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REVIEW 5 major objections 4 minor 21 references

Multiprecision computing for multistage fractional physics-informed neural networks

T0 review · 5 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read A two-stage, multi-scale fractional PINN brings subdiffusion solution errors down from about $10^{-3}$ to $10^{-7}$--$10^{-8}$ on both uniform and graded nonuniform meshes.

desk verdict The multistage fPINN extension is real and worth a look, but the advertised 10^-7 accuracy is unsupported by the paper's own 40-point L1 discretization, which puts a ~10^-3 floor on how close any trained network can get to the exact solution. read the letter →

arxiv 2505.22377 v1 pith:U33FCDX3 submitted 2025-05-28 math.NA cs.NA

Na Xue , Minghua Chen This is my paper
classification math.NAcs.NA MSC 26A3326A3065L20
keywords fractionalphysics-informedneuralnetworkssubdiffusionequationsmultistagePINNmulti-scalenetworkL1schemegradedmeshCaputoderivativemultiprecisioncomputing
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 claims that staging a fractional physics-informed neural network through two networks turns the usual fPINN accuracy ceiling of about $10^{-3}\sim10^{-4}$ into $10^{-7}\sim10^{-8}$ relative L2 error for subdiffusion equations. The first network is an ordinary fPINN, and the second is a multi-scale network trained on the residual left by the first, with its amplitude set by the residual's dominant frequency. This multistage strategy is tested on uniform and graded nonuniform time meshes with exponential and polynomial solutions, using $40\times40$ collocation points in the second stage. If the claim holds, high-accuracy fractional PDE solving becomes practical without the very large collocation sets normally associated with fPINNs.

What carries the argument

The load-bearing object is the composite approximation $u_g=u_0+\epsilon_1 u_1$ together with the correction equation $\epsilon_1(D_t^\alpha u_1-\Delta u_1)=-r(x,u_0)$, where $r=D_t^\alpha u_0-\Delta u_0-f$ is the first-stage residual. The amplitude-frequency relation $\epsilon_1=\epsilon_{r1}/(2\pi f_d^{(1)})$, justified by the Fourier scaling $(2\pi f_d)^\alpha$ of fractional derivatives, sets the second stage's prefactor from residual properties. A multi-scale network with a sinusoidal first hidden layer and scaling factors $a_i$ carries the high spatial frequencies that single-scale tanh networks cannot fit.

What would settle it

Recompute the trained two-stage network's prediction against the analytic solution $u(x,t)=x(1-x^2)^2e^{-t}$ on a fine grid while keeping the 40-point L1 residual as the training loss; if the exact L2 error is near $4\times10^{-3}$ instead of $10^{-7}$, the reported accuracy is an artifact of the coarse discrete norm.

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Extended reading notes

Core claim

The central discovery, stated on the paper's terms, is that the residual-correction idea of multistage PINNs transfers to nonlocal fractional operators. Writing the composite solution as $u_g=u_0+\epsilon_1 u_1$ and substituting into $D_t^\alpha u-\Delta u=f$ gives the correction equation $\epsilon_1(D_t^\alpha u_1-\Delta u_1)=-r(x,u_0)$, where $r$ is the first-stage residual. The amplitude $\epsilon_1$ is estimated from the residual's RMS and dominant frequency as $\epsilon_1=\epsilon_{r1}/(2\pi f_d^{(1)})$, and a multi-scale second-stage network with a sinusoidal first hidden layer and scaling factors $a_i$ fits the high-frequency components the first stage misses. In the reported experiments, two-stage errors drop roughly three to four orders of magnitude below single-stage errors, to $10^{-7}$ for exponential solutions and to $10^{-8}$ for linear solutions in the favorable cases.

Load-bearing premise

The whole result rests on the assumption that the finite-difference formula used for the fractional time derivative during training is itself accurate to far better than the claimed $10^{-7}$; on the paper's 40-step time grid the standard formula has truncation error around $10^{-3}$, so the claim requires an unstated finer time resolution or a comparison against the discrete solution rather than the exact one.

