Pith. sign in

REVIEW 4 major objections 4 minor 1 cited by

An LSTM-PINN Hybrid Method to the specific problem of population forecasting

T0 review · 4 major / 4 minor · reviewed 2026-08-16 · deepseek-v4-flash

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

Pith's one-line read The paper claims that a PINN and an LSTM-PINN hybrid can solve the age-structured McKendrick-von Foerster equation with policy-driven fertility encoded directly in the birth term, producing 2024–2054 population projections under three…

desk verdict A preliminary synthetic demo whose central equation is mis-scaled, so the claimed McKendrick-von Foerster solutions are not established; deserves a rewrite rather than a referee. read the letter →

arxiv 2505.01819 v1 pith:EV5F62IW submitted 2025-05-03 cs.LG

Ze Tao This is my paper · ORCID
classification cs.LG
keywords Longshort-termmemoryPopulationforecastingPhysics-informedneuralnetworkHybridmethodMcKendrick-vonFoersterequationAge-structuredFertilitypolicy
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

This paper proposes two neural-network solvers for the age-structured population equation: a standard physics-informed neural network (PINN) and an LSTM-augmented variant (LSTM-PINN). Both embed age- and time-dependent fertility functions directly into the McKendrick-von Foerster transport-reaction PDE, so the choice of child-bearing policy changes the equation the network learns to satisfy, not just the data fed to it. The paper claims that the LSTM layer lets the hybrid retain long-range temporal dependencies across the age-time domain, with all three loss components—the PDE residual, the boundary birth integral, and the initial age condition—converging stably. On this basis it projects population density from 2024 to 2054 under three fertility-policy scenarios and reports that the projected age distributions differ substantially across policies. A sympathetic reader would take the contribution to be a template: when policy interventions can be written as functions inside a PDE, the same hybrid framework can encode them and produce a policy-sensitive forecast without fitting a new statistical model.

What carries the argument

The machinery is the composite loss function built from three residuals: the PDE residual $\mathcal{L}_1$ enforces the McKendrick-von Foerster equation at 5,000 collocation points, the initial-condition loss $\mathcal{L}_2$ matches the starting age distribution at 2,000 points, and the boundary-condition loss $\mathcal{L}_3$ enforces the birth integral $P(0,t)=\int_0^{a_0} b(a,t)P(a,t)\,da$ at 2,000 points. The McKendrick-von Foerster equation is the transport-reaction PDE $\partial P/\partial t + \alpha\,\partial P/\partial a = -\mu(a)\,P$, with $\alpha$ set to the time-age scaling factor $(t_{\max}-t_{\min})/a_0$, which here equals 0.3. Policy content enters through $b(a,t)$, defined as a base age-specific fertility rate multiplied by indicator-gated boost factors and capped at 0.20 or 0.25. In the LSTM-PINN, the stacked LSTM layers are the memory mechanism that carries information across time steps, and their input, forget, and output gates decide what to keep, which the paper uses to argue for stable long-range dependence.

What would settle it

Recompute the 2024–2054 evolution with a classical characteristics or finite-difference scheme using the same initial age distribution, mortality, fertility functions, and $\alpha=0.3$, then compare the age-time surfaces; disagreement beyond training error would show that the neural networks are not solving the stated McKendrick-von Foerster equation.

Watch

Extended reading notes

Core claim

The central claim is that solutions of an age-structured demographic PDE with policy-dependent birth terms can be learned by a neural network whose loss simultaneously enforces the PDE, the birth boundary condition, and the initial age profile. Concretely, the paper solves $\partial P/\partial t + \alpha\,\partial P/\partial a = -\mu(a)\,P$ on $(a,t)\in[0,100]\times[2024,2054]$, with $P(0,t)=\int_0^{a_0} b(a,t)P(a,t)\,da$ and $P(a,2024)=P_{\mathrm{data}}(a,0)$, where $b(a,t)$ is a base fertility curve multiplied by policy-boost factors that switch on in 2014, 2016, and 2021 under the three-child policy and in 2024 under the two-child variants. The LSTM-PINN replaces the PINN's feedforward trunk with stacked LSTM layers so that the network retains prior state across time steps; the paper reports that both networks converge and that the fertility policy visibly changes the predicted age-time population surface. The paper does not claim that these projections match real census data; it claims that the hybrid solver is a coherent, extensible way to fold policy-dependent fertility into demographic forecasting.

