin output satisfies ~y ∈ C^1(It, ~V_IO) should instead read ~y ∈ C^1(It, ~V'_IO), with the usual identification in the finite-dimensional setting.","section":"§2.4"},{"comment":"In the discrete energy balance displayed after (2.8), the integrand uses x(t), u(t), and φ|_{x(t)} rather than the discrete curves x_N(t), u_N(t); this should be corrected so that the discrete and continuous arguments are not confused.","section":"§2.5 (displayed EBE)"},{"comment":"It should be stated explicitly that L|x is skew-symmetric with respect to the duality pairing (3.2), which is the property that makes the graph D_es a Dirac structure; the current presentation leaves this essential verification implicit.","section":"§3.3 (Eq. (3.6))"},{"comment":"The port-variable convention is reversed between the iso-pH and es-pH Dirac formulations: in §3.2 one sets f_FIO = y and e_FIO = u, while in §3.3 one sets f_FIO = u and e_FIO = y. The authors should warn the reader explicitly, since this switch affects the signs of the feedthrough terms and the interpretation of passivity.","section":"§3.2 and §3.3"},{"comment":"The paper states that a numerical study is future work. For a methods paper this is acceptable, but a small illustrative example verifying the temporal admissibility condition would substantially increase confidence in the claimed structure preservation in time.","section":"Conclusion"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is within scope for math.NA, and the core structural results (Theorem 2.1, the spatial Galerkin preservation, and the Dirac-structure embedding) are sound. The main risk is that the abstract's claim of structure preservation in time is currently unsupported because of the §2.5 admissibility gap. This is repairable within the manuscript's scope, so I recommend major revision rather than rejection. The related-work coverage is adequate, but the authors should carefully distinguish the finite-dimensional Dirac-structure result from the infinite-dimensional Banach-space formulation."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nThe short version: this paper defines energy-stable port-Hamiltonian (es-pH) systems by attaching an input-output port to the energy-stable framework [3], proves that every solution satisfies a power balance and dissipation inequality, and shows that Galerkin projection preserves the structure. The core calculation is correct. But the paper's headline advantage for temporal discretization is not established, and the new class is smaller than the pH systems it claims to unify.\n\nWhat is actually new is the es-pH definition itself: the output is defined through the state velocity and the input, rather than through the Hamiltonian derivative. That choice makes the PBE/DI proof (Theorem 2.1) immediate: evaluate the variational form at (dx/dt, u), and skew-symmetry plus positive semidefiniteness does the rest. It also means any Galerkin projection of an es-pH system is again es-pH, which is a genuinely structure-preserving spatial discretization and MOR property. The Dirac-structure section is a faithful embedding of es-pH systems into the standard pH framework. The writing is clear and the authors do not oversell (they note that existence and uniqueness are not addressed, and that numerics are future work).\n\nThe first soft spot is in Section 2.5. The claim is that if dx_N/dt ∈ W_N and y ∈ W_IO,N, then the Petrov–Galerkin time discretization inherits an EBE/DI. To reproduce the proof of Theorem 2.1, you need to insert the test pair (dx_N/dt, u_N) into (2.8). The first inclusion is assumed, but u_N is only in V_IO,N, not necessarily in W_IO,N; the stated condition on y does not make u_N admissible. No example of spaces satisfying these hypotheses is given. As written, the discrete EBE does not follow. This is not a wild accusation; it is just that the test function for the input is missing from the assumptions.\n\nThe second soft spot is coverage. Section 3.3 shows es-pH implies pH. The converse is never shown, and for good reason: most standard iso-pH systems have an output that depends on ∂H/∂x and u, and cannot be rewritten in the es-pH form, where the output depends on dx/dt and u. So the framework does not unify pH systems; it describes a subclass. That is fine if presented as such, but the intro's phrasing suggests a general combination.\n\nThere is also no numerical evidence for the claimed practical advantage. The authors say it is future work, so this is a missing support rather than a contradiction.\n\nOverall, the machinery is sound and the paper is worth a serious referee, but the time-discretization part needs repair and the scope needs to be stated more honestly. I would send it to review with a request for major revision.