{"id":"3ace6910-e169-4dfd-b6b9-dea8cd8d0092","arxiv_id":"2411.17553","paper_version":2,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Structural identifiability of linear-in-parameter parabolic PDEs is characterized by the kernel of an auxiliary elliptic operator, with non-identifiability occurring exactly for initial conditions in that kernel.","lead":"This paper gives a new way to decide whether the parameters in a class of reaction-diffusion models can be recovered from data, by turning the question into a spectral problem for an auxiliary elliptic equation. The approach pinpoints which initial conditions cause parameters to become indistinguishable, and shows these failures have practical consequences for data fitting.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Example 4.1's logistic global identifiability claim is not established: the proof assumes uniqueness or discreteness of all nontrivial solutions of the auxiliary elliptic equation, but the b=0 case is an untreated continuum and the cited uniqueness is for positive solutions, not sign-changing ones.","rationale":"The reader's weakest assumption identifies the same load-bearing point: Theorem 4.1's logistic conclusion depends on the auxiliary elliptic equation having a unique or discrete solution set, and the b=0 case is an untreated continuum. I agree with that diagnosis. My reading adds that even the b≠0 uniqueness assertion is overbroad, because sign-changing solutions are not covered by the cited positive-solution uniqueness theorems, so the proof of ANI=∅ is unsupported in two places. Neither issue appears to invalidate the paper's main framework: Theorem 3.7 and the linear spectral reduction are internally coherent, the Fredholm-alternative argument is standard, and the numerical demonstration of practical identifiability is independent evidence for the linear analysis. The nonlinear example is the paper's headline claim of global unconditional identifiability, so the gap matters for the strength of the conclusions, but it is localized to Example 4.1 and likely repairable. A conditional acceptance with a request for a corrected proof of Example 4.1 is therefore appropriate, and this does not change the reader's verdict.","tokens_in":22726,"tokens_out":14022,"duration_ms":134150,"concrete_test":"Fix the logistic model with b1=b2>0, d1>d2, and a1-a2=(d1-d2)λ_n for a Dirichlet eigenvalue λ_n. The auxiliary equation then forces any common solution u(·,t) to lie in the one-dimensional eigenspace, so write u(x,t)=γ(t)ψ_n(x). Substitute this form into either original PDE; the term b1γ(t)²ψ_n(x)² cannot be balanced by the remaining terms unless γ≡0, so no nonzero time-dependent common solution exists. This check settles whether the b=0 continuum actually produces non-identifiability: if γ≡0 is forced, Example 4.1 can be repaired by a separate short argument for b=0, while any nonzero γ would refute the global identifiability claim.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 4.1 only proves identifiability when every parameter difference (d,B) lies in R or AI, i.e. when the auxiliary elliptic equation has a unique or discrete solution set. Example 4.1 attempts to show ANI is empty by asserting that for a,b with b>0 the equation -dΔψ=aψ-bψ² has a unique nontrivial solution, and by symmetry for b<0. This is not established. The classical uniqueness results cited (Pao, Zhao) are for positive solutions; sign-changing solutions exist, for example in the one-dimensional Dirichlet problem -ψ''=aψ-ψ² there are branches bifurcating from each eigenvalue λ_n, so for a>4π² there is a positive solution and a sign-changing solution. More decisively, the case b=0 is never treated. Then -dΔψ=aψ is linear, and if a/d is an eigenvalue its solution set is a vector space, a continuum, which falls squarely in the paper's own ANI defined in (4.3). This case corresponds to b1=b2, which is permitted even when both quadratic coefficients are positive. Consequently, the claimed global unconditional identifiability of the logistic model does not follow from Theorem 4.1 as written. The conclusion may still be true — the nonlinear term may prevent the linear continuum from generating indistinguishable solutions — but the proof has a genuine, localized gap in the central nonlinear example.