Pith. sign in

REVIEW 4 major objections 5 minor 34 references

Fragment size density estimator for shrinkage-induced fracture based on a physics-informed neural network

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 ↗

Pith's one-line read A physics-informed neural network maps model parameters alpha and gamma directly to the scaled fragment-size density and scale constant, bypassing numerical solution of the governing IDE, and is accurate and fast enough for Monte Carlo…

desk verdict A credible, well-engineered PINN surrogate for the fragmentation IDE, with the main gap being thin benchmark coverage at intermediate parameters rather than a fatal flaw. read the letter →

arxiv 2507.11799 v3 pith:XULAG2WY submitted 2025-07-15 physics.comp-ph cs.AI

Shin-ichi Ito This is my paper · ORCID
classification physics.comp-phcs.AI
keywords physics-informedneuralnetworkintegro-differentialequationfragmentsizedistributionshrinkage-inducedfracturedynamicalscalingBayesianinferenceMonteCarlosurrogatemodel
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 sets out to show that a physics-informed neural network can stand in for a costly integro-differential-equation solver in shrinkage-induced fracture models: instead of solving for the fragment-size density $p(x)$ and scale constant $a$ from scratch for each parameter pair $(\alpha,\gamma)$, the network learns the whole parameter-to-solution map once. If that claim holds, repeated evaluations inside Monte Carlo Bayesian inference become cheap: the paper reports about a 50-fold speedup over a finite-difference solver and a 1,000-sample posterior that finishes in about 2 hours instead of 100. The same surrogate also stays accurate in stiff parameter regimes where the finite-difference scheme drifts to a spurious solution.

What carries the argument

The carrier of the argument is the multiplicative ansatz $p_\theta(x;\alpha,\gamma)=\exp[(\alpha-1)\,\mathrm{NN}_p(x,\alpha,\gamma)]\,\hat p_{\alpha,\gamma}(x)$, where $\hat p_{\alpha,\gamma}$ is a generalized gamma distribution that already has the correct $x\to0$ power-law and $x\to\infty$ stretched-exponential tails; a second network outputs the scale constant $a_\phi(\alpha,\gamma)=\exp[(\alpha-1)\mathrm{NN}_a(\alpha,\gamma)]c_{\alpha,\gamma}$. The network therefore learns only a smooth multiplicative correction to the asymptotically correct shape, which accelerates convergence and stabilizes optimization. The loss combines the squared IDE residual with a $1/x$ weight, plus the normalization constraints, averaged over log-uniform sample parameters; all integrals are evaluated with double-exponential quadrature, and training uses a hybrid L-BFGS and Levenberg-Marquardt optimizer.

What would settle it

Using the released implementation, compute the residual $\mathcal I[x,p_\theta,a_\phi,\alpha,\gamma]$ on an independent quadrature with, say, 256 segments and wider truncation, and compare $p_\theta$ against a high-resolution finite-difference solution for several parameter pairs in $[1.2,6]^2$ with $x$ up to $10^3$; if the error at those off-training points is orders of magnitude larger than the reported test losses, the Monte Carlo likelihood in Eq. (21) is biased.

Watch

Extended reading notes

Core claim

On its own terms, the paper's central discovery is that the stationary scaled fragment-size density solves an integro-differential equation that lives on a smooth two-parameter manifold, and a physics-informed neural network can learn that manifold directly. The network outputs $p_\theta(x;\alpha,\gamma)$ and $a_\phi(\alpha,\gamma)$, and the loss enforces the IDE residual together with the two normalization conditions, with training pairs drawn log-uniformly from $(\alpha,\gamma)\in[1.2,6]^2$. Against exact solutions at $(\alpha,\gamma)=(2,1)$ and $(2,2)$, the surrogate matches or beats a finite-difference solver on test loss, and for $(\alpha,\gamma)=(3,6)$ it keeps the expected power-law and stretched-exponential asymptotics where the finite-difference solution collapses. The paper reads this as evidence that the surrogate has captured the physical solution manifold rather than merely interpolating training data.