\n\nBest.","headline":"A clean but narrow extension of energy-stable systems to port-Hamiltonian ones; the main theorem is correct, but the time-discretization proof has a genuine gap and the class does not cover most standard pH systems.","tokens_in":11001,"tokens_out":2605,"would_cite":true,"duration_ms":23425,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["37M15","65P10"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper introduces energy-stable port-Hamiltonian (es-pH) systems, a reformulation of port-Hamiltonian systems in which Galerkin projection and Petrov-Galerkin time discretization automatically preserve the system's power balance and…","keywords":["energy-stable systems","port-Hamiltonian systems","structure-preserving discretization","model reduction","Dirac structure","dissipation inequality","power balance","Galerkin projection"],"falsifier":"Take a standard input-state-output port-Hamiltonian system whose output depends on $\\partial H/\\partial x$ and $u$, for example $y = G^\\top\\partial H/\\partial x + S u$, and check whether there exist coefficient blocks $\\omega,\\rho,\\gamma,\\pi,\\mu,\\sigma$ that cast it into the es-pH form (1.2); a concrete system for which this is provably impossible would refute the claim that the es-pH formulation offers natural structure-preserving discretization for the pH systems one wants to simulate.","tokens_in":10026,"feed_emoji":"⚡","tokens_out":17026,"duration_ms":132171,"temperature":0.7,"pith_summary":"The paper tries to establish a new formalism, energy-stable port-Hamiltonian (es-pH) systems, that merges the energy-stable framework with port-Hamiltonian (pH) systems by adding an input-output port. The payoff is structural: a Galerkin projection in space or a Petrov-Galerkin projection in time of an es-pH system is again an es-pH system, and the power balance inequality $\\frac{d}{dt}H(x)\\le\\langle y,u\\rangle$ is inherited automatically. The paper proves this inequality for the infinite-dimensional formulation and shows that every es-pH system is a port-Hamiltonian system via a Dirac structure. If the framework covers the pH systems one wants to simulate, it would make structure-preserving discretization and model reduction essentially free, since no special integrator or reduction algorithm is needed beyond projection.","feed_headline":"Projection preserves energy guarantees in new port-Hamiltonian form","feed_subtitle":"Reduced models inherit the original system's power balance and dissipation inequality with no extra steps.","key_machinery":"The central object is the es-pH system — a port-Hamiltonian system whose structure and dissipation matrices act on $(\\dot{x},u)$ rather than on $(\\partial H/\\partial x,u)$ — in its variational form (2.5): for every test pair $(v,v_{IO})$, $\\langle [-\\partial H/\\partial x; y] + (\\lambda-\\phi)[\\dot{x};u], [v;v_{IO}]\\rangle = 0$. The mechanism is to test this equation with $v=\\dot{x}$ and $v_{IO}=u$; the skew-symmetric $\\lambda$ term then vanishes by symmetry, the positive-semidefinite $\\phi$ term contributes a nonpositive dissipation, and the chain rule turns the left side into $\\frac{d}{dt}H(x)$, giving the power balance and dissipation inequality. Since the entire structure is captured by this bilinear form, restricting to subspaces or temporal test spaces preserves the form, so any Galerkin or compatible Petrov-Galerkin projection is automatically structure-preserving.","core_discovery":"The central discovery is a new class of open, dissipative systems, the energy-stable port-Hamiltonian (es-pH) systems, defined by pairing the dynamics $(-\\omega(x)+\\rho(x))\\dot{x} = -\\partial H/\\partial x + (\\gamma(x)-\\pi(x))u$ with the output equation $y = (\\gamma(x)^\\top+\\pi(x)^\\top)\\dot{x} + (-\\mu(x)+\\sigma(x))u$, where the combined block matrices $\\lambda$ and $\\phi$ are pointwise skew-symmetric and symmetric positive semidefinite, respectively. Theorem 2.1 shows that every classical solution obeys the power balance identity $\\frac{d}{dt}H(x) = -\\langle \\phi[\\dot{x};u],[\\dot{x};u]\\rangle + \\langle y,u\\rangle$ and hence the dissipation inequality $\\frac{d}{dt}H(x) \\le \\langle y,u\\rangle$. Because the system is encoded as a single variational equation in $(\\dot{x},u)$, a Galerkin projection onto closed subspaces of the state and input-output spaces produces another es-pH system, and the same structure is shown to survive a Petrov-Galerkin time discretization when the temporal subspaces satisfy a compatibility condition. Section 3.3 reformulates the es-pH system on a Dirac structure, proving it is a port-Hamiltonian system in the established sense.","pith_inferences":["The paper proves only that every es-pH system is a pH system; whether every standard pH system can be rewritten in es-pH form is left open, so the practical reach of the framework depends on the size of this subclass.","The time-discretization result relies on a compatibility condition between temporal subspaces that the paper assumes rather than verifies; a concrete map from standard time-stepping methods (discrete gradients, average vector field, collocation methods) to subspaces satisfying it would turn the framework into an algorithm.","Because the output in es-pH form is driven by $(\\dot{x},u)$ rather than by $(\\partial H/\\partial x,u)$, applying the framework to a mechanical or electrical system may require reinterpreting what the output port measures; this