{"id":"fbcc4423-c9c0-487c-a59f-653a17837195","arxiv_id":"2501.11955","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"Under restrictive admissibility conditions, a quadratic time-dependent mean field game is uniquely identifiable from full lateral boundary Cauchy data of its perturbed stationary states.","lead":"This mathematics paper proves a uniqueness result: for a quadratic time-dependent mean field game, the unknown stationary state, running cost, and Hamiltonian can all be recovered from full boundary Cauchy data. The result extends the authors' earlier theorem, but it rests on restrictive assumptions and the proof has gaps that would need to be closed.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 5.1 as stated asserts nonzero CGO solutions of a homogeneous parabolic IBVP with zero lateral and initial data, which contradicts parabolic uniqueness; the q1=q2 step therefore rests on an impossible premise.","rationale":"The paper aims to prove a strong uniqueness theorem for an inverse MFG problem by linearizing around an unknown stable stationary state and using CGO solutions plus unique continuation. The high-level strategy is coherent and extends the authors' earlier framework, and the admissibility restrictions in Definition 2.2, while restrictive, are stated as assumptions rather than being hidden. The reader's conditional verdict is justified, but I do not think the most load-bearing problem is the restrictive form of F; it is that Theorem 5.1, the main CGO tool, is internally inconsistent as displayed. A nonzero solution of a homogeneous linear parabolic equation with zero lateral boundary and zero initial data cannot exist on a bounded domain, so the asserted CGO solutions cannot be the ones used in the proof of q1=q2. This is stronger than a missing estimate: the proof as written uses an impossible object. The same boundary-condition confusion appears in Lemma 5.2, where the function class X has zero lateral boundary but the argument later treats the lateral trace as arbitrary. Both issues are likely repairable by importing the correct Runge-approximation and CGO statements from [55] and [43] and by choosing boundary data so that the first-order linearized solution is the required CGO solution, but that repair requires a nontrivial rewrite of Section 5.1. The rest of the proof, including the recovery of F from higher-order linearizations and Theorem 5.3, is sketched but plausible. For these reasons I keep the reader's CONDITIONAL verdict rather than moving to ACCEPT or REJECT: the central claim may be true, but the written proof has a load-bearing gap that must be fixed.","tokens_in":19202,"tokens_out":11333,"duration_ms":126566,"concrete_test":"Compare the preprint's Theorem 5.1 with the exact statement in the cited reference [55, Sahoo and Vashisth, Inverse Probl. Imaging 14 (2020)]. If the reference constructs CGO solutions with a nonzero boundary trace or with an inhomogeneous equation, then the preprint's zero-boundary version is a misstatement; recompute (5.16)-(5.18) using the correct boundary condition and verify that the boundary terms from the first-order solution with the measured Cauchy data vanish or are controlled at order o(rho) after division by rho. Independently, in the 1D domain Omega=(0,1) with constant q1 != q2, solve the homogeneous forward and backward parabolic IBVPs with zero lateral and initial/final data; any numerical solver will return the zero solution, directly contradicting the asserted nontrivial CGO form (5.2)/(5.5). This settles whether Theorem 5.1 can be used as stated.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central recovery step in Section 5.1 is the extraction of q1 = q2 from the CGO asymptotic (5.17)–(5.18). That step uses Theorem 5.1 to assert existence of a solution u2^(1) of the backward parabolic equation of the form (5.5), and a solution w of the adjoint forward equation of the form (5.2). But Theorem 5.1, exactly as displayed, requires w to satisfy the homogeneous forward parabolic equation with w(x,t)=0 on Sigma and w(x,0)=0 in Omega, and similarly v to satisfy the homogeneous backward equation with v(x,T)=0. For a bounded domain with smooth coefficients, the forward/backward parabolic initial-boundary value problem has at most one solution, and the zero data force w = 0 and v = 0 identically. The asserted exponential form e^{±psi}(...) cannot vanish on the lateral boundary for arbitrary chi in C_c^∞(0,T), so the theorem is internally inconsistent. Consequently, Section 5.1 cannot actually obtain a nontrivial CGO solution u2^(1) with the measured boundary data from Theorem 5.1 as stated. If the intended CGO solutions instead carry nonzero boundary trace, the proof must additionally show that the first-order solution corresponding to the measured boundary data can be chosen equal to such a CGO solution, and must compute the resulting boundary terms in (5.16); neither is done. Lemma 5.2 has a related flaw: X is defined as solutions W of (5.1) with W|Sigma = 0, but the Hahn-Banach argument later states W|Sigma can be arbitrary; under the actual definition the boundary integral vanishes identically and the conclusion partial_nu W~ = 0 does not follow. These are not merely missing estimates: as written, the key asymptotic argument for q1=q2 is unsupported, and the recovery of U, A and F depends on it.