{"id":"9c263f5a-2a5f-4bca-b773-2e7bd4d73da1","arxiv_id":"2506.24101","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":5,"one_line_summary":"WSINDy is extended to identify heterogeneous model ingredients and boundary birth processes in age- and size-structured population models from noisy data.","lead":"This paper presents an extension of the WSINDy algorithm to learn the growth, death, and birth terms of structured population models from noisy histogram data. The method stacks a weak form of the population PDE with a weak form of the total population ODE to recover both interior dynamics and the boundary birth process.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Boundary terms enter only through the scalar ODE (Eq. 6); without boundary-local weak forms, β is identifiable only up to its projection onto the observed density, so the claim of recovering true boundary processes is not generally supported.","rationale":"The reader's weakest_assumption is the exact-sparsity/library-inclusion assumption. That is a real limitation, but the paper explicitly acknowledges it and frames the method as selecting effective terms from a given library, so it is not the most load-bearing weakness for the stated claim. The more serious issue is structural: the boundary process β is never directly constrained by the interior PDE weak form, and the only coupling is through the scalar total-population ODE. This makes the boundary identification problem underdetermined in general, independent of library completeness. The boundary-bagging procedure is a heuristic that refits the same scalar ODE residual, so it cannot resolve the identifiability gap. Because the paper's central claim explicitly includes learning the boundary process from data, and because the synthetic and real-data evaluations do not test a case where two different boundary shapes are indistinguishable from the total-population trajectory, the concern is load-bearing. I still recommend keeping the reader's CONDITIONAL verdict: the method may work well when the boundary function is well separated in the library and the data are information-rich, but the paper should either add boundary-local weak forms (test functions supported at the boundary) or explicitly characterize the identifiability conditions under which Eq. (9) can recover β⋆. The proposed concrete test would settle whether the concern actually lands in a representative setting.","tokens_in":16658,"tokens_out":3762,"duration_ms":57447,"concrete_test":"Construct a two-term boundary library containing the true β⋆(s) = Gaussian(10,5) and a perturbed β2(s) = Gaussian(12,5) for Example L.2, with all other library terms fixed to the true source/transport terms. Simulate the model to the same T and binning as in the paper, and compute the two corresponding ODE matrix columns Ξβ,1 and Ξβ,2 from Eq. (8). If the normalized columns satisfy ∥Ξβ,1 − Ξβ,2∥ / ∥Ξβ,1∥ is below the noise-induced residual level (or the condition number of [Ξβ,1 Ξβ,2] is very large), then the stacked system is numerically unable to distinguish the two boundary shapes; report which shape WSINDyStructuredPop selects and whether the selection is stable across noise realizations.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is that the stacked system Eq. (9) recovers the true boundary process β⋆ from noisy histogram data. However, β⋆ appears nowhere in the interior weak form (4): the test functions are compactly supported inside Ω (Section 2.2), so the boundary condition (3b) is absent. The only information about β⋆ enters through the total-population ODE (6), whose weak form (7) contains only the scalar integral ∫_Ω β(s)n(t,s) ds. Consequently, the stacked system can at best identify the projection of β onto the range of the observed density n(t,·), not β itself. If two library candidates β1 and β2 produce nearly identical values of ∫_Ω β_i(s)n(t,s) ds over the sampled times—for example, when n converges rapidly to a stable shape, or when the candidates differ mainly in regions where n is small—then the ODE rows cannot distinguish them, and any selection is driven by noise or numerical rounding. The boundary-bagging heuristic in Section 2.2.1 refits β from the same ODE residual (b_ode − Ξ_f w_f) and therefore cannot create information that is absent from the scalar projection. The synthetic experiments report aggregate TPR/E2 metrics but do not isolate this identifiability failure; the elephant-data comparison also shows only broad agreement with a matrix-model fertility schedule, not a resolution of the boundary-shape ambiguity. This is a load-bearing gap because the paper's claimed novelty includes learning the boundary process directly from data.