Load-bearing premise

The projected curves inherit a starting age distribution and an aging-rate coefficient that the paper asserts rather than sources or derives; if the initial density $P_{\mathrm{data}}(a,0)$ is not the real population or $\alpha=0.3$ does not correspond to the actual time-age scaling, every forecast in the paper shifts.

Editorial extensions

If this is right

  • The same three-term loss can be run with any analytically expressible fertility or mortality schedule, so new policy scenarios become a matter of editing $b(a,t)$ and $\mu(a)$ rather than rebuilding the model.
  • Because all three loss components decrease together, the method offers a consistency diagnostic: a solver that satisfies the PDE, boundary, and initial-condition constraints simultaneously is less likely to drift into implausible age structures over a 30-year horizon.
  • If the LSTM-PINN genuinely retains long-range temporal dependencies, the architecture should generalize to longer horizons and sparser data than the vanilla PINN, making it the better candidate for real demographic series.
  • The projected age distributions differ across the three fertility policies, so the model could be used to inspect which age cohorts are most affected by each policy design.

Reading between the lines

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

  • The initial density $P_{\mathrm{data}}(a,0)$ is never displayed or sourced in the paper; if the released code contains it, re-running the same framework with a swapped-in census curve would immediately test whether the approach is usable operationally rather than only as a solver demonstration.
  • The coefficient $\alpha=0.3$ equals the ratio of the time window to the age window, so the equation should be invariant to rescaling age by $1/\alpha$; a natural check is to run with a rescaled age variable and coefficient 1 and confirm the same age-time surfaces.
  • A stronger validation than loss curves would be a held-out temporal test: train on 2024–2039, predict 2040–2054, and compare with a model trained on the full window, which would separate the LSTM's memory contribution from the PDE's inductive bias.
  • The text alternates between naming the third scenario 'Two-child policy' and 'Separate two-child policy,' and the two two-child fertility definitions in Section 4.1 differ only in the cap; resolving this label mismatch is a prerequisite for reproducing the scenario comparison.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 4 minor

Summary. This paper proposes two physics-informed neural network formulations—a standard PINN and an LSTM-augmented PINN—for solving the McKendrick–von Foerster equation with age- and time-dependent fertility on the domain [0,100] x [2024,2054]. The models are trained with a composite loss consisting of a PDE residual, an initial-condition term, and a birth-boundary integral term, under three hand-specified fertility policy schedules. The reported results are training-loss curves and age-time population-density heatmaps; the authors conclude that the LSTM-PINN captures long-range temporal dependencies and that the simulations reveal policy-sensitive demographic shifts. The code is publicly available.

Significance. If the method worked as claimed, a PINN/LSTM-PINN solver for age-structured population equations with policy-dependent fertility would be a useful complement to classical numerical schemes, and the public code is a strength. However, the paper's evidence does not establish the central claim: no initial population distribution is specified, no comparison is made to a reference solution or to real demographic data, and the loss function is not the residual of the stated PDE under a consistent normalization. The empirical contribution is therefore currently a demonstration of training convergence on a synthetic problem rather than a validated forecasting method.