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies an inverse boundary value problem for a quadratic mean field game (MFG) system. It claims that from boundary Cauchy measurements of small perturbations around a stable stationary state, one can uniquely recover the stationary state U = (u0, m0), the running cost F, and the Hamiltonian metric A up to a conformal class. The proof strategy is high-order linearization around the unknown stationary solution, combined with complex geometric optics (CGO) solutions and unique continuation principles. The main result is Theorem 2.5, and the proof occupies Section 5.","tokens_in":19463,"tokens_out":3915,"duration_ms":42059,"significance":"If the main theorem were correctly established, the result would be a substantial advance for inverse problems of nonlinear PDEs and for mean field games in particular, going beyond the prior work [46] by recovering the unknown stationary state and Hamiltonian together with the running cost. The paper also contains useful structural observations, such as the staged recovery via the vector field q = 2A∇u0. However, the proof as written relies on inconsistent auxiliary statements and on admissibility assumptions that are tailored to the method, so the claimed uniqueness theorem is not currently supported.","major_comments":[{"comment":"Theorem 5.1 as stated is internally inconsistent. It asserts the existence of nontrivial solutions w of the homogeneous forward parabolic equation with w = 0 on Σ and w(x,0) = 0 in Ω, and v of the homogeneous backward equation with v(x,T) = 0, in the exponential form (5.2) and (5.5). For a bounded domain with smooth coefficients, such initial-boundary value problems have at most one solution, and the zero data force w ≡ 0 and v ≡ 0. The displayed exponential factors with χ ∈ C_c^∞(0,T) cannot vanish on the lateral boundary in the required way. Therefore the CGO solutions used in Section 5.1 do not exist under the stated hypotheses, and the subsequent extraction of (5.18) has no valid basis.","section":"Section 5, Theorem 5.1 (Eqs. (5.1), (5.4)–(5.6))"},{"comment":"The proof of Lemma 5.2 contains a false statement that voids the Hahn-Banach argument. The set X is defined as solutions W of (5.1), which includes the condition W|Σ = 0. Later the proof asserts “Since W|Σ can be arbitrary function, which is compactly supported on Σ.” This is false under the given definition of X, so the conclusion that ∂νW = 0 on Σ does not follow. Consequently, the Runge approximation property stated in Lemma 5.2 is not proved, and the paper cannot legitimately pass from H^1-regular CGO solutions to the C^{2+α,1+α/2} solutions needed in the inverse argument.","section":"Section 5, Lemma 5.2"},{"comment":"The step from boundary data equality to the estimate (5.16) is not demonstrated. The inequality (5.16) is attributed to “the argument of Section 5 of [55]”, but the manuscript does not show how the measured Cauchy data imply this estimate for solutions of the admissible class. Moreover, the solution u_2^(1) used in the product term is taken from (5.5), which imposes no lateral boundary condition, while the measurement map provides boundary data; the proof must show that the first-order solution corresponding to the measured data can be chosen equal to such a CGO solution, and it must compute the resulting boundary contributions in the integration by parts leading to (5.16). Neither is done, so the extraction of q1 = q2 from (5.17) to (5.18) is not justified.","section":"Section 5.1, derivation of (5.16)–(5.18)"},{"comment":"The admissibility conditions F(x, m0) = 0 and F^{(1)}(x) = 0 are load-bearing for the proof. They remove F from the stationary system (2.8) and remove the F^{(1)}m^{(1)} source term from the first-order linearized u-equation (4.8), so the staged recovery of q = 2A∇u0, then U, then A, and finally F can proceed. Since F and m0 are both unknown, this is a strong a priori restriction on the model class that is not verifiable from the boundary data. The paper does not quantify how