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops a structural identifiability framework for fully observed parabolic PDEs that are linear in their parameters. The central idea is to reduce indistinguishability of two parameter points to the existence of nontrivial solutions of an auxiliary elliptic equation obtained by subtracting the two PDEs. For linear homogeneous equations, the kernel of the operator L[A1-A2] characterizes potentially indistinguishable initial conditions, and Fredholm theory is used to show that this kernel is finite-dimensional for nondegenerate elliptic operators. Explicit non-identifiable separable solutions are constructed for Dirichlet, Neumann, Robin, and periodic boundary conditions, and the authors show that initial conditions close to a dominant eigenfunction lead to practical non-identifiability in a profile-likelihood study. For nonlinear reaction terms that are linear in parameters, Theorem 4.1 claims unconditional identifiability when the auxiliary elliptic equation has a unique or discrete solution set, and Example 4.1 applies this to logistic growth to conclude global unconditional identifiability. Section 5 presents numerical experiments with reproducible code illustrating how structural non-identifiability affects parameter inference.","tokens_in":54,"tokens_out":10849,"duration_ms":161549,"significance":"The linear theory in Sections 2 and 3 is a genuine and useful contribution: it connects PDE identifiability to spectral properties of an auxiliary elliptic operator, yields explicit non-identifiable solution constructions, and gives a clean sufficient condition for identifiability from a given initial condition. The numerical demonstration in Section 5 is also valuable and is supported by publicly available code. However, the flagship nonlinear result--global unconditional identifiability of the logistic model--is not established by the arguments as written. The gap is localized and likely fixable, but it currently affects the paper's main advertised nonlinear claim. If repaired, the paper would be a strong contribution to the emerging theory of PDE structural identifiability.","major_comments":[{"comment":"The claim that for a in R and b > 0 'any nontrivial solution is unique' is not supported by the cited references, which concern positive solutions. For the one-dimensional Dirichlet problem -psi'' = a psi - psi^2, sign-changing solutions bifurcate from each eigenvalue of -d^2/dx^2; for instance, for a > 4 pi^2 on (0, ell) there exist both a positive solution and sign-changing solutions. Thus the set of all nontrivial solutions is neither a singleton nor discrete, and the conclusion ANI = empty is not established. The symmetry argument for b < 0 merely maps sign-changing solutions to sign-changing solutions and does not remove them. This is load-bearing because Theorem 4.1's 'global unconditional identifiability' for the logistic model depends on ANI being empty.","section":"Section 4.1, Example 4.1"},{"comment":"The case b = 0 is never treated. When b_1 = b_2 (so b = 0), the auxiliary equation is the linear equation -d Delta psi = a psi; if a/d is an eigenvalue of the Laplacian with the chosen boundary conditions, the solution set is a vector space, i.e., a continuum. This falls squarely within the paper's own definition of ANI in (4.3). This parameter difference is admissible even when both quadratic coefficients are positive. The authors must either prove that solutions from this continuum cannot satisfy the original parabolic problem (e.g., by showing that the quadratic term prevents the time-dependent separation-of-variables ansatz when b_1 = b_2 > 0) or restrict the statement of Theorem 4.1 and Example 4.1 accordingly.","section":"Example 4.1 and Eq. (4.3)"},{"comment":"Theorem 4.1 assumes u_t not identically zero but concludes 'unconditionally identifiable,' which is defined in Definition 3.5 as a property for all nontrivial initial conditions. Initial conditions that are steady states give u_t identically zero and are not covered by the theorem's hypothesis; moreover, such initial conditions can genuinely be indistinguishable. For example, with Neumann boundary conditions and logistic nonlinearity, every constant K > 0 is a solution for any parameter triple (d, a, b) with a/b = K, so the model is not unconditionally identifiable in the sense of Definition 3.5. The theorem and the concluding sentence of Example 4.1 should be restated as identifiability from initial conditions whose solution has nonconstant time dependence, or a separate definition of identifiability within the class of nonstationary solutions should be introduced.","section":"Theorem 4.1 and Definition 3.5"}],"minor_comments":[{"comment":"Because f(x, 0; B) = 0, the zero function is always a solution of the auxiliary elliptic equation; the definition of R as 'has a unique solution' should specify whether the trivial solution is included, since otherwise R excludes every case with a nonzero solution and the intended meaning of 'unique nontrivial solution' is lost.","section":"Definitions (4.2)-(4.3)"},{"comment":"The inference from 'the solution set of the auxiliary elliptic equation is discrete' to 'u(x, t) = psi(x) for some fixed psi for all t' should be justified explicitly: it uses continuity of t maps to u(., t), the connectedness of (0, T), and the fact that a connected subset of a discrete set is a singleton.","section":"Proof of Theorem 4.1"},{"comment":"Example 4.1 refers to 'model (3.7)' where model (4.1) is meant, and its concluding sentence attributes 'unconditionally identifiable' to Definition 3.4, although Definition 3.4 is identifiability from a fixed initial condition and Definition 3.5 is unconditional identifiability.","section":"Example 4.1 and Section 4"},{"comment":"The 'in