major comments (4)
  1. [Section 2 and Section 4.2, Eq. (1a) and Eq. (3a)] The advection coefficient alpha = 0.3 is never justified in units of years. If (a,t) are dimensional years, the equation implies that a 30-year-old in 2024 becomes 39 in 2054, contradicting both the McKendrick–von Foerster equation and the paper's stated age range. If, as stated in Section 4.2, the network inputs are normalized, then the non-dimensionalized transport equation is dP/dtau + (T/a0) dP/dx = -T mu(a0 x) P, so the mortality term must be multiplied by T = 30; Eq. (3a) does not do this. The residual that is minimized is therefore not the residual of the claimed demographic PDE, and the numerical solutions cannot be presented as solutions of the McKendrick–von Foerster problem.
  2. [Section 2, Eq. (1c), and Section 4, Eq. (3b)] The initial population density Pdata(a,0) is required by the initial-condition loss but is never stated, plotted, or sourced. All projected age structures inherit this unspecified input, and the paper gives no way to reproduce or evaluate the forecasts. This is a load-bearing omission for a paper whose title promises population forecasting.
  3. [Section 4.1, policy functions b(a,t)] The differences among the three policy scenarios are inserted by construction through the indicator-function multipliers and caps in b(a,t). Reporting that the resulting solutions differ is a check that the solver responds to its prescribed forcing, not evidence that the models 'reflect' policy sensitivity or that demographic dynamics are sensitive to the named policies. The conclusion's claim in Section 5 goes beyond what this experimental design can show.
  4. [Sections 4.2 and 4.3, Figs. 3-6] No quantitative error metric is reported. The convergence curves show that the training losses decrease, but they do not demonstrate accuracy with respect to any reference solution (for example, a method-of-lines or finite-volume solution of the same equation) or to observed population data. The assertion that LSTM-PINN 'effectively captured long-range temporal dependencies' is therefore unsupported; no comparison of PINN versus LSTM-PINN accuracy is provided.
minor comments (4)
  1. [Section 3.1, Eqs. (3b) and (3c)] The text immediately after Eq. (3) describes the loss terms with the labels reversed: Eq. (3b) is the initial-condition term, not the boundary condition, and Eq. (3c) is the birth-boundary integral term.
  2. [Abstract and Sections 4.1 and 5] Policy names are inconsistent: the abstract and conclusion refer to a 'Separate two-child policy' while Section 4.1 defines a 'two-child policy'; also, the 'Universal two-child' and 'two-child' definitions differ only in the cap (0.25 versus 0.20), which should be stated explicitly.
  3. [Throughout] Several editorial problems remain: 'can be can be written' in Section 3.1, the dangling phrase 'denoted by and for the aging rate' after Eq. (1), and the unresolved figure references 'Fig.??' in Section 4.2.
  4. [Section 4.3] The LSTM-PINN input sequence is not specified: the paper says the input at each time step is a two-dimensional vector, but it does not explain how age points are ordered into sequences for the LSTM layers.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the numerical solutions are forward simulations of the stated PDE with prescribed policy inputs; the lone self-citation is not load-bearing.

full rationale

The paper's derivation chain is self-contained in the sense that it defines a transport-reaction PDE (Eq. 1), assigns a composite loss (Eq. 2) whose residual terms are the PDE, initial condition, and birth boundary condition, and then minimizes that loss. The fertility policies enter only through the prescribed functions b(a,t) in Section 4.1; because b is an input and is not fitted to the resulting age distributions, reporting that different b produce different solutions is a standard forward-simulation check, not a circular prediction. No fitted parameter is later renamed as a prediction, and no uniqueness theorem or ansatz is imported from the authors' prior work. The only self-citation ([15], to the author's heat-conduction paper) appears in a cursory sentence on recent PINN advances and is not load-bearing for the population model, so it does not raise the circularity score. The paper itself labels the study preliminary and lists missing real-data integration, which further supports that the claims are about numerical behavior of the stated model rather than an empirical forecast. Issues such as the unexplained initial population density Pdata(a,0), the ambiguous alpha=0.3 aging coefficient without mortality rescaling, and the swapped labels of Eqs. (3b)-(3c) are correctness and transparency concerns, not circularity, because the model is solved as written rather than being equivalent to its inputs by construction.

