Pith. sign in

REVIEW 4 major objections 5 minor 1 cited by

Learning Structured Population Models from Data with WSINDy

T0 review · 4 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Adding a total-population ODE lets WSINDy recover birth, death, and growth terms of structured population models from noisy binned data.

desk verdict 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. read the letter →

arxiv 2506.24101 v1 pith:QK3QWNTW submitted 2025-06-30 q-bio.PE math.DS

Rainey Lyons , Vanja Dukic , David M. Bortz This is my paper · ORCID
classification q-bio.PEmath.DS MSC 92D2535L60
keywords structuredpopulationmodelsWSINDyweak-formsparseidentificationage-structuredsize-structuredequationdiscoveryboundaryprocessdynamics
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

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.

What carries the argument

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.

What would settle it

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.

Watch

Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

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

  • 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.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 5 minor

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.

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 (4)
  1. [Section 2.2, Eqs. (6)-(9)] 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.
  2. [Section 2.2.1, Algorithm 1] 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.
  3. [Section 3.2.2 and Figure 3] 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.
  4. [Section 4, Discussion versus Figure 1] 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.
minor comments (5)
  1. [Section 3.2.1 and Figure 2] 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.
  2. [Section 1 and Section 4] 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.
  3. [Figure 8 caption] The caption uses 'respectfully' where 'respectively' is meant.
  4. [Section 3.1] 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.
  5. [Table 3] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the stacked weak-form system is a regression fit to data, and the prediction error is evaluated on an out-of-sample time interval; self-citations are provenance, not load-bearing assumptions.

full rationale

The paper's derivation chain is self-contained as a data-fitting and model-selection method. Equation (4) is the standard weak form of the interior PDE, and because the compactly supported test functions omit the boundary condition, the authors introduce the total-population ODE (6) obtained by integrating (3), with weak form (7). Stacking the interior PDE system (5) with the ODE system (8) gives the full regression problem (9). The coefficients w are fit to the observed histogram data via sparse regression, and the reported prediction error Ep is defined by simulating the learned model over the holdout interval (Ttest, T) and comparing to the true solution; it is not the training residual, so the prediction is not a disguised fit. The boundary-bagging refinement in Algorithm 1 refits only the boundary weights from the ODE residual bode - Ξf wf; this is a model-selection heuristic and does not make the learned boundary process an input to itself. The paper also explicitly acknowledges the distinguishability and identifiability limitations of the boundary term, noting that the ODE rows contain only the averaged quantity integral of beta times n, so this is an identifiability caveat rather than a circular construction. Citations to prior WSINDy work by the same group provide the algorithmic foundation of MSTLS and weak-form integration, but the present extension, including heterogeneous dynamics and boundary-process learning, is derived from the model equations and evaluated on independent synthetic and real data. No equation in the paper is defined in terms of a fitted output, no fitted parameter is renamed as a prediction, and no load-bearing argument reduces to a self-citation. Therefore the appropriate circularity score is 0.

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

The central claims rest on the assumptions that the data are generated by a model of the form (3), that the true ingredients are in the candidate libraries, that the total-population ODE is valid, and that the test-function quadrature is accurate. These are standard for SINDy-type methods but are untested for new applications. No ad hoc entities are introduced.