Editorial extensions

If this is right

  • A second residual-fitting stage lowers fPINN relative L2 error by about three to four orders of magnitude for the tested subdiffusion problems, from $10^{-4}$ to $10^{-7}$.
  • For exponential solutions, graded nonuniform meshes with $r=(2-\alpha)/\alpha$ outperform uniform meshes by roughly an order of magnitude at every tested $\alpha$.
  • For linear solutions the optimal mesh depends on $\alpha$: uniform grids are better at $\alpha=0.1$, nonuniform grids at $\alpha=0.5$, and both reach around $10^{-8}$ near $\alpha=0.9$.
  • The claimed accuracy is reached with only $40\times40$ collocation points in the second stage, far fewer than the $1000\times400$ points used by the baseline fPINN to reach $10^{-4}$.
  • Keeping equation and data losses equally weighted and skipping auxiliary schemes such as gradient-enhanced PINNs or residual-adaptive refinement is sufficient in these experiments.

Reading between the lines

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

  • If the reported errors are measured against the exact solution, the stated $40$-step L1 grid cannot by itself deliver them: the L1 truncation error for $\alpha=0.5$ and $\tau=1/40$ is about $4\times10^{-3}$, so reproducing $10^{-7}$ accuracy requires either an unstated finer temporal quadrature inside the residual or an error metric taken against the discrete finite-difference solution.
  • The same residual-correction construction with amplitude set by $(2\pi f_d)^\alpha$ scaling should transfer to other nonlocal operators, such as Riesz space-fractional derivatives or the fractional Laplacian, where single-scale networks face the same frequency-spread bottleneck.
  • A practical follow-up is to make the stage count and scale factors adaptive, choosing the next stage only when the current residual's dominant frequency is separable from noise, which would turn the method into a black-box refinement loop.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

5 major / 4 minor

Summary. The paper proposes a multistage fractional physics-informed neural network that combines the multistage PINN of Wang and Lai with multiscale DNN architectures and an amplitude-frequency estimation rule, and applies it to a time-fractional subdiffusion equation with Caputo derivative. The fractional derivative in the loss is evaluated by the L1 finite-difference scheme (2.13) on uniform or graded time grids. Numerical experiments report that two-stage training reduces relative L2 errors from about 10^{-3}-10^{-4} to 10^{-7}-10^{-8} for exponential and polynomial solutions using 40x40 collocation points.

Significance. If the reported accuracies were reproducible, the method would be a noteworthy advance for fPINNs, which typically achieve 10^{-3}-10^{-4}. The paper provides a concrete algorithmic protocol, explicit equations for the correction stage, and comparisons across alpha and grid-grading strategies. However, the central numerical claim is currently unsupported because the reported errors lie below the truncation floor of the L1 discretization used in the loss, and the experimental reporting selects the best of three trials without distribution. The 'multiprecision' framing in the title is also not reflected in the content.