Load-bearing premise

The load-bearing premise is that the discretized loss, evaluated on the chosen quadrature points with fixed segment counts and truncation bounds, is a faithful proxy for the continuous integro-differential equation wherever the posterior likelihood will be evaluated.

Editorial extensions

If this is right

  • Bayesian inversion of fragment-size data becomes routine: a 1,000-sample posterior that costs at least 100 hours with the finite-difference solver completes in about 2 hours with the surrogate.
  • The surrogate is roughly 50 times faster than the fastest finite-difference evaluation, and reaching comparable accuracy with the finite-difference scheme requires a grid beyond $2^{13}$ points, which is more than 1,000 times slower.
  • The PINN remains stable in stiff regimes where the finite-difference scheme converges to a spurious solution, such as $(\alpha,\gamma)=(3,6)$ at $k=2^{13}$, so repeated evaluations inside an MCMC chain do not silently leave the true solution manifold.
  • Because $p_\theta$ is differentiable with respect to both parameters, gradient-based samplers can replace random-walk Metropolis-Hastings, further cutting inference cost.
  • Training costs at most about 6 hours, which is negligible next to the 100-hour-plus FD-based inference run it replaces.

Reading between the lines

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

  • The paper leaves implicit that the same parameter-to-density map could be retrained for other fragmentation kernels: the network only corrects the generalized-gamma asymptotics, so the hard-wired division-ratio kernel and power-law lifetime are modeling assumptions rather than architectural constraints.
  • The existence of exact closed-form solutions at $(2,1)$ and $(2,2)$ suggests that a broader analytic family of solutions may exist; if so, the network's speed is a practical stopgap and the underlying analytic structure would be the deeper result.
  • A user should re-check quadrature convergence before trusting the surrogate far beyond the training nodes, especially for $x\gtrsim300$; that check is cheap and independent of the solver itself.
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 paper proposes a physics-informed neural network (PINN) surrogate for the integro-differential equation (IDE) describing the scaled fragment-size distribution in a stochastic shrinkage-induced fragmentation model. The surrogate maps the model parameters (alpha, gamma) directly to the density p(x) and the scale constant a, without solving the IDE, by minimizing a loss that combines the IDE residual and the two normalization constraints (Eqs. (9)-(13)). The network representation in Eq. (14) embeds a generalized-gamma baseline motivated by the asymptotic behaviors derived in Appendix C. The paper reports test losses at two exact-solution parameter pairs, a comparison with a finite-difference (FD) baseline, a solution profile at (alpha, gamma) = (3,6), and a Metropolis-Hastings Bayesian inversion on synthetic data. The central claims are that the surrogate is orders of magnitude faster than FD at comparable accuracy and that it enables practical Monte Carlo posterior inference.

Significance. If the central claim is established, the paper offers a genuinely useful computational tool: a surrogate that evaluates the fragment-size density at arbitrary parameters in about 1/50 of the time of the FD baseline and that is differentiable with respect to the model parameters, making gradient-based Bayesian inference possible. The paper has clear strengths: the governing IDE and the exact benchmark solutions are made explicit; the ansatz in Eq. (14) respects the known asymptotic behavior; the training loss directly enforces the residual and normalization, not the target density; and the authors state that the code is available. The loss formulation and the fast-evaluation property are the main contributions. However, the current evidence for the 'any parameter in the training domain' claim is not yet commensurate with the strength of that claim, because validation relies on only two exact-solution points and on a comparison at a parameter where the FD baseline is unstable.