free parameters (5)
  • Test function support radii r_s and r_t = 0.5
    Set to 0.5 for all numerical results. These control the compact support of the weak-form test functions and were chosen through numerical experiments (Section 2.2). They affect quadrature accuracy and term distinguishability.
  • Test function polynomial order p = q = 14
    Chosen for the numerical results (Section 2.2). The smoothness of the test functions influences the convergence of the trapezoidal rule and the conditioning of the weak-form matrix.
  • Sparsity threshold / lambda in loss function = not specified
    The loss function (Eq. 10) includes a sparsity parameter lambda, but the paper does not state how lambda is selected for the MSTLS algorithm. This directly controls which terms are selected.
  • Real-data library shape parameters = Gaussian fertility mean=33, sd=11; mortality rate 0.06, shift 85
    For the elephant data, these nonlinear parameters are fixed by hand to construct the candidate library (Section 3.4). The linear coefficients (alpha, b, c) are learned, but the shape parameters are not. If they are wrong, the recovered model is constrained.
  • Loess smoothing parameters for variance estimation = not specified
    Appendix B uses a loess fit to estimate the noise variance sigma^2 for bias correction. The smoothing span and degree are not specified, so this is an unstated user choice.
assumptions (5)
  • domain assumption Data are generated by a hyperbolic structured population PDE of the form (3) with smooth or globally Lipschitz coefficients.
    Section 2.1 states this assumption for the noise-free data. If the true dynamics involve shocks or nonlocal terms outside this class, the weak form may not be valid.
  • domain assumption True model ingredients are sparse linear combinations of the trial functions in the user-selected library.
    Section 2.2: 'we assume the true model ingredients g*, f*, and beta* can be represented as a (sparse) linear combination of a given set of trial functions.' This is the classic SINDy library assumption.
  • standard math Trapezoidal rule convergence lemma from Messenger and Bortz [14] applies to the weak-form integrals with the chosen test functions.
    Section 2.2 cites Lemma 2 of [14] for rapid convergence. This requires the test functions to be sufficiently smooth and the data to be sufficiently resolved.
  • domain assumption The total population ODE (Eq. 6) is exact for the data, i.e., the boundary conditions (3b)-(3c) hold in aggregate.
    The ODE is derived by integrating the PDE over Omega and using the boundary conditions. This assumption is needed to couple the boundary process to the interior dynamics via the stacked system (Eq. 9).
  • domain assumption Multiplicative lognormal noise model and the bias correction using loess-estimated variance.
    Section 3 assumes n_j = epsilon_j n*_j with log-normal epsilon, and Appendix B corrects the bias in N using a loess variance estimate. If the noise is not log-normal, the bias correction may be invalid.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Learning Structured Population Models from Data with WSINDy." pith.science (2026). https://pith.science/paper/QK3QWNTW

@misc{pith2026250624101,
  author       = {Pith},
  title        = {Pith review of: Learning Structured Population Models from Data with WSINDy},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QK3QWNTW}},
  note         = {Machine review of arXiv:2506.24101}
}
read the original abstract

In the context of population dynamics, identifying effective model features, such as fecundity and mortality rates, is generally a complex and computationally intensive process, especially when the dynamics are heterogeneous across the population. In this work, we propose a Weak form Scientific Machine Learning-based method for selecting appropriate model ingredients from a library of scientifically feasible functions used to model structured populations. This method uses extensions of the Weak form Sparse Identification of Nonlinear Dynamics (WSINDy) method to select the best-fitting ingredients from noisy time-series histogram data. This extension includes learning heterogeneous dynamics and also learning the boundary process of the model directly from the data. We additionally provide a cross-validation method which helps fine tune the recovered boundary process to the data. Several test cases are considered, demonstrating the method's performance for different previously studied models, including age and size-structured models. Through these examples, we examine both the advantages and limitations of the method, with a particular focus on the distinguishability of terms in the library.

Figures

Figures reproduced from arXiv: 2506.24101 by the authors.

Figure 1
Figure 1. Top: Average performance metrics for the linear models in [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. Typical results of using the WSINDy algorithm for example problem L.3 in Table [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. TPR and condition number over different choices of library parameters for case 1 (top) and [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: A typical run of Example NL.2 with σNR = 0.66 and 50 points in time. The prediction error and TPR of this learned model are 0.004 and 1, respectively. 11 [PITH_FULL_IMAGE:figures/full_fig_p011_4.png]
Figure 5
Figure 5. Figure 5: Left: The learned and true dynamics of the total population, N, plotted with the training data with noise ratio σNR ≈ 0.02. Right: The learned nonlinear structure (of the form a + bN3 ) plotted with the true structure over the training time interval [PITH_FULL_IMAGE:f…
Figure 6
Figure 6. Figure 6: From left to right, we present the learned linear birth kernel [PITH_FULL_IMAGE:figures/full_fig_p012_6.png]
Figure 7
Figure 7. Figure 7: Top: WSINDy-reconstructed age-structured population dynamics of the semi-captive Asian elephant population over the 31-year period. Bottom: Estimated age-specific fertility and survival functions inferred by WSINDy (solid lines), shown alongside those derived from clas…
Figure 8
Figure 8. Figure 8: Typical results of using the WSINDy algorithm for the linear models in Table [PITH_FULL_IMAGE:figures/full_fig_p019_8.png]

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

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

  1. Weak Form Scientific Machine Learning: Test Function Construction for System Identification

    math.NA 2025-07 conditional novelty 6.0 of 10

    A new algorithm picks the support radius of weak-form test functions by locating the changepoint of an estimated integration error curve, and this radius lands near the parameter-error minimum in most tested cases.