particular' statement that identifiability follows from u0 not in ker(L[A]) for all A in A requires passing to the limit L[A]u(., t) -> L[A]u0 as t -> 0; this needs u0 to belong to the domain of L[A] (e.g., C^2 or H^2 satisfying the boundary condition), not merely the stated compatibility condition Bu0 = 0.","section":"Theorem 3.7(b)"},{"comment":"For the bivariate profile likelihood over (c, d), the 95% likelihood-ratio threshold should be approximately 5.991 (chi-square with 2 degrees of freedom), not 2.997, which is the one-degree-of-freedom threshold; please correct the threshold and check whether the displayed confidence regions change materially.","section":"Section 5 and Figure 4"}],"recommendation":"major_revision","confidential_remarks":"The nonlinear example gap is localized and fixable: the authors need to treat the b = 0 continuum, clarify the positive-solution versus sign-changing solution issue, and align the statement of Theorem 4.1 with the hypothesis u_t not identically zero. The linear theory is sound and is the paper's main contribution. I see no concerns about novelty disclosure; the authors' prior work [20] is cited appropriately. The paper is a good fit for an applied PDE or mathematical biology journal."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nRead Salmaniw and Browning. The reduction of structural identifiability for a class of linear-in-parameter parabolic PDEs to kernel analysis of an auxiliary elliptic operator is genuinely useful, and the linear parts are largely correct. For the homogeneous case, the paper gives a clean spectral characterization of non-identifiability for Dirichlet, Neumann, Robin, and periodic boundary conditions, constructs the non-identifiable solutions explicitly, and shows how the initial condition interacts with parameter combinations via a quotient space. The numerical section connecting structural to practical identifiability is a real plus, and the code is public. I buy the central mechanism: subtract two candidate parameter sets, get an elliptic equation, and classify identifiability by uniqueness of its solutions.\n\nThe weaknesses are concentrated in the nonlinear example. Theorem 4.1 proves unconditional identifiability if the auxiliary elliptic equation has a unique or discrete solution set. Example 4.1 claims the logistic model is globally unconditionally identifiable by asserting the auxiliary equation has only one nontrivial solution. That is not established: the cited uniqueness results are for positive solutions, while sign-changing solutions exist, and the b=0 case, where the auxiliary equation becomes linear and its solution set can be a continuum, is not treated. The stress-test note is right on both counts. The claim may still be true, but the proof as written has a real gap. That is a localized flaw in the flagship nonlinear example, not a collapse of the method. I would also flag a smaller technical point: Theorem 3.7(b) uses convergence of L[A]u(·,t) to L[A]u0 at t=0, which needs more initial regularity than the stated compatibility condition; likely fixable with a standard smoothing argument.\n\nWho benefits: anyone doing identifiability for reaction-diffusion models in biology, and people working on structural identifiability for PDEs more generally. The paper is readable, honestly engages with prior work, and the main framework is sound. It deserves a serious referee, but Example 4.1 needs to be fixed or qualified. I would send it to peer review, asking the authors to address the logistic gap and the regularity point.\n\nBest.","headline":"A useful reduction of parabolic identifiability to elliptic spectral theory, with a genuine but localized gap in the flagship logistic example.","tokens_in":23498,"tokens_out":1484,"would_cite":true,"duration_ms":14355,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35K55","35J25","35R30","93B30"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that for fully observed parabolic PDEs linear in their parameters, structural identifiability reduces to the kernel of an auxiliary elliptic operator, and that logistic-type nonlinearities make the model globally…","keywords":["structural identifiability","partial differential equations","linear-in-parameter models","auxiliary elliptic operator","Fredholm alternative","reaction-diffusion equations","practical identifiability","logistic growth"],"falsifier":"Take the logistic model with $b_1 = b_2$, choose any Dirichlet eigenpair $(\\lambda_1, \\sin(\\pi x))$ on $(0,1)$, and pick $(a_1,d_1) \\neq (a_2,d_2)$ with $a_1 - d_1\\lambda_1 = a_2 - d_2\\lambda_1$; starting from $u_0 = \\sin(\\pi x)$, both parameter sets produce the identical non-stationary solution $u(x,t) = e^{(a_1 - d_1\\lambda_1)t}\\sin(\\pi x)$, which would contradict the paper's unconditional identifiability claim for the logistic model unless $b=0$ is excluded.","tokens_in":22498,"feed_emoji":"📐","tokens_out":10690,"duration_ms":106478,"temperature":0.7,"pith_summary":"The paper tries to establish that for parabolic PDEs that are linear in their parameters, the