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript extends the Weak SINDy (WSINDy) framework to structured population models of the form (3). Given noisy histogram observations of the population density n(t,s), the method builds a weak-form linear system from an interior PDE test-function equation (4), stacks it with a weak form of the total-population ODE (7), and uses sparse regression to select transport, source, and boundary terms from a user-supplied library. A 'boundary bagging' heuristic (Algorithm 1) refits the boundary coefficients from the ODE residual. The method is tested on four linear and two nonlinear synthetic examples under multiplicative lognormal noise, on a library-distinguishability study, and on an Asian elephant age-structured dataset, with metrics including coefficient errors, true positivity ratio, and temporal holdout prediction error.","tokens_in":16994,"tokens_out":6141,"duration_ms":68788,"significance":"If the claims are substantiated, the method would provide a fast, interpretable alternative to repeated forward-solve parameter estimation for structured population models, and the temporal holdout prediction error provides an out-of-sample check that strengthens the empirical case. The authors are appropriately explicit about limitations such as library exactness, test-function hyperparameters, and the distinguishability of similar library terms. The code release and the real-data comparison to an established matrix model are also useful contributions. However, the central claim that the boundary process is learned 'directly from the data' is currently stronger than the information content of the stacked system supports, and the synthetic experiments do not isolate that failure mode.","major_comments":[{"comment":"The boundary term β* appears in the stacked system only through the scalar integral ∫_Ω β(s)n(t,s) ds in Eq. (7), while the PDE weak form (4) uses test functions compactly supported in the interior of Ω and therefore contains no boundary information. Consequently, the stacked system can identify β only up to its projection onto the span of the observed densities n(t,·) over the sampled times; two library functions that produce nearly identical values of this integral at the sampled times cannot be distinguished by any regression on Eq. (9). The manuscript does not state or verify identifiability conditions (e.g., sufficient variation of n(t,s) in s and t, or boundary-local test functions), and the aggregate TPR/E2 metrics in Section 3.2 do not isolate the boundary component. This gap is load-bearing for the abstract's claim of learning the boundary process directly from the data.","section":"Section 2.2, Eqs. (6)-(9)"},{"comment":"The proposed 'boundary bagging' procedure is not a cross-validation method: it uses the same ODE rows and the already-fitted source weights to refit β from the residual b_ode − Ξ_f w_f, with no held-out data. It therefore cannot create information about β that is absent from Eq. (7). Calling it cross-validation in the abstract and in Section 2.2.1 overstates what the procedure does. The support-intersection heuristic may be a reasonable stability check, but the manuscript should either describe it as such or compare it with an actual data-splitting scheme.","section":"Section 2.2.1, Algorithm 1"},{"comment":"The distinguishability experiments fix the transport and the non-target component to the true terms and vary only the spacing of candidate functions; they report TPR and condition numbers, but they do not quantify how much information about the boundary term is present in the observed density snapshots. As a result, the experiments do not resolve the projection-identifiability issue raised by Eq. (7). Please add either an information-theoretic or coherence analysis of the stacked system, or a demonstration with intentionally information-poor data (e.g., densities that quickly converge to a stable shape) showing when boundary recovery succeeds and fails.","section":"Section 3.2.2 and Figure 3"},{"comment":"The Discussion states that WSciML methods are 'highly robust to noise' and cites Figure 1, but Figure 1 shows the true positivity ratio dropping from 1 to roughly 0.6-0.8 as σNR increases, with prediction error remaining low. This is not necessarily a contradiction, but the robustness claim should be qualified to mean robust predictive performance rather than robust term selection; otherwise the text promises stronger support than the reported metrics provide.","section":"Section 4, Discussion versus Figure 1"}],"minor_comments":[{"comment":"The text says the TPR = 0.8 example is problem L.2, while the Figure 2 caption says example L.3; please reconcile this inconsistency.","section":"Section 3.2.1 and Figure 2"},{"comment":"Section 1 contains the duplicated phrase 'the the Weak-form Estimation'; Section 4 also repeats the sentence 'It is of course, still necessary to carefully curate a library...' verbatim in consecutive paragraphs.","section":"Section 1 and Section 4"},{"comment":"The caption uses 'respectfully' where 'respectively' is meant.","section":"Figure 8 caption"},{"comment":"The statement that nonlinear parameters must be given in the library is important and should appear earlier, perhaps in the abstract or introduction, since it defines the scope of 'selection' and directly affects the practical applicability of the method.","section":"Section 3.1"},{"comment":"The boundary library notation for L.2 uses {f_Gauss(s; 