Reference graph

Works this paper leans on

42 extracted references · 40 canonical work pages · cited by 1 Pith paper

  1. [1]

    J. A. J. Metz, O. Diekmann, and S. Levin, eds., The Dynamics of Physiologically Structured Populations, vol. 68 of Lecture Notes in Biomathematics . Berlin, Heidelberg: Springer Berlin Heidelberg, 1986

    work page 1986

  2. [2]

    Some Remarks on Changing Populations,

    H. Von Foerster, “Some Remarks on Changing Populations,” in The Kinetics of Cellular Prolif- eration (F. Stohlman, ed.), pp. 382–407, Grune & Stratton, 1959

    work page 1959

  3. [3]

    The McKendrick partial differential equation and its uses in epi- demiology and population study,

    B. Keyfitz and N. Keyfitz, “The McKendrick partial differential equation and its uses in epi- demiology and population study,” Mathematical and Computer Modelling , vol. 26, pp. 1–9, Sept. 1997

    work page 1997

  4. [4]

    Applications of Mathematics to Medical Problems,

    A. G. McKendrick, “Applications of Mathematics to Medical Problems,” Proc. Edinb. Math. Soc., vol. 44, p. 98, Feb. 1925

    work page 1925

  5. [5]

    A New Model For Age-Size Structure of a Population,

    J. W. Sinko and W. Streifer, “A New Model For Age-Size Structure of a Population,” Ecology, vol. 48, pp. 910–918, Nov. 1967

    work page 1967

  6. [6]

    Rapidly converging numerical algorithms for models of population dynamics,

    F. A. Milner and G. Rabbiolo, “Rapidly converging numerical algorithms for models of population dynamics,” J. Math. Biol. , vol. 30, Aug. 1992

    work page 1992

  7. [7]

    Determining a Coefficient in a First-Order Hyperbolic Equation,

    M. Pilant and W. Rundell, “Determining a Coefficient in a First-Order Hyperbolic Equation,” SIAM J. Appl. Math. , vol. 51, pp. 494–506, Apr. 1991

    work page 1991

  8. [8]

    Non-linear age-dependent population dynamics,

    M. E. Gurtin and R. C. Maccamy, “Non-linear age-dependent population dynamics,” Arch. Ra- tional Mech. Anal. , vol. 54, pp. 281–300, Sept. 1974

    work page 1974

Show all 42 references
  1. [9]

    Some competition models for size-structured populations,

    J. Cushing, “Some competition models for size-structured populations,” Rocky Mountain J. Math., vol. 20, Dec. 1990

    1990

  2. [10]

    Inverse Problems and Structured-Population Dynamics,

    S. N. Wood, “Inverse Problems and Structured-Population Dynamics,” in Structured-Population Models in Marine, Terrestrial, and Freshwater Systems (S. Tuljapurkar and H. Caswell, eds.), pp. 555–586, Boston, MA: Springer US, 1997

    1997

  3. [11]

    Discovering governing equations from data by sparse identification of nonlinear dynamical systems,

    S. L. Brunton, J. L. Proctor, and J. N. Kutz, “Discovering governing equations from data by sparse identification of nonlinear dynamical systems,” Proc. Natl. Acad. Sci. , vol. 113, pp. 3932–3937, Apr. 2016. 15

    2016

  4. [12]

    Data-driven discovery of partial differential equations,

    S. H. Rudy, S. L. Brunton, J. L. Proctor, and J. N. Kutz, “Data-driven discovery of partial differential equations,” Sci. Adv., vol. 3, p. e1602614, Apr. 2017

    2017

  5. [13]

    Weak SINDy For Partial Differential Equations,

    D. A. Messenger and D. M. Bortz, “Weak SINDy For Partial Differential Equations,” J. Comput. Phys., vol. 443, p. 110525, Oct. 2021

    2021

  6. [14]

    Weak SINDy: Galerkin-Based Data-Driven Model Selection,

    D. A. Messenger and D. M. Bortz, “Weak SINDy: Galerkin-Based Data-Driven Model Selection,” Multiscale Model. Simul. , vol. 19, no. 3, pp. 1474–1497, 2021

    2021

  7. [15]

    Direct Estimation of Parameters in ODE Mod- els Using WENDy: Weak-form Estimation of Nonlinear Dynamics,

    D. M. Bortz, D. A. Messenger, and V. Dukic, “Direct Estimation of Parameters in ODE Mod- els Using WENDy: Weak-form Estimation of Nonlinear Dynamics,” Bull. Math. Biol. , vol. 85, no. 110, 2023

    2023

  8. [16]

    WENDy for Nonlinear-in- Parameter ODEs,

    N. Rummel, D. A. Messenger, S. Becker, V. Dukic, and D. M. Bortz, “WENDy for Nonlinear-in- Parameter ODEs,” arXiv:2502.08881, Feb. 2025

    2025

  9. [17]

    Weak form-based data-driven modeling: Compu- tationally Efficient and Noise Robust Equation Learning and Parameter Inference,

    D. M. Bortz, D. A. Messenger, and A. Tran, “Weak form-based data-driven modeling: Compu- tationally Efficient and Noise Robust Equation Learning and Parameter Inference,” in Numerical Analysis Meets Machine Learning (S. Mishra and A. Townsend, eds.), vol. 25 of Handbook of Num...