Assumptions & free parameters 5 free parameters · 6 assumptions · 0 invented entities

The paper introduces no new physical entities, but it relies on several hand-chosen numeric inputs: the scaling factor alpha, mortality parameters, the base fertility schedule, policy multipliers, and loss weights. The most serious unstated input is Pdata(a,0), the initial age distribution, without which the forecast is not reproducible from the text.

free parameters (5)
  • alpha (time-age scaling factor) = 0.3
    Introduced in Eq. (1a) without derivation; the standard McKendrick-von Foerster equation uses coefficient 1 when age and time share units, and no normalization is provided.
  • mortality parameters mu0 and B = mu0 = 0.006805083, B = 0.0003
    Defined in Section 4.1 with no source or fitting procedure; all age-specific survival depends on these values.
  • base ASFR parameters = coefficient 0.0022, age window 20 to 35
    base_asfr(a) in Section 4.1 is a hand-chosen quadratic fertility schedule with no empirical source.
  • policy multipliers and caps = 1 + 0.2 indicator terms, caps 0.20 and 0.25
    The three policy scenarios are encoded by hand-chosen multipliers and caps; the scenario outputs are forced by these choices.
  • loss weights lambda1, lambda2, lambda3 = not stated; text says empirically selected, then says equally weighted
    Section 3.1 says values are empirically selected, while Section 4.3 says each component is equally weighted; no values or tuning procedure are provided.
assumptions (6)
  • domain assumption The McKendrick-von Foerster equation is the correct model for the population dynamics.
    The paper assumes Eq. (1) governs the system without deriving it or validating it against demographic data.
  • ad hoc to paper A known initial population density Pdata(a,0) exists and is supplied to the initial-condition loss.
    Eq. (1c) and Section 4.2 require Pdata(a,0), but no function or dataset is reported anywhere in the paper.
  • ad hoc to paper alpha = 0.3 correctly rescales the age-time relationship.
    No normalization is given; if age and time are both in years, this changes the speed at which cohorts age.
  • ad hoc to paper The hand-coded fertility functions b(a,t) represent the named policy scenarios.
    The Three-child, Universal two-child, and Separate two-child policies are encoded as ad hoc indicator bumps with caps, with no demographic evidence or calibration.
  • ad hoc to paper The piecewise mortality schedule mu(a) approximates real survival.
    Section 4.1 gives a linear-then-exponential schedule with chosen constants and no source or fitting procedure.
  • domain assumption Minimizing the composite loss yields an accurate solution of the PDE.
    PINN training assumes that low residual loss implies an accurate PDE solution; the paper provides no convergence guarantee or independent accuracy check.

how reviews work

0 comments
Cite this review

Pith. "Pith review of An LSTM-PINN Hybrid Method to the specific problem of population forecasting." pith.science (2026). https://pith.science/paper/EV5F62IW

@misc{pith2026250501819,
  author       = {Pith},
  title        = {Pith review of: An LSTM-PINN Hybrid Method to the specific problem of population forecasting},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EV5F62IW}},
  note         = {Machine review of arXiv:2505.01819}
}
read the original abstract

Deep learning has emerged as a powerful tool in scientific modeling, particularly for complex dynamical systems; however, accurately capturing age-structured population dynamics under policy-driven fertility changes remains a significant challenge due to the lack of effective integration between domain knowledge and long-term temporal dependencies. To address this issue, we propose two physics-informed deep learning frameworks--PINN and LSTM-PINN--that incorporate policy-aware fertility functions into a transport-reaction partial differential equation to simulate population evolution from 2024 to 2054. The standard PINN model enforces the governing equation and boundary conditions via collocation-based training, enabling accurate learning of underlying population dynamics and ensuring stable convergence. Building on this, the LSTM-PINN framework integrates sequential memory mechanisms to effectively capture long-range dependencies in the age-time domain, achieving robust training performance across multiple loss components. Simulation results under three distinct fertility policy scenarios-the Three-child policy, the Universal two-child policy, and the Separate two-child policy--demonstrate the models' ability to reflect policy-sensitive demographic shifts and highlight the effectiveness of integrating domain knowledge into data-driven forecasting. This study provides a novel and extensible framework for modeling age-structured population dynamics under policy interventions, offering valuable insights for data-informed demographic forecasting and long-term policy planning in the face of emerging population challenges.