major comments (4)
  1. [Sec. 4.1, Eqs. (9)-(13)] The discretized loss is used as the sole enforcement mechanism for the IDE, but the quadrature and parameter-grid choices are not validated by any convergence study. The integrals are truncated to y in [-3,2] and z in [-3,3] with 64 segments for l1 and 32 segments for the other integrals, and the expectation over parameters is approximated on 16 x 16 vertices with bilinear interpolation. A small residual at these quadrature nodes does not guarantee a small residual on a refined grid or at intermediate parameter values. The paper should report a convergence check versus segment count, truncation width, and parameter-grid spacing, and ideally a refined-grid residual evaluation, to show that the loss is a faithful proxy for the continuous IDE.
  2. [Secs. 4.2-4.4, Fig. 5] The validation of the parameter-to-solution map rests on only two exact solutions, (alpha, gamma) = (2,1) and (2,2), and on the comparison at (3,6) where the FD solution is explicitly unstable. The (3,6) comparison cannot serve as an independent check because the PINN's architecture already encodes the x^(alpha-1) and exp(-x^gamma) asymptotic structure used as the reference, and the FD baseline is not trustworthy there. Since the surrogate is later used in Bayesian inference over the full training domain, a held-out parameter sweep against a stable FD reference (or against a more accurate solver) at several interior points is needed to map the error and support the claim of accuracy across the domain.
  3. [Sec. 4.2, Fig. 3] The performance numbers in Secs. 4.3-4.5 are reported for the 'best-performing PINN model', selected by the smallest final training loss among ten seeds. Figure 3 shows that final outcomes depend strongly on random initialization, so the best-seed numbers overstate the performance of a typical training run. The paper should report the distribution of test losses across the ten seeds (mean, standard deviation, worst case) and state whether the speed and accuracy claims hold for a typical seed, not just for the best one.
  4. [Sec. 4.5, Eq. (21)] The Monte Carlo likelihood in Eq. (21) evaluates p_theta(x_i; alpha, gamma) at arbitrary data points and arbitrary parameter values inside the training domain, but the posterior is not validated against an exact or FD-based likelihood. If the surrogate error is localized or correlated with the parameters, the Metropolis-Hastings chain can be silently biased. The authors should add a posterior-calibration check, for example by comparing the surrogate-based posterior for a smaller synthetic dataset against a posterior obtained with the exact density or with a high-accuracy FD solver at a few parameter settings.
minor comments (5)
  1. [Abstract] There is a typo: 'Validatation' should be 'Validation'.
  2. [Sec. 4.1, Fig. 2] The random Fourier feature frequencies are stated to be drawn from a standard normal distribution, but it is not specified whether they are fixed once at initialization or resampled during training; please clarify.
  3. [Fig. 5] The bottom-row axis labels are confusing: the x-axis is labeled 'log(x)' but the tick values run from 0 to 4, while the rows are described as log-log and log[-log] plots; please relabel the axes consistently with the plotted quantities.
  4. [Sec. 4.1] The use of y in [-3,2] for the double-exponential transformation is asymmetric, and the manuscript does not explain why the positive tail is truncated more aggressively than the negative tail; a brief justification would help assess the accuracy on the large-x tail.
  5. [References] Reference [26] gives a repository name but no URL or DOI; if the code is intended to be reproducible, please provide a permanent link or DOI.

Circularity Check

1 steps flagged · score 2.0 of 10

Central PINN claim is not circular: the surrogate is trained on the IDE residual and independently checked against closed-form exact solutions at (2,1) and (2,2); only the qualitative (3,6) asymptotic check partly restates the architecture's own ansatz.

  1. self definitional [Sec. 3, Eqs. (14)-(15) (network ansatz); Sec. 4.4, Fig. 5 discussion for (alpha,gamma)=(3,6)]
    "The probability density p̂α,γ serves as an approximate solution to Eq. (5), which captures the asymptotic behaviors as x → 0 and x → ∞ . ... the PINN solution successfully captures the anticipated asymptotic structure. Although the exact solution is not available for this parameter set, the ability of the PINN to reproduce the expected asymptotic behavior suggests that it closely approximates the true solution."