    2024

  10. [18]

    The Weak Form Is Stronger Than You Think,

    D. A. Messenger, A. Tran, V. Dukic, and D. M. Bortz, “The Weak Form Is Stronger Than You Think,” SIAM News , vol. 57, Oct. 2024

    2024

  11. [19]

    Measure-valued solutions for a hierarchically size-structured population,

    A. S. Ackleh and K. Ito, “Measure-valued solutions for a hierarchically size-structured population,” J. Differ. Equ. , vol. 217, pp. 431–455, Oct. 2005

    2005

  12. [20]

    A structured coagulation-fragmentation equation in the space of radon measures: Unifying discrete and continuous models,

    A. S. Ackleh, R. Lyons, and N. Saintier, “A structured coagulation-fragmentation equation in the space of radon measures: Unifying discrete and continuous models,” ESAIM: M2AN, vol. 55, pp. 2473–2501, Sept. 2021

    2021

  13. [21]

    D¨ ull, P

    C. D¨ ull, P. Gwiazda, A. Marciniak-Czochra, and J. Skrzeczkowski,Spaces of Measures and Their Applications to Structured Population Models . Cambridge University Press, 2021

    2021

  14. [22]

    The ultimate-sustainable-yield problem in nonlinear age-structured populations,

    W. M. Getz, “The ultimate-sustainable-yield problem in nonlinear age-structured populations,” Mathematical Biosciences, vol. 48, pp. 279–292, Apr. 1980

    1980

  15. [23]

    Perthame, Transport Equations in Biology

    B. Perthame, Transport Equations in Biology. Frontiers in Mathematics, Basel: Birkh¨ auser Basel, 2007

    2007

  16. [24]

    Influence of four major plant traits on average height, leaf-area cover, net primary productivity, and biomass density in single-species forests: A theoretical investigation,

    D. S. Falster, ˚A. Br¨ annstr¨ om, U. Dieckmann, and M. Westoby, “Influence of four major plant traits on average height, leaf-area cover, net primary productivity, and biomass density in single-species forests: A theoretical investigation,” Journal of Ecology , vol. 99, pp. 1...

    2011

  17. [25]

    On the dynamics of chemically stressed populations: The deduction of population consequences from effects on individuals,

    S. Kooijman and J. Metz, “On the dynamics of chemically stressed populations: The deduction of population consequences from effects on individuals,” Ecotoxicology and Environmental Safety, vol. 8, pp. 254–274, June 1984

    1984

  18. [26]

    Learning Weather Models from Data with WSINDy,

    S. Minor, D. A. Messenger, V. Dukic, and D. M. Bortz, “Learning Weather Models from Data with WSINDy,” arXiv:2501.00738, Jan. 2025

    2025 arXiv

  19. [27]

    Weak-form inference for hybrid dynamical systems in ecology,

    D. A. Messenger, G. Dwyer, and V. Dukic, “Weak-form inference for hybrid dynamical systems in ecology,” J. R. Soc. Interface. , vol. 21, p. 20240376, Dec. 2024

    2024

  20. [28]

    Ensemble-SINDy: Robust sparse model discovery in the low-data, high-noise limit, with active learning and control,

    U. Fasel, J. N. Kutz, B. W. Brunton, and S. L. Brunton, “Ensemble-SINDy: Robust sparse model discovery in the low-data, high-noise limit, with active learning and control,” Proc. R. Soc. A. , vol. 478, p. 20210904, Apr. 2022

    2022

  21. [29]

    On the heterogeneity of human populations as reflected by mortality dynamics,

    D. Avraam, S. Arnold, O. Vasieva, and B. Vasiev, “On the heterogeneity of human populations as reflected by mortality dynamics,” Aging, vol. 8, pp. 3045–3064, Nov. 2016

    2016

  22. [30]