Figures

Figures reproduced from arXiv: 2505.01819 by the authors.

Figure 1
Figure 1. PINN structure diagram differential equation are calculated using automatic differentiation. The initial age distribution and the boundary birth condition are incorporated into the loss function as penalty terms. The total loss, comprising PDE residuals and constraint violations, is minimized using stochastic gradient descent for iterative optimization of the neural network parameters 𝜃 = [𝑊 , 𝑏]. The training proce… view at source ↗
Figure 2
Figure 2. LSTM-PINN structure diagram capabilities of the LSTM. The total loss is minimized using stochastic gradient descent (SGD), which iteratively updates the network’s parameters 𝜃 = [𝑊 , 𝑏] through backpropagation of gradients obtained via automatic differentiation. Similar to the traditional PINN framework, the training process terminates when the loss function falls below a predefined threshold 𝜖 or when the number of… view at source ↗
Figure 3
Figure 3. The loss function of PINN with different policy shown in Fig.??, where the total loss, PDE residual loss, initial condition loss, and boundary condition loss all exhibit steady convergence throughout training, demonstrating the model’s ability to capture the underlying dynamics. The corresponding population projection results for three distinct fertility policy scenarios—the "Three-child policy", the "Two-child poli… view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: PINN results with different policy 4.3. LSTM-PINN Approach to Numerical Solutions In our LSTM-PINN framework, the neural network model employs a stacked LSTM architecture with 4 layers, each consisting of 64 units. The input to the LSTM at each time step is a two-dimen…
Figure 5
Figure 5. Figure 5: The loss function of PINN with different policy and Universal two-child policy—are presented in Fig.6, highlighting notable variations in age-time dynamics and population growth trajectories driven by varying fertility policies. 5. Conclusion In this study, we develope…
Figure 6
Figure 6. Figure 6: LSTM-PINN results with different policy in the context of population challenges faced by China. Furthermore, more rigorous mathematical formulations and refined modeling assumptions are anticipated to produce more accurate and interpretable results. The complete source…

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. Effects of fuel and soot concentrations on the inception and development of contrails

    physics.flu-dyn 2026-03 conditional novelty 6.0 of 10

    A laboratory contrail tunnel shows ice nucleation across turbulent shear layers and finds contrail scattering more sensitive to exhaust water vapor than to soot number concentration.

Reference graph

Works this paper leans on

16 extracted references · 15 canonical work pages · cited by 1 Pith paper

  1. [1]

    International Monetary Fund, 2024

    Kohei Asao, Danila Smirnov, and Ms TengTeng Xu.Japan’s Fertility: More Children Please. International Monetary Fund, 2024

    work page 2024

  2. [2]

    Population aging as a global issue

    Jennifer D Sciubba. Population aging as a global issue. InOxford Research Encyclopedia of International Studies. 2020

    work page 2020

  3. [3]

    Social interactions and fluctuations in birth rates.Population Studies, 54(2):223–237, 2000

    Hans-Peter Kohler. Social interactions and fluctuations in birth rates.Population Studies, 54(2):223–237, 2000

    work page 2000

  4. [4]

    Changing fertility patterns in china.Chinese Journal of Sociology, 9(4):497–521, 2023

    Wei Chen. Changing fertility patterns in china.Chinese Journal of Sociology, 9(4):497–521, 2023

    work page 2023

  5. [5]

    Heterogeneously structured compartmental models of epidemiological systems: from individual-level processes to population-scale dynamics

    Emanuele Bernardi,Tommaso Lorenzi, MattiaSensi, andAndrea Tosin. Heterogeneously structured compartmentalmodels of epidemiolog- ical systems: from individual-level processes to population-scale dynamics.arXiv preprint arXiv:2503.11225, 2025

    work page Pith review arXiv 2025

  6. [6]

    A systematic procedure for incorporating separable static heterogeneity into compartmental epidemic models