restrictive this is or provide an example showing that the theorem covers a natural class of running costs beyond those constructed to satisfy (ii)–(iii). The central claim should be stated with this limitation made explicit, or the proof should be extended to treat F with a nonzero linear term.","section":"Section 2.1, Definition 2.2(ii)–(iii)"}],"minor_comments":[{"comment":"In the statement of Theorem 2.5, “A1(x) = A2(x) in Q” should refer to Ω rather than Q, since the metric depends only on x.","section":"Section 2.2, Theorem 2.5"},{"comment":"Definition 2.2 states Y : R^n × C → C, but F in the MFG system is real-valued; this domain/codomain discrepancy should be clarified.","section":"Section 2.1, Definition 2.2"},{"comment":"In the integration by parts displayed in the proof of Lemma 5.2, the boundary term is written as ∫Σ ∂νW W dS dt, but it should involve the adjoint solution and the function W or w in a way that is consistent with the vanishing of W|Σ; as written the notation is confusing and the subsequent claim about arbitrary W|Σ is not supported.","section":"Section 5, Lemma 5.2 proof"},{"comment":"The estimate (5.16) is written with a “≤” that has no preceding quantity on the left; the intended statement should be made precise.","section":"Section 5.1, Eq. (5.16)"},{"comment":"The paper cites [55] for the CGO construction and Lemma 5.2 follows [43], but the exact hypotheses under which the results of [55] apply to the coupled MFG linearization are not stated; this makes it difficult for the reader to locate the missing details.","section":"Section 1.2"}],"recommendation":"reject","confidential_remarks":"The manuscript is part of a series with heavy self-citation, and the proof reuses the framework of [46] and [55] with minimal adaptation. The central problem is that Theorem 5.1, as displayed, is not merely incomplete but internally inconsistent: homogeneous parabolic IBVPs with zero data have only the zero solution, so the CGO solutions cannot exist under the stated boundary conditions. Lemma 5.2 has a similar decisive flaw. These are not local typos; the main uniqueness theorem rests on them. In my assessment this is a load-bearing error that cannot be repaired by small edits within the manuscript's current scope, so I recommend rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThe paper claims a genuinely new result—simultaneous recovery of the stationary state U, the running cost F, and the Hamiltonian A for a quadratic time-dependent mean field game from boundary Cauchy data. That’s a meaningful extension of Liu–Lo–Zhang [46] and Liu–Zhang [51], and the overall strategy (high-order linearization around an unknown stationary state, then CGO probes) is a coherent research line. The admissibility conditions on F are stated clearly, and the forward well-posedness section is useful.\n\nBut the proof as written does not work. Theorem 5.1, quoted from Sahoo–Vashisth, claims nonzero CGO solutions of a homogeneous parabolic initial-boundary value problem with zero lateral data and zero initial/final data. On a bounded domain with smooth coefficients, that IBVP has only the zero solution. The displayed theorem is therefore false, and the main recovery step for q1=q2 rests on it. The same problem infects Lemma 5.2: the set X is defined via (5.1), which forces W|Σ=0, yet the Hahn-Banach argument says 'W|Σ can be arbitrary.' The boundary integral vanishes identically, so the conclusion ∂νW=0 doesn't follow. Those are load-bearing, not cosmetic. Additionally, the proof does not show that the measured boundary perturbations can be chosen so that the first-order solution u2^(1) equals a CGO of the form (5.5), and the estimates in (5.16)-(5.17) are summarized with 'by following the argument of Section 5 of [55]' rather than shown. The higher-order induction for F is also only sketched.\n\nI'm not saying the result is false. The strategy is plausible, and the flaws look like misstatements/gaps that could be repaired—if the CGO construction is corrected to allow nonzero boundary traces and the Runge approximation is properly stated. But as it stands, the central theorem is unproven, and a referee would need substantial revisions to take it further.\n\nThis paper is for specialists in inverse problems for MFGs. They might learn from the formulation and the high-order linearization setup, but they should not rely on the uniqueness theorem yet. I would send it to peer review because the question is important and the approach has merit, but I would flag the CGO step as the decisive issue. If the authors cannot fix it, the paper should not be published in its current form.