    Pareto-based optimization of sparse dynam- ical systems,

    G. Ducci, M. Kouyate, K. Reuter, and C. Scheurer, “Pareto-based optimization of sparse dynam- ical systems,” J. Chem. Phys. , vol. 162, p. 114118, Mar. 2025. 16

    2025

  23. [31]

    Finite difference schemes for a structured population model in the space of measures,

    A. S. Ackleh, R. Lyons, and N. Saintier, “Finite difference schemes for a structured population model in the space of measures,” Math. Biosci. Eng. , vol. 17, no. 1, pp. 747–775, 2020

    2020

  24. [32]

    Data from: Changes in age-structure over four decades were a key determinant of population growth rate in a long-lived mammal,

    J. Jackson, K. Mar, W. Htut, D. Childs, and V. Lummaa, “Data from: Changes in age-structure over four decades were a key determinant of population growth rate in a long-lived mammal,” July 2020

    2020

  25. [33]

    Changes in age-structure over four decades were a key determinant of population growth rate in a long-lived mammal,

    J. Jackson, K. U. Mar, W. Htut, D. Z. Childs, and V. Lummaa, “Changes in age-structure over four decades were a key determinant of population growth rate in a long-lived mammal,” Journal of Animal Ecology, vol. 89, pp. 2268–2278, Oct. 2020

    2020

  26. [34]

    Asymptotic consistency of the WSINDy algorithm in the limit of continuum data,

    D. A. Messenger and D. M. Bortz, “Asymptotic consistency of the WSINDy algorithm in the limit of continuum data,” IMA J. Numer. Anal. , p. drae086, Dec. 2024

    2024

  27. [35]

    Influence of initial conditions on data-driven model identification and information entropy for ideal mhd problems,

    G. Vasey, D. Messenger, D. Bortz, A. Christlieb, and B. O’Shea, “Influence of initial conditions on data-driven model identification and information entropy for ideal mhd problems,” Journal of Computational Physics , vol. 524, p. 113719, Mar. 2025

    2025

  28. [36]

    DoE-integrated Sparse Identification of Nonlinear Dynamics for Au- tomated Model Generation and Parameter Estimation in Kinetic Studies,

    W. Lyu and F. Galvanin, “DoE-integrated Sparse Identification of Nonlinear Dynamics for Au- tomated Model Generation and Parameter Estimation in Kinetic Studies,” in Computer Aided Chemical Engineering, vol. 53, pp. 169–174, Elsevier, 2024

    2024

  29. [37]

    Estimation of growth and survival in size-structured cohort data: An application to larval striped bass (Morone saxatilis),

    H. T. Banks, L. W. Botsford, F. Kappel, and C. Wang, “Estimation of growth and survival in size-structured cohort data: An application to larval striped bass (Morone saxatilis),” J. Math. Biol., vol. 30, pp. 125–150, 1991

    1991

  30. [38]

    Estimation of Cell Proliferation Dynamics Using CFSE Data,

    H. T. Banks, K. L. Sutton, W. Clayton Thompson, G. Bocharov, D. Roose, T. Schenkel, and A. Meyerhans, “Estimation of Cell Proliferation Dynamics Using CFSE Data,” Bull. Math. Biol. , vol. 73, pp. 116–150, Jan. 2011

    2011

  31. [39]

    Finite difference approximations for measure- valued solutions of a hierarchicallysize-structured population model,

    A. S. Ackleh, V. K. Chellamuthu, and K. Ito, “Finite difference approximations for measure- valued solutions of a hierarchicallysize-structured population model,” Math. Biosci. Eng. , vol. 12, no. 2, pp. 233–258, 2015

    2015

  32. [40]

    High resolution finite difference schemes for a size structured coagulation-fragmentation model in the space of radon measures,

    A. S. Ackleh, R. Lyons, and N. Saintier, “High resolution finite difference schemes for a size structured coagulation-fragmentation model in the space of radon measures,” MBE, vol. 20, no. 7, pp. 11805–11820, 2023

    2023

  33. [41]

    A Second-Order High-Resolution Scheme for a Juvenile-Adult Model of Amphibians,

    A. S. Ackleh and B. Ma, “A Second-Order High-Resolution Scheme for a Juvenile-Adult Model of Amphibians,” Numer. Funct. Anal. Optim. , vol. 34, pp. 365–403, Apr. 2013

    2013

  34. [42]

    F. E. Harrell, Regression Modeling Strategies: With Applications to Linear Models, Logistic and Ordinal Regression, and Survival Analysis . Springer Series in Statistics, Cham: Springer Interna- tional Publishing, 2015. A Simulation details Due to the histogram-style interpret...

    2015

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.