5k, 5)}_{k=1,2,3}; the later text refers to f_Gauss(s; 10, 5) as the true term in Case 2, which matches k=2, but the notation is easy to misread and should be clarified.","section":"Table 3"}],"recommendation":"major_revision","confidential_remarks":"For the editor: this is a useful application and extension of WSINDy to structured population models, with a reasonable set of synthetic and real-data experiments. The main risk is that the paper claims to learn the boundary process directly from data, while the stacked ODE weak form only provides a projected view of the boundary term; I believe this can be addressed by adding identifiability conditions, softening the claims, and adding targeted experiments, so I recommend major revision rather than rejection. I do not see a novelty-disclosure problem, though the phrase 'first to use SciML methods to learn this class of population models' is somewhat broad in light of existing weak-form parameter estimation literature. The journal scope is appropriate if the paper is framed as a methods paper with a real-data demonstration."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is a solid, useful methods paper that extends WSINDy to a class of hyperbolic structured population models. The stacked PDE/ODE weak form is a natural trick for bringing the boundary condition into the regression, and the synthetic tests are honest about distinguishability limits. But the headline claim about learning the boundary process directly from data is not fully supported. The birth function β enters the total-population ODE only through the integral ∫β(s)n(t,s)ds, so the data identify at best the projection of β onto the observed density. Two different β candidates that produce nearly equal values of that integral over the sampled times are indistinguishable, and the boundary-bagging heuristic refits from the same residual, so it cannot create information that is absent from the scalar projection. The paper should either prove that the projection uniquely determines β under the library assumptions, or soften the claim to learning the boundary term’s effect on the total population. The synthetic experiments don’t isolate this failure; a simple example with two β functions that yield the same N(t) would be convincing.\n\nWhat the paper does well: the extension to heterogeneous ingredients (functions of s and N) is new relative to prior WSINDy work, and the treatment of nonlocal nonlinearities like N(t)-dependent terms is careful. The examples cover age- and size-structured models with different transport, source, and boundary terms. The prediction error on temporal holdouts provides a legitimate out-of-sample check. The real elephant data application is a nice touch, even if it mainly serves as a sanity check rather than rigorous validation.\n\nSoft spots beyond the boundary issue: the noise robustness claim is a bit loose. TPR drops off as noise increases, but prediction error stays moderate because the method finds effective but incorrect terms. The authors acknowledge this, but the Discussion phrasing is stronger than the results. The real-data section lacks uncertainty quantification, and the code is promised but not yet released, so the numerical results aren’t independently reproducible today. Finally, the method assumes the true ingredients are sparse in the library, which is standard for WSINDy but should be stated more prominently as a selection condition, not just a modeling convenience.\n\nBottom line: I’d send this to review. The method is practical, the boundary identifiability issue is real but fixable with a proper analysis or a softer claim, and the paper would be a useful reference for anyone doing data-driven population modeling. It’s not groundbreaking, but it’s a genuine step forward for WSINDy applications.","headline":"Solid extension of WSINDy to structured population models, but the boundary-learning claim is stronger than the math supports: β is only identified up to its projection onto the observed density.","tokens_in":17519,"tokens_out":4395,"would_cite":true,"duration_ms":46854,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["92D25","35L60"],"pacs":[],"model":"deepseek-v4-flash","headline":"Adding a total-population ODE lets WSINDy recover birth, death, and growth terms of structured population models from noisy binned data.","keywords":["structured population models","WSINDy","weak-form sparse identification","age-structured models","size-structured models","equation discovery","boundary process","population dynamics"],"falsifier":"Simulate a structured population model whose true birth rate is a Gaussian with mean lying between two library Gaussians, add low-level log-normal noise, and run the algorithm; because the true term is absent from the library, exact recovery is impossible and the returned coefficients will be systematically biased, demonstrating that the central recovery claim is conditional on the library containing the true ingredients.","tokens_in":16453,"feed_emoji":"🐘","tokens_out":8798,"duration_ms":93774,"temperature":0.7,"pith_summary":"The