modeling step is not addressed in the paper.","A natural next step is to test the framework numerically on one of the listed applications, such as shallow water or magneto-quasistatics, and compare the discrete energy balance of the projected es-pH scheme against a standard pH discretization."],"forward_implications":["Spatial discretization of an es-pH system by a Galerkin projection onto closed subspaces yields a reduced model that is again es-pH, so the discrete model inherits the power balance and dissipation inequality without adding stabilization or correction terms.","Temporal discretization by a Petrov-Galerkin method, under the subspace condition $\\frac{dx_N}{dt}\\in W_N$ and $y_N\\in W_{IO,N}$, preserves a discrete energy balance, extending discrete-gradient and average-vector-field ideas to systems with an input-output port.","Because the es-pH class is a subclass of port-Hamiltonian systems via the Dirac-structure construction, all existing pH analysis applies while the new form adds projection-based structure preservation.","The Banach-space formulation covers PDEs, ODEs, and DAEs, so the same structure-preserving projection framework applies to wave and elasticity equations, shallow water equations, and magneto-quasistatics.","Model reduction of parametric es-pH systems is obtained by the same Galerkin projection, as noted in the paper, although the parametric details are left for future work."],"supporting_citations":[{"why":"supplies the energy-stable systems formalism and the projection-based structure-preserving discretization idea that es-pH systems extend","marker":"[3]"},{"why":"defines port-Hamiltonian systems and Dirac structures, the framework the paper connects to","marker":"[13]"},{"why":"provides the input-state-output port-Hamiltonian form (1.1) and the control-theoretic context for pH systems","marker":"[9]"},{"why":"gives energy-consistent Petrov-Galerkin time discretizations for port-Hamiltonian systems, which the temporal discretization extends","marker":"[4]"},{"why":"exemplifies structure-preserving numerical methods for Hamiltonian PDEs, an application area the es-pH framework targets","marker":"[1]"}],"fun_headline_variants":["Structure-preserving merger of energy stability and port-Hamiltonian","New es-pH systems keep energy bounds under reduction","Port-Hamiltonian form gains energy stability, preserves structure","Energy-stable ports: new systems that survive discretization","Reduced models inherit power balance via es-pH formulation"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the port-Hamiltonian systems one actually wants to simulate can be written in the es-pH form; the paper shows every es-pH system is a port-Hamiltonian system, but it does not show that every port-Hamiltonian system admits an es-pH representation.","fun_headline_variants_meta":{"raw":{"variants":["Structure-preserving merger of energy stability and port-Hamiltonian","New es-pH systems keep energy bounds under reduction","Port-Hamiltonian form gains energy stability, preserves structure","Energy-stable ports: new systems that survive discretization","Reduced models inherit power balance via es-pH formulation"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000184,"raw_usage":{"total_tokens":1284,"prompt_tokens":879,"completion_tokens":405,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":495,"completion_tokens_details":{"reasoning_tokens":324}},"tokens_in":495,"tokens_out":405,"duration_ms":4105,"temperature":1.0,"reasoning_tokens":324,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T05:56:53.258356+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a standard input-state-output port-Hamiltonian system whose output depends on $\\partial H/\\partial x$ and $u$, for example $y = G^\\top\\partial H/\\partial x + S u$, and check whether there exist coefficient blocks $\\omega,\\rho,\\gamma,\\pi,\\mu,\\sigma$ that cast it into the es-pH form (1.2); a concrete system for which this is provably impossible would refute the claim that the es-pH formulation offers natural structure-preserving discretization for the pH systems one wants to simulate.","supporting_citations":[{"cited_title":"Egger, O","cited_arxiv_id":null,"evidence_quote":"supplies the energy-stable systems formalism and the projection-based structure-preserving discretization idea that es-pH systems extend"},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"defines port-Hamiltonian systems and Dirac structures, the framework the paper connects to"},{"cited_title":"Mehrmann and B","cited_arxiv_id":null,"evidence_quote":"provides the input-state-output port-Hamiltonian form (1.1) and the control-theoretic context for pH systems"},{"cited_title":"Giesselmann, A","cited_arxiv_id":null,"evidence_quote":"gives energy-consistent Petrov-Galerkin time discretizations for port-Hamiltonian systems, which the temporal discretization extends"},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"exemplifies structure-preserving numerical methods for Hamiltonian PDEs, an application area the es-pH framework targets"}],"review_version":1}