    The 'expected asymptotic behavior' used as evidence at (3,6) — p(x)∝x^{α−1} as x→0 and p(x)∝exp[−(x/a)^γ] as x→∞ — is hard-wired into the network representation by construction: pθ(x;α,γ)=exp[(α−1)NNp(x,α,γ;θ)]·p̂α,γ(x) with p̂α,γ the generalized gamma carrying exactly these tails. With no exact solution and an unstable FD baseline at (3,6), the match to these tails largely restates the ansatz, so the inference 'closely approximates the true solution' is partly self-definitional. This is a minor qualitative check: the central quantitative evidence at (2,1) and (2,2) is independent, since the generalized gamma is not the exact solution there and exp[(α−1)NNp] must (and does) perform nontrivial correction.

full rationale

No significant circularity in the central derivation chain. The surrogate pθ(x;α,γ) and aφ(α,γ) are trained by minimizing the residual of Eq. (5) plus the two normalization constraints (Eqs. (9)-(13)) — not by regression onto precomputed densities or onto the exact solutions used for testing. The held-out points are genuine: the training loss is discretized on a 16×16 log-uniform grid over [1.2,6]^2, so (2,2) is interior to a training cell and (2,1) lies outside the training range; both are checked against closed-form exact solutions (17)-(18), which are parameter-free and machine-checkable by substitution into Eq. (5). The generalized-gamma ansatz (15) is derived in-paper (Appendix C) from the equation's own asymptotics, rather than imported by citation; for α=1 it is exact by construction, but at α=2 the exact solution differs from the ansatz, so the reported test losses (orders of magnitude below the FD test losses at comparable resolution) certify genuine learned corrections, not ansatz recall. The MC benchmark draws synthetic data from the exact density (18), not from the surrogate, so the posterior concentration near (2,2) is a legitimate end-to-end check. Self-citations ([11], [12], [14], [18]) establish the model and background, but the model equations are stated in full and the benchmark solutions are closed forms, so the citations carry no unverifiable load. One minor self-referential element exists: the qualitative (3,6) validation in Sec. 4.4 judges the PINN against asymptotic exponents that Eqs. (14)-(15) encode in the architecture by construction, limiting the independence of that particular sanity check; this does not affect the main accuracy and speed claims, which rest on the (2,1)/(2,2) test losses and the FD comparison.

Assumptions & free parameters 5 free parameters · 4 assumptions · 0 invented entities

The paper contributes a PINN solver on top of a pre-existing physical model. Everything physical is pulled from earlier work by the same author (the IDE, the scaling construction, and the exact solutions used for validation), and the solver's working depends on hand-chosen quadrature and architecture settings and on unproven well-posedness of the scaled IDE. No new physical entities are introduced.

free parameters (5)
  • Double-exponential quadrature limits = x: y in [-3,2]; r: z in [-3,3]
    Hand-chosen truncation of infinite domains; the paper reports no convergence test, yet the residual loss is evaluated on this grid.
  • Quadrature segment counts = 64 segments for l1; 32 for l2, l3 and Eq. (6) integrals; 16-point Gauss-Legendre per segment
    Chosen for cost reasons; the authors state a sparse residual grid suffices because NN interpolation is smooth, but give no numerical evidence.
  • Loss weights = w2 = w3 = 1
    Normalization constraints and residual are weighted equally without discussion; the balance affects how exactly the output remains a normalized density during Bayesian likelihood evaluation.
  • Training parameter domain and grid = alpha,gamma in [1.2,6]^2, 16x16 log-uniform grid with bilinear interpolation
    Hand-chosen domain; one benchmark (gamma=1) lies outside it by extrapolation, so the generalization claim is not fully covered.
  • Network architecture hyperparameters = RFF 128 features; 3-layer MLP width 128; Swish; weight variance 1e-6
    Selected via an architecture study in the supplement; not claimed to be globally optimal.
assumptions (4)
  • domain assumption The long-time scaling limit of the Ito-Yukawa model yields the closed IDE Eq. (5) with a time-invariant scaled density p(x) and a scale constant a.
    Adopted from prior work (Refs. 11, 12); the paper does not prove the time derivative is negligible for the parameter range studied.
  • domain assumption Fragment division ratios follow a symmetric Beta(alpha,alpha) distribution and fragment lifetimes follow T(s) proportional to s^{-gamma}.
    Fixes the class of models solved; all results, including future inverse applications, are conditional on these constitutive choices.
  • standard math Equation (5) with normalization conditions (7)-(8) has a unique positive solution for each (alpha,gamma) in [1.2,6]^2.
    The paper relies on this well-posedness but provides no existence or uniqueness argument.
  • ad hoc to paper The ansatz p_theta = exp[(alpha-1) NN_p] times the generalized-gamma baseline is expressive enough to represent the exact solution on (0,infinity).
    The baseline gives the correct asymptotic functional forms but not the exact tail scale for alpha != 1; the NN must learn that correction, and no approximation-error bound is given.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Fragment size density estimator for shrinkage-induced fracture based on a physics-informed neural network." pith.science (2026). https://pith.science/paper/XULAG2WY