paper extends the weak-form sparse identification method WSINDy to structured population models, the hyperbolic PDEs used for age- and size-structured population dynamics. Its central claim is that stacking the weak form of the interior PDE with a weak form of the total-population ODE produces one sparse linear system from which transport (growth or aging), source (mortality), and boundary (birth) terms can be recovered simultaneously from noisy time-series histogram data. The authors add a cross-validation boundary-bagging step that refits the boundary terms against the ODE residual and keeps only terms both fits agree on, which improves recovery of the birth process. If the method works as claimed, researchers can compare many candidate fecundity and mortality hypotheses against data at once without repeatedly simulating the PDE forward.","feed_headline":"Stacked weak-form equations recover birth and death from noisy data","feed_subtitle":"Adding a total-population ODE lets sparse regression infer growth, mortality, and fecundity without PDE simulation.","key_machinery":"The load-bearing object is the stacked weak-form linear system $b = Gw$ of Equation (9). The PDE block is assembled from inner products of compactly supported piecewise-polynomial test functions with transport and source library terms, while the ODE block is assembled from the weak form of $dN/dt = \\int \\beta n + \\int f$, with the total population $N$ computed from the same binned data. Sparse weights $w = (w_g, w_f, w_\\beta)$ are found by minimizing $\\|b - Gw\\|^2 + \\lambda\\|w\\|_0$ with modified sequential thresholding least squares. Because the PDE block has far more rows, the boundary-bagging step fixes $w_f$ and refits $w_\\beta$ from the ODE residual, then prunes the boundary library to the support intersection (or union) of the two fits before refitting. This combination moves boundary information into the regression and keeps the sparse solution from pushing all error into the birth term.","core_discovery":"The paper claims that the true transport, source, and boundary ingredients of a structured population model can be recovered simultaneously from noisy histogram data by solving one stacked sparse linear system. The interior weak form of the hyperbolic PDE yields rows that constrain the growth term and the mortality term, but the compactly supported test functions erase the birth process at the inflow boundary. The paper therefore adds the weak form of the total-population ODE obtained by integrating the PDE over the whole structural domain, whose rows carry the boundary information through the birth kernel and the source term. Concatenating the two blocks and applying modified sequential thresholding least squares, with the boundary-bagging step to counter the imbalance between the large PDE block and the small ODE block, returns sparse coefficient vectors that select the correct library terms; in the synthetic tests the method recovers the true coefficients at low noise, and in misspecified libraries it can still return effective models with low prediction error.","pith_inferences":["A direct extension suggested by the paper's own discussion would close the loop between library construction and selection: after a first regression selects a family, re-estimate that family's nonlinear parameters and re-run the selection until the active matrix's condition number drops.","The boundary-bagging trick is a general recipe for weak-form discovery: when compactly supported test functions erase a process, append an aggregated conservation equation for that process and arbitrate the two fits by support intersection; the same construction should apply to conservation laws, moment systems, and flux boundary conditions beyond population biology.","The distinguishability experiments imply a testable screening rule: success rate on a given library and dataset should track the condition number of the weak-form matrix, so candidate libraries could be rejected a priori when that condition number exceeds a threshold, before any regression is run."],"forward_implications":["With a library that contains the true model ingredients, the stacked system recovers transport, source, and boundary coefficients from noisy histogram data without forward simulation; the paper reports per-realization run times under ten seconds on a laptop.","If the true ingredients are not in the library, the learned model can still predict the population well even when its term support is wrong, so the output is best interpreted as an effective hypothesis over the supplied library rather than the true mechanism.","Library distinguishability is a practical bottleneck: as candidate functions are spaced closer together, the condition number of the weak-form matrix grows and the true-positive ratio drops sharply, so nearly redundant trial functions should be excluded.","For density-dependent models, data that converge quickly to equilibrium make different functions of total population $N$ look alike, and the algorithm may substitute a sparser linear or lower-order approximation; this