question of structural identifiability—whether two different parameter sets can produce the same observed solution—is decided by an auxiliary elliptic operator obtained by subtracting the two equations. For linear equations, identifiability from a given initial condition holds unless the evolving solution lies in the kernel of that operator at some time; the Fredholm alternative then makes the set of problematic initial conditions finite-dimensional and rare. For nonlinear reactions such as logistic growth, uniqueness of solutions to the auxiliary elliptic equation implies unconditional global identifiability as long as the observed solution changes in time. This matters because spatial biological data are increasingly common, while identifiability theory for PDEs is underdeveloped; the paper's reduction turns an infinite-dimensional inverse problem into classical spectral theory and gives explicit parameter combinations that can never be separated.","feed_headline":"One elliptic operator decides if PDE parameters are identifiable","feed_subtitle":"Initial conditions that hit the kernel make parameters indistinguishable when the model is structurally identifiable.","key_machinery":"The load-bearing object is the auxiliary elliptic operator $L[A] = d\\Delta + b\\cdot\\nabla + c$ for the linear case, and the auxiliary elliptic equation $-d\\Delta\\psi = f(x,\\psi;B)$ for the nonlinear case. Subtracting the two PDEs that the same solution would satisfy eliminates the time derivative and leaves a time-independent equation that the solution must solve at every instant; therefore, if the solution ever leaves the kernel (or, in the nonlinear case, if the elliptic problem has a unique or discrete solution set), the two parameter points are forced to coincide. The Fredholm alternative is used to show that for elliptic $L[A]$ the kernel is finite-dimensional, so non-identifiability is confined to a thin set of initial conditions, while uniqueness theorems for the logistic elliptic problem carry the unconditional identifiability result.","core_discovery":"The central claim is Theorem 3.7: for the homogeneous fully observed model $u_t = d\\Delta u + b\\cdot\\nabla u + cu$ with Dirichlet, Neumann, Robin, or periodic boundary conditions, two parameter points $A_1$ and $A_2$ are distinguishable unless the difference $A = A_1 - A_2$ belongs to the set $\\mathcal{A}$ where the auxiliary elliptic operator $L[A] = d\\Delta + b\\cdot\\nabla + c$ has a nontrivial kernel. The model is unconditionally identifiable on the quotient set $\\mathcal{R}$, identifiable from an initial condition $u_0$ whenever $u_0$ avoids $\\ker(L[A])$ for every $A \\in \\mathcal{A}$, and never unconditionally identifiable over the whole parameter space because some initial condition always lands in a kernel. For the logistic reaction-diffusion model $u_t - d\\Delta u = au - bu^2$, Theorem 4.1 establishes global unconditional identifiability whenever the solution is nontrivial and $u_t \\not\\equiv 0$, because the auxiliary elliptic equation $-d\\Delta\\psi = a\\psi - b\\psi^2$ has only discrete (in fact unique) nontrivial solutions for the parameter differences considered. The paper works out the precise non-identifiable parameter sets for Dirichlet, Neumann, Robin, and periodic conditions, and shows numerically that initial conditions close to the dominant eigenfunction make parameters practically unidentifiable even when structural identifiability holds.","pith_inferences":["Beyond the paper, the same subtraction argument gives a template for any linear-in-parameter parabolic model: write the parameter difference as an auxiliary operator and compute its kernel; this could be automated for polynomial reaction terms with symbolic linear algebra and spectral solvers.","The paper's Example 4.1 silently passes over $b = 0$ in the logistic auxiliary equation; since the equation then becomes linear and its solution set is a vector space whenever $a/d$ is an eigenvalue, the claimed global unconditional identifiability needs either an explicit exclusion of $b=0$ or a separate argument.","The practical-identifiability analysis suggests a design rule the authors do not state: the closer an experiment's initial condition is to the dominant eigenfunction of the diffusion operator, the wider the confidence region in the diffusion-growth plane, so spatial initial conditions should be chosen to maximize projection on higher modes.","For boundary conditions with repeated eigenvalues, such as periodic conditions on a symmetric domain, the kernel is multidimensional, so more than one initial condition is needed to break non-identifiability; a similar counting rule should apply to any PDE model whose elliptic operator has eigenvalues of multiplicity greater than one."],"forward_implications":["For the linear reaction-diffusion equation with Dirichlet conditions, non-identifiable parameter pairs are exactly the curves $c = d(n\\pi/\\ell)^2$ (and their drift-generalized surfaces $c = \\lambda_n(d,b)$); the only structurally non-identifiable