@misc{pith2026250711799,
  author       = {Pith},
  title        = {Pith review of: Fragment size density estimator for shrinkage-induced fracture based on a physics-informed neural network},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XULAG2WY}},
  note         = {Machine review of arXiv:2507.11799}
}
read the original abstract

This paper presents a neural network (NN)-based solver for an integro-differential equation that models shrinkage-induced fragmentation. The proposed method directly maps input parameters to the corresponding probability density function without numerically solving the governing equation, thereby significantly reducing computational costs. Specifically, it enables efficient evaluation of the density function in Monte Carlo simulations while maintaining accuracy comparable to or even exceeding that of conventional finite difference schemes. Validatation on synthetic data demonstrates both the method's computational efficiency and predictive reliability. This study establishes a foundation for the data-driven inverse analysis of fragmentation and suggests the potential for extending the framework beyond pre-specified model structures.

Figures

Figures reproduced from arXiv: 2507.11799 by the authors.

Figure 1
Figure 1. (Color online) Schematic illustration of the proposed method. The neural network (NN) model serves as a surrogate for the solution of a stochastic model describing the shrinkage-induced fragmentation process, thereby enabling the construction of the fragment size distribution. parisons with finite differences and exact solutions. It includes a demonstration of Bayesian inference using synthetic data. Section 5 concl… view at source ↗
Figure 2
Figure 2. (Color online) Schematic architecture of the neural networks used in this study. 10−6 . Weight scaling23) is introduced to each Linear to facilitate learning. This architecture was selected because it achieved the best performance across our experiments, which varied both the layer width and the number of skip-connected layers. A comprehensive analysis of this architecture-dependency study is provided in Section 1 o… view at source ↗
Figure 3
Figure 3. (Color online) Optimization history of each loss as a function of actual training time. Panels (a), (b), and (c) show the train loss L, the test loss L test 2,1 , and L test 2,2 , respectively. The line colors indicate different optimization strategies. Each strategy underwent ten trials with different random seeds to initialize the network parameters. 18 [PITH_FULL_IMAGE:figures/full_fig_p018_3.png] view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: (Color online) Comparison with the finite difference (FD) scheme. Panel (a) shows the test losses of the best-performing PINN model obtained from the analysis in Sect. 4.2 and those of the FD model as functions of the number of grid points. Panel (b) presents the predi…
Figure 5
Figure 5. Figure 5: (Color online) Detailed solution profiles. The first and second columns correspond to (α, γ) = (2, 2) and (3, 6), respectively. Each row displays the profiles of the probability density function (PDF) on different scales: linear (top), log–log (middle), and log–log(− l…
Figure 6
Figure 6. Figure 6: (Color online) Histogram view of the joint posterior distribution of α and γ obtained from an MC simulation applied to 1,000 synthetic x-samples drawn from the exact density for (α, γ) = (2, 2) (Eq. (18)). The pixels in this color map correspond to the histogram bins. …