\n\nRegards.","headline":"New inverse MFG result on recovering stationary state and parameters, but the proof's CGO lemma is misstated and the Runge argument is void—as written the main theorem doesn't follow.","tokens_in":20128,"tokens_out":4065,"would_cite":false,"duration_ms":40188,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35Q89","35R30","91A16","35R35"],"pacs":[],"model":"deepseek-v4-flash","headline":"For a quadratic mean field game, boundary Cauchy data of small perturbations around an unknown stable stationary state uniquely determine the stationary state, the running cost, and the Hamiltonian metric up to a fixed conformal class.","keywords":["mean field games","inverse problems","Cauchy dataset","unique identifiability","complex geometric optics","high-order linearization","stationary solutions","parabolic systems"],"falsifier":"Find two admissible triples $(A_1,F_1,U_1)\\ne(A_2,F_2,U_2)$ satisfying the hypotheses of Theorem 2.5 whose boundary Cauchy maps coincide; either a numerical search over quadratic MFG stationary states or an explicit construction would refute the claimed injectivity. A more targeted check is to retain the term $F^{(1)}(x)m^{(1)}(x,t)$ in the first-order linearized $u$-equation and see whether the complex-geometric-optics step can still isolate $q=2A\\nabla u_0$; if not, the admissibility condition is doing essential work.","tokens_in":18874,"feed_emoji":"🎯","tokens_out":6688,"duration_ms":65771,"temperature":0.7,"pith_summary":"This paper proves a unique-identifiability theorem for quadratic mean field games: if two configurations produce the same boundary Cauchy data from small perturbations around an unknown stable stationary state, then the stationary states and running costs coincide and the kinetic Hamiltonians agree within the fixed conformal class $C_g$ (each $A_i=\\kappa_i g$ for a known metric $g$). The result matters because an observer with access only to the boundary of the state domain can in principle decode the interior equilibrium population density and value function together with the interaction cost, without knowing either in advance. The proof combines high-order linearization around the unknown stationary solution with complex geometric optics solutions and unique continuation principles. This settles the open problem left by an earlier study that recovered only the running cost while assuming the stationary state was fixed.","feed_headline":"Boundary data pin down hidden mean-field game states","feed_subtitle":"The stationary state, running cost and Hamiltonian are all uniquely recoverable from boundary data.","key_machinery":"The key object is the measurement map $M_{F,U,A}$, which records the boundary traces of $u$, $m$ and their gradients for small perturbations around the stationary state $U=(u_0,m_0)$. The argument is carried by successive high-order linearization around this unknown stationary solution: the first-order system exposes the recoverable drift $q=2A\\nabla u_0$; elliptic unique continuation then fixes $u_0$ and $m_0$; and the second- and higher-order linearized systems isolate the Taylor coefficients $F^{(2)},F^{(3)},\\dots$ one by one, with the parabolic unique continuation principle forcing each difference to vanish.","core_discovery":"The central discovery is Theorem 2.5: for two admissible configurations $(A_i,F_i,U_i)$ of the quadratic mean field game system with stable stationary solutions, equality of the boundary Cauchy maps $M_{F_1,A_1,U_1}=M_{F_2,A_2,U_2}$ forces $F_1=F_2$, $U_1=U_2$, and $A_1=A_2$ within the given conformal class $C_g$. The proof first recovers the drift vector $q=2A\\nabla u_0$ from the first-order linearized system using complex geometric optics solutions, then uses elliptic unique continuation to recover $u_0$ and $m_0$ separately, and then identifies the Hamiltonian. Once $U$ and $A$ are known, the higher-order linearized systems become forced equations whose discrepancies are controlled by a parabolic unique continuation principle, yielding $F^{(k)}$ for every $k$ and hence the full analytic running cost $F$.","pith_inferences":["Editorial inference: because the proof only uses the affine-in-$p$ structure of the Hamiltonian through the drift $q=2A\\nabla u_0$, a similar uniqueness statement may hold for more general Hamiltonians whose momentum derivative is an unknown coefficient, provided the same admissibility conditions hold.","Editorial inference: the assumptions $F(x,m_0)=0$ and $F^{(1)}(x)=0$ are tailored to decouple the first-order linearized equation; a natural next problem is whether the uniqueness persists when $F$ has a nonzero linear density dependence, where the coupling would enter at first order.","Editorial inference: since the theorem is about injectivity, a practical