initial condition is an eigenfunction, and the parameter combination $c - d\\pi^2$ is identifiable even when $c$ and $d$ individually are not.","Any initial condition that is not an element of the kernel of the auxiliary elliptic operator restores structural identifiability for the homogeneous linear model, so non-identifiability is a finite-dimensional, measure-zero phenomenon.","For the logistic model, no nontrivial time-varying solution can be generated by two distinct parameter sets, so the model is globally unconditionally identifiable under any of the four boundary conditions considered.","In full parameter space the linear model is always identifiable in the weak sense that some initial condition separates any two parameters, yet always fails unconditional identifiability because some initial condition falls in the kernel.","Experiments initialized near the dominant eigenfunction can be practically non-identifiable: the 95% confidence region for $(c,d)$ follows the theoretically indistinguishable set and can span several orders of magnitude in $d$."],"supporting_citations":[{"why":"Supplies the authors' preceding differential-algebra framework for linear reaction-advection-diffusion models, whose loss of linear independence motivates the kernel-based approach.","marker":"[20]"},{"why":"Provides the recent algebraic identifiability framework for PDE models that this work positions itself against and complements.","marker":"[22]"},{"why":"Gives the identifiability definitions used and the known necessary and sufficient conditions for the linear ODE analogue.","marker":"[28]"},{"why":"States the Fredholm alternative used to conclude the auxiliary elliptic operator's kernel is finite-dimensional.","marker":"[30]"},{"why":"Supplies the uniqueness result for positive solutions of the elliptic logistic equation that carries Theorem 4.1.","marker":"[33]"},{"why":"Provides the subhomogeneity condition that guarantees uniqueness of the nontrivial logistic solution in Example 4.1.","marker":"[34]"},{"why":"Supplies uniqueness of positive solutions in the spatially heterogeneous logistic case of Example 4.2.","marker":"[37]"}],"fun_headline_variants":["Elliptic kernel decides which initial conditions hide parameters","Kernel of an elliptic operator predicts PDE parameter identifiability","Bad initial conditions can make identifiable PDE parameters invisible","Identifiability of PDE parameters collapses for special initial data"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The strongest results rely on the observed solution being regular enough that the auxiliary elliptic operator can be applied at $t=0$, and on the auxiliary elliptic equation having at most finitely many solutions for every parameter difference; in the logistic case the unexamined $b=0$ limit makes the equation linear, where solutions can form a continuum, and that single case would break the claim of global unconditional identifiability.","fun_headline_variants_meta":{"raw":{"variants":["Elliptic kernel decides which initial conditions hide parameters","Kernel of an elliptic operator predicts PDE parameter identifiability","Bad initial conditions can make identifiable PDE parameters invisible","Identifiability of PDE parameters collapses for special initial data"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.0009,"raw_usage":{"total_tokens":3941,"prompt_tokens":1080,"completion_tokens":2861,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":696,"completion_tokens_details":{"reasoning_tokens":2809}},"tokens_in":696,"tokens_out":2861,"duration_ms":16970,"temperature":1.0,"reasoning_tokens":2809,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T11:59:42.959568+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take the logistic model with $b_1 = b_2$, choose any Dirichlet eigenpair $(\\lambda_1, \\sin(\\pi x))$ on $(0,1)$, and pick $(a_1,d_1) \\neq (a_2,d_2)$ with $a_1 - d_1\\lambda_1 = a_2 - d_2\\lambda_1$; starting from $u_0 = \\sin(\\pi x)$, both parameter sets produce the identical non-stationary solution $u(x,t) = e^{(a_1 - d_1\\lambda_1)t}\\sin(\\pi x)$, which would contradict the paper's unconditional identifiability claim for the logistic model unless $b=0$ is excluded.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the recent algebraic identifiability framework for PDE models that this work positions itself against and complements."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the identifiability definitions used and the known necessary and sufficient conditions for the linear ODE analogue."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"States the Fredholm alternative used to conclude the auxiliary elliptic operator's kernel is finite-dimensional."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the subhomogeneity condition that guarantees uniqueness of the nontrivial logistic solution in Example 4.1."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies uniqueness of positive solutions in the spatially heterogeneous logistic case of Example 4.2."}],"review_version":1}