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

34 extracted references · 27 canonical work pages

  1. [1]

    K. A. Shorlin, J. R. de Bruyn, M. Graham, and S. W. Morris, Ph ys. Rev. E 61, 6950 (2000). 10.1103/PhysRevE.61.6950

    work page doi:10.1103/physreve.61.6950 2000

  2. [2]

    Y use and M

    A. Y use and M. Sano, Nature 362, 329 (1993). 10.1038 /362329a0

    work page 1993

  3. [3]

    Giorgiutti-Dauphin´ e and L

    F. Giorgiutti-Dauphin´ e and L. Pauchard, J. Appl. Phys. 120, 065107 (2016). 10.1063/1.4960438

    work page doi:10.1063/1.4960438 2016

  4. [4]

    Goehring, A

    L. Goehring, A. Nakahara, T. Dutta, S. Kitsunezaki, and S. Tarafdar, Desiccation cracks and their patterns: F ormation and Modelling in Science and N ature (John Wiley & Sons, 2015). 10.1002 /9783527671922

    work page 2015

  5. [5]

    Bacchin, D

    P . Bacchin, D. Brutin, A. Davaille, E. Di Giuseppe, X. D. Ch en, I. Gergianakis, F. Giorgiutti-Dauphin´ e, L. Goehring, Y . Hallez, R. Heyd, R. Jeantet, C. Le Floch-Fou´ er´ e, M. Meireles, E. Mittelstaedt, C. Nicloux, L. Pauchard, and M .-L. Saboungi, Eur. Phys. J. E 41, 94 (2018). 10.1140 /epje/i2018-11712-x

    work page 2018

  6. [6]

    Groisman and E

    A. Groisman and E. Kaplan, Europhys. Lett. 25, 415 (1994). 10.1209/0295-5075/25/6/004

    work page doi:10.1209/0295-5075/25/6/004 1994

  7. [7]

    Allain and L

    C. Allain and L. Limat, Phys. Rev. Lett. 74, 2981 (1995). 10.1103/PhysRevLett.74.2981

    work page doi:10.1103/physrevlett.74.2981 1995

  8. [8]

    Lecocq and N

    N. Lecocq and N. V andewalle, Eur. Phys. J. E8, 445 (2002). 10.1140/epje/i2002-10040-2

    work page doi:10.1140/epje/i2002-10040-2 2002

Show all 34 references
  1. [9]

    Akiba, J

    Y . Akiba, J. Magome, H. Kobayashi, and H. Shima, Phys. Rev. E 96, 023003 (2017). 10.1103/PhysRevE.96.023003

    2017 doi

  2. [10]

    Lilin, M

    P . Lilin, M. Ibrahim, and I. Bischofberger, Sci. Adv. 10, eadp3746 (2024). 10.1126/sciadv.adp3746

    2024 doi

  3. [11]

    Ito and S

    S. Ito and S. Y ukawa, Phys. Rev. E 90, 042909 (2014). 10.1103 /PhysRevE.90.042909

    2014

  4. [12]

    Ito and S

    S. Ito and S. Y ukawa, J. Phys. Soc. Jpn. 83, 124005 (2014). 10.7566 /JPSJ.83.124005

    2014

  5. [13]

    Hal´ asz, A

    Z. Hal´ asz, A. Nakahara, S. Kitsunezaki, and F. Kun, Phys . Rev. E 96, 033006 (2017). 10.1103/PhysRevE.96.033006

    2017 doi

  6. [14]

    S. Ito, A. Nakahara, and S. Y ukawa, arXiv:2009.13691. 10 .48550/arXiv.2009.13691

    2009 doi arXiv

  7. [15]

    Raissi, P

    M. Raissi, P . Perdikaris, and G. Karniadakis, J. Comput. Phys. 378, 686 (2019). 10.1016/j.jcp.2018.10.045

    2019 doi

  8. [16]