is a documented limitation rather than a resolved issue.","When the aging rate is known a priori for age-structured data, moving that transport term into the data vector focuses the regression entirely on death and birth, which is how the elephant-data application is handled."],"supporting_citations":[{"why":"Defines the classical age-structured PDE whose boundary birth integral the method must recover.","marker":"[4]"},{"why":"Defines the size-structured model class with transport and boundary flux used in the test problems.","marker":"[5]"},{"why":"Supplies the weak-form sparse regression framework and the modified sequential-thresholding solver used to minimize the sparse loss.","marker":"[13]"},{"why":"Supplies the Galerkin weak-form construction, piecewise-polynomial test functions, and the trapezoidal-rule convergence lemma behind the integral computations.","marker":"[14]"},{"why":"Provides the weak-form parameter estimation context and noise-robustness comparison that motivate the method's computational efficiency claim.","marker":"[15]"},{"why":"Inspires the boundary-bagging cross-validation heuristic for pruning the boundary library.","marker":"[28]"},{"why":"Supplies the Asian elephant dataset and the matrix-model survival and fertility comparisons used in the real-data application.","marker":"[33]"}],"fun_headline_variants":["Stacked weak forms pull birth and death rates from noisy histograms","Sparse regression learns population model terms without PDE simulation","Weak-form stacking recovers fecundity and mortality from noisy data","Structured population models identified via stacked weak equations","Boundary-aware weak learning: birth rates from noisy histograms"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The true growth, death, and birth functions must be exactly sparse linear combinations of the user-supplied library trial functions, and any nonlinear parameters in those functions must be fixed in advance.","fun_headline_variants_meta":{"raw":{"variants":["Stacked weak forms pull birth and death rates from noisy histograms","Sparse regression learns population model terms without PDE simulation","Weak-form stacking recovers fecundity and mortality from noisy data","Structured population models identified via stacked weak equations","Boundary-aware weak learning: birth rates from noisy histograms"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000505,"raw_usage":{"total_tokens":2441,"prompt_tokens":899,"completion_tokens":1542,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":515,"completion_tokens_details":{"reasoning_tokens":1458}},"tokens_in":515,"tokens_out":1542,"duration_ms":11186,"temperature":1.0,"reasoning_tokens":1458,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T21:24:46.190120+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Simulate a structured population model whose true birth rate is a Gaussian with mean lying between two library Gaussians, add low-level log-normal noise, and run the algorithm; because the true term is absent from the library, exact recovery is impossible and the returned coefficients will be systematically biased, demonstrating that the central recovery claim is conditional on the library containing the true ingredients.","supporting_citations":[{"cited_title":"Applications of Mathematics to Medical Problems,","cited_arxiv_id":null,"evidence_quote":"Defines the classical age-structured PDE whose boundary birth integral the method must recover."},{"cited_title":"A New Model For Age-Size Structure of a Population,","cited_arxiv_id":null,"evidence_quote":"Defines the size-structured model class with transport and boundary flux used in the test problems."},{"cited_title":"Weak SINDy For Partial Differential Equations,","cited_arxiv_id":null,"evidence_quote":"Supplies the weak-form sparse regression framework and the modified sequential-thresholding solver used to minimize the sparse loss."},{"cited_title":"Weak SINDy: Galerkin-Based Data-Driven Model Selection,","cited_arxiv_id":null,"evidence_quote":"Supplies the Galerkin weak-form construction, piecewise-polynomial test functions, and the trapezoidal-rule convergence lemma behind the integral computations."},{"cited_title":"Direct Estimation of Parameters in ODE Mod- els Using WENDy: Weak-form Estimation of Nonlinear Dynamics,","cited_arxiv_id":null,"evidence_quote":"Provides the weak-form parameter estimation context and noise-robustness comparison that motivate the method's computational efficiency claim."},{"cited_title":"Ensemble-SINDy: Robust sparse model discovery in the low-data, high-noise limit, with active learning and control,","cited_arxiv_id":null,"evidence_quote":"Inspires the boundary-bagging cross-validation heuristic for pruning the boundary library."},{"cited_title":"Changes in age-structure over four decades were a key determinant of population growth rate in a long-lived mammal,","cited_arxiv_id":null,"evidence_quote":"Supplies the Asian elephant dataset and the matrix-model survival and fertility comparisons used in the real-data application."}],"review_version":1}