    Odo Diekmann and Hisashi Inaba. A systematic procedure for incorporating separable static heterogeneity into compartmental epidemic models. Journal of Mathematical Biology, 86(2):29, 2023

    work page 2023

  7. [7]

    A comparative study of compartmental models for covid-19 transmission in ontario, canada.Scientific Reports, 13(1):15050, 2023

    Yuxuan Zhao and Samuel WK Wong. A comparative study of compartmental models for covid-19 transmission in ontario, canada.Scientific Reports, 13(1):15050, 2023. Z.Tao et al.:Preprint submitted to Elsevier Page 8 of 9 LSTM-PINN for population forecasting

    work page 2023

  8. [8]

    Boundaryvalueproblemfortheloadedmckendrickvonfoersterequation of fractional order.Adyghe International Scientific Journal, 23(4):28–33, 2023

    FatimaMukhamedovnaLosanovaandRaisaOsmanovnaKenetova. Boundaryvalueproblemfortheloadedmckendrickvonfoersterequation of fractional order.Adyghe International Scientific Journal, 23(4):28–33, 2023

    work page 2023

Show all 16 references
  1. [9]

    From individual-based epidemic models to mckendrick-von foerster pdes: A guide to modeling and inferring covid-19 dynamics.Journal of Mathematical Biology, 85(4):43, 2022

    Félix Foutel-Rodier, François Blanquart, Philibert Courau, Peter Czuppon, Jean-Jil Duchamps, Jasmine Gamblin, Élise Kerdoncuff, Rob Kulathinal, Léo Régnier, Laura Vuduc, et al. From individual-based epidemic models to mckendrick-von foerster pdes: A guide to modeling and infer...

    2022

  2. [10]

    Numerical solution to a nonlinear mckendrick-von foerster equation with diffusion.Numerical Algorithms, 92(2):1007–1039, 2023

    Joydev Halder and Suman Kumar Tumuluri. Numerical solution to a nonlinear mckendrick-von foerster equation with diffusion.Numerical Algorithms, 92(2):1007–1039, 2023

    2023

  3. [11]

    Ahigherordernumericalschemetoanonlinearmckendrick–vonfoersterequationwithsingular mortality

    JoydevHalderandSumanKumarTumuluri. Ahigherordernumericalschemetoanonlinearmckendrick–vonfoersterequationwithsingular mortality. Applied Numerical Mathematics, 202:21–41, 2024

    2024

  4. [12]

    O. A. Malafeyev, T. R. Nabiev, and N. D. Redinskikh. Modeling a demographic problem using the leslie matrix, 2024

    2024

  5. [13]

    A data-driven crowd simulation framework integrating physics-informed machine learning with navigation potential fields.IEEE Transactions on Computational Social Systems, 2025

    Bin Chen, Runkang Guo, Qi Zhang, Yong Zhao, Xiao Wang, and Zhengqiu Zhu. A data-driven crowd simulation framework integrating physics-informed machine learning with navigation potential fields.IEEE Transactions on Computational Social Systems, 2025

    2025

  6. [14]

    Data-driven machine learning approach based on physics-informed neural network for population balance model.Advances in Continuous and Discrete Models, 2025(1):12, 2025

    Ishtiaq Ali. Data-driven machine learning approach based on physics-informed neural network for population balance model.Advances in Continuous and Discrete Models, 2025(1):12, 2025

    2025

  7. [15]

    Analytical and neural network approaches for solving two-dimensional nonlinear transient heat conduction.arXiv preprint arXiv:2504.02845, 2025

    Ze Tao, Fujun Liu, Jinhua Li, and Guibo Chen. Analytical and neural network approaches for solving two-dimensional nonlinear transient heat conduction.arXiv preprint arXiv:2504.02845, 2025

    2025 arXiv

  8. [16]

    Krzysztof Zarzycki and Maciej Ławryńczuk. Long short-term memory neural networks for modeling dynamical processes and predictive control: A hybrid physics-informed approach.Fuzzy Systems and Neural Networks for Engineering Applications, 2023, 2025. Z.Tao et al.:Preprint submit...

    2023

Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.