    G. E. Karniadakis, I. G. Kevrekidis, L. Lu, P . Perdikaris , S. Wang, and L. Y ang, Nat. Rev. Phys. 3, 422 (2021). 10.1038 /s42254-021-00314-5 22

    2021

  9. [17]

    Kovachki, Z

    N. Kovachki, Z. Li, B. Liu, K. Azizzadenesheli, K. Bhatta charya, A. Stuart, and A. Anandkumar, J. Mach. Learn. Res. 24, 1 (2023). http://jmlr.org/papers/v24/21-1524.html

    2023

  10. [18]

    S. Ito, Dr. Thesis, Department of Earth and Space Science , Osaka University (2015). 10.18910/52296

    2015 doi

  11. [19]

    Dosovitskiy and J

    A. Dosovitskiy and J. Djolonga, Proc. 7th Int. Conf. on Le arning Representations, 2020. https://openreview.net/forum?id=HyxY6JHKwr

    2020

  12. [20]

    J. Bae, M. R. Zhang, M. Ruan, E. Wang, S. Hasegawa, J. Ba, an d R. Grosse, Proc. 11th Int. Conf. on Learning Representation s, 2023. https://openreview.net/forum?id=OJ8aSjCaMNK

    2023

  13. [21]

    Tancik, P

    M. Tancik, P . Srinivasan, B. Mildenhall, S. Fridovich-K eil, N. Raghavan, U. Singhal, R. Ramamoorthi, J. Barron, and R. Ng, Adv. Neural Inf. Proces s. Syst., 2020, 33, p. 7537–7547. 10.5555 /3495724.3496356

    2020

  14. [22]

    Ramachandran, B

    P . Ramachandran, B. Zoph, and Q. V . Le., arXiv:1710.0594 1. 10.48550/arXiv.1710.05941

    doi

  15. [23]

    S. Wang, H. Wang, J. H. Seidman, and P . Perdikaris., arXiv :2210.01274. 10.48550/arXiv.2210.01274

    doi arXiv

  16. [24]

    (Supplemental Materials) The architecture-dependenc y study and the details of the Levenberg–Marquardt algorithm are provided online

  17. [25]

    Takahasi and M

    H. Takahasi and M. Mori, Publ. Res. Inst. Math. Sci. 9, 721 (1974). 10.2977/prims/1195192451

    1974

  18. [26]

    Fragment-Size-Density-Estimator-for-Shrinkage-In duced-Fracture-Based-on-PINN

  19. [27]

    Nocedal, Math

    J. Nocedal, Math. Comput. 35, 773 (1980). 10.1090 /S0025-5718-1980-0572855-7

    1980

  20. [28]

    A. S. Krishnapriyan, A. Gholami, S. Zhe, R. M. Kirby, and M . W. Mahoney, Adv. Neural Inf. Process. Syst., 2021, 34, p. 26548–26560. 10.5555 /3540261.3542294

    2021

  21. [29]

    L. Lu, X. Meng, Z. Mao, and G. E. Karniadakis, SIAM rev., 63, 208 (2021). 10.1137/19M1274067

    2021 doi

  22. [30]

    Levenberg, Q

    K. Levenberg, Q. Appl. Math. 2, 164 (1944). 10.1090 /qam/10666

    1944

  23. [31]

    D. W. Marquardt, SIAM J. Appl. Math. 11, 431 (1963). 10.1137 /0111030

    1963

  24. [32]

    Jnini and F

    A. Jnini and F. V ella, arXiv:2505.21404. 10.48550 /arXiv.2505.21404

    doi

  25. [33]

    Betancourt, arXiv:1701.02434

    M. Betancourt, arXiv:1701.02434. 10.48550 /arXiv.1701.02434

    doi arXiv

  26. [34]

    Corbella, S

    A. Corbella, S. E. F. Spencer, and G. O. Roberts, Stat. Com put. 32, 107 (2022). 10.1007/s11222-022-10142-x 23

    2022 doi

Pith tools

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