follow-up is to ask whether the map is stably invertible and how much boundary data is genuinely needed; numerical experiments on random quadratic MFG instances could test the observable stability of the inversion."],"forward_implications":["If the theorem is correct, the measurement map is injective on the admissible class: no two distinct unknown states, costs, and Hamiltonians can produce the same boundary Cauchy data.","An agent who can perturb the system near its stationary state and measure boundary traces can decode the interior equilibrium density, and therefore anticipate the population's aggregate behavior, without interior access.","The identifiability holds for the fully time-dependent quadratic MFG system, not just for stationary problems, and no probability-density normalization or Neumann boundary condition is imposed.","All Taylor coefficients of the analytic running cost are recoverable, so the whole interaction cost function, not just a finite-dimensional projection, is determined by the data."],"supporting_citations":[{"why":"Supplies the predecessor result recovering F with a fixed unknown stationary state and the parabolic unique continuation principle used in the higher-order step.","marker":"[46]"},{"why":"Provides the complex geometric optics solutions for parabolic convection-diffusion equations whose asymptotic limit is used to recover the drift q.","marker":"[55]"},{"why":"Supplies the elliptic unique continuation principle used to recover u0 and m0 from boundary equality.","marker":"[36]"},{"why":"Provides the Runge approximation and denseness lemma that converts weak CGO solutions into the Hölder-regular solutions needed for the inverse argument.","marker":"[43]"}],"fun_headline_variants":["MFG inverse problem: boundary data fix state, cost, Hamiltonian","Simultaneous MFG recovery: state and parameters from boundary info","Boundary Cauchy maps decode mean field games fully","From boundary data alone: MFG state, cost, and Hamiltonian"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the running cost vanishes at the unknown equilibrium density and has zero first derivative there, so the first-order linearized equation for the value perturbation does not contain a term coupling F to the density perturbation; if a real model has linear density dependence at equilibrium, this staged recovery no longer goes through.","fun_headline_variants_meta":{"raw":{"variants":["MFG inverse problem: boundary data fix state, cost, Hamiltonian","Simultaneous MFG recovery: state and parameters from boundary info","Boundary Cauchy maps decode mean field games fully","From boundary data alone: MFG state, cost, and Hamiltonian"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000738,"raw_usage":{"total_tokens":3260,"prompt_tokens":869,"completion_tokens":2391,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":485,"completion_tokens_details":{"reasoning_tokens":2320}},"tokens_in":485,"tokens_out":2391,"duration_ms":18206,"temperature":1.0,"reasoning_tokens":2320,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T17:41:24.015727+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Find two admissible triples $(A_1,F_1,U_1)\\ne(A_2,F_2,U_2)$ satisfying the hypotheses of Theorem 2.5 whose boundary Cauchy maps coincide; either a numerical search over quadratic MFG stationary states or an explicit construction would refute the claimed injectivity. A more targeted check is to retain the term $F^{(1)}(x)m^{(1)}(x,t)$ in the first-order linearized $u$-equation and see whether the complex-geometric-optics step can still isolate $q=2A\\nabla u_0$; if not, the admissibility condition is doing essential work.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the predecessor result recovering F with a fixed unknown stationary state and the parabolic unique continuation principle used in the higher-order step."},{"cited_title":"A partial data inverse problem for the convection-diffusion equation","cited_arxiv_id":null,"evidence_quote":"Provides the complex geometric optics solutions for parabolic convection-diffusion equations whose asymptotic limit is used to recover the drift q."},{"cited_title":"Carleman estimates and unique continuation for second-order elliptic equations with nonsmooth coefficients","cited_arxiv_id":null,"evidence_quote":"Supplies the elliptic unique continuation principle used to recover u0 and m0 from boundary equality."},{"cited_title":"Simultaneous recoveries for semilinear parabolic systems","cited_arxiv_id":null,"evidence_quote":"Provides the Runge approximation and denseness lemma that converts weak CGO solutions into the Hölder-regular solutions needed for the inverse argument."}],"review_version":1}