major comments (5)
  1. [§4.1, Eq. (2.13)] The reported two-stage errors in Tables 5 and 6 (e.g., 3.9e-6 for alpha=0.5 and r=1, and 3.3e-7 for alpha=0.9 and r=(2-alpha)/alpha) are smaller than the L1 discretization truncation error of the scheme used to define the loss. With N=40 time points, tau=1/40, the L1 approximation has error O(tau^{2-alpha}), i.e., about 4e-3 for alpha=0.5 and 1.7e-2 for alpha=0.9 for smooth solutions. Hence the exact solution does not satisfy the discrete residual equation (2.14) at the training nodes; its discrete residual is already O(10^{-3}). A network trained to minimize that residual approximates the solution of the discrete scheme, so its L2 error against the exact solution cannot be expected to fall below this discretization floor unless a finer fractional-derivative quadrature is used. The manuscript states fixed 40x40 collocation points and does not disclose any subgrid refinement, so the central accuracy claim is unexplained.
  2. [§4.1] The experimental protocol states that each stage runs three independent trials and the result with the smallest loss is selected as the initial value for the next stage. Reported errors are therefore best-of-three, not typical performance. Since the same data are used for selection and evaluation, the reported 10^{-7}-10^{-8} numbers may reflect favorable initialization seeds. The paper should report the mean, standard deviation, and worst case over all trials, and should state clearly whether the tables list the selected best runs.
  3. [§4.1] The textual report that L2 error decreases from 10^{-4} to 10^{-7} while absolute errors reduce from 10^{-3} to 10^{-4} is internally inconsistent for the O(1) solution u(x,t)=x(1-x^2)^2 e^{-t}: a relative L2 error of 10^{-7} corresponds to an absolute L2 error of about 10^{-7}, not 10^{-4}. The paper should specify the norm used for 'absolute errors' and report error maps or pointwise maxima; Figures 4-9 lack labeled axes and quantitative values, making the claimed convergence impossible to verify.
  4. [§3.1, Eqs. (2.7)-(2.8), (3.7)] The amplitude-frequency relations that configure the second stage are asserted without a derivation for the fractional case. Eq. (3.7) states that second-stage networks must fit functions with amplitude scaling 1/(2*pi*f_x)^2, which appears to assume that the Laplacian term dominates the Caputo term in (3.6); this is not justified for all alpha and f_x. The paper should either derive these relations from (3.6) or validate them empirically across the parameter range used in Tables 5 and 6.
  5. [Title and Abstract] The title and abstract claim 'multiprecision computing,' but the manuscript never uses or discusses multiprecision arithmetic (e.g., double vs. extended precision or mixed precision). All experiments are described with standard optimizers and default floating-point settings. The contribution as stated in the title is therefore not supported by the content; either implement and benchmark multiprecision computations or drop the term.
minor comments (4)
  1. [Abstract and Section 4.2] There are typos throughout: 'nouniform' should be 'nonuniform', 'T raining' should be 'Training', 'W eight' should be 'Weight', 'F orward propagation' should be 'Forward propagation', and 're´ gime' should be 'regime'.
  2. [Section 4.2] The text says 'for identical grid points' when comparing uniform and nonuniform schemes, but the first and second stages use different resolutions (10x10 and 40x40); please clarify which grid is used for the reported comparisons.
  3. [General] No code or data are provided, which hampers reproduction of the numerical claims; please include a reproducibility statement or a link to the implementation.
  4. [Section 4.2, Table 6] The discussion explains the non-uniform grid's poor performance for small alpha by 'round-off errors,' but no numerical evidence (e.g., condition numbers or precision tests) is given for this explanation.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the two-stage correction solves the PDE residual, not the target solution; only a minor, non-load-bearing self-citation is present.

full rationale

The central derivation, Eq. (3.4)-(3.6), constructs the second stage from the residual r(x,u0)=D_t^alpha u0 - Delta u0 - f, so the correction network u1 is trained to satisfy the physical equation, not fitted to the known exact solution. The amplitude-frequency estimates (2.7)-(2.8) are heuristic parameter choices based on the residual's spectral content; they do not define the reported error in terms of the target. The method's numerical claim (10^-7 to 10^-8) is an empirical result, not a tautology. The paper's only self-citation, reference [20] by X. Xu and M. Chen, appears in the bibliography and is not invoked in the derivation, so it is not load-bearing. A separate concern is that the reported accuracy appears hard to reconcile with the O(tau^{2-alpha}) discretization error of the L1 scheme (2.13) on the stated 40x40 grid; however, that is an internal-consistency or validation issue, not a circularity. No step in the paper reduces, by its own equations or by self-citation, to its own inputs.

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

The central claim rests on an unproved frequency-amplitude relation for fractional derivatives, a coarse L1 discretization whose error is not reconciled with the claimed accuracy, and the assumption that multistage error correction transfers from integer to fractional order.

free parameters (5)
  • dominant frequency f_d
    Used to set the scale of the second-stage network, estimated from the first-stage residual rather than from the exact solution.
  • residual amplitude epsilon_r1
    Computed as RMS of the first-stage residual and enters Eq. (2.8) to determine the error amplitude epsilon_1.
  • scale factors a_i
    Chosen based on the dominant spatial frequency f_x; the selection rule in Section 3.3 is heuristic.
  • grid grading parameter r = r=1 and r=(2-alpha)/alpha
    User-selected; the paper tests these two values in Tables 5 and 6.
  • loss weight w = 0.5
    Set to equal weight for data and physics losses; the paper states this maintains stability without further justification.
assumptions (4)
  • standard math Caputo fractional derivative definition (2.9) is the correct model for subdiffusion.
    Standard definition used throughout the paper.
  • domain assumption The L1 discretization (2.13) approximates the Caputo derivative with negligible error relative to the claimed 10^-7 accuracy.
    The paper relies on L1 discretization in the loss but does not account for its truncation error, which is O(tau^(2-alpha)).
  • ad hoc to paper For zero-initial-condition functions, the Caputo derivative preserves the dominant frequency and scales amplitude by (2*pi*f_d)^alpha.
    Stated in Section 3.1 without proof or citation; load-bearing for the second-stage network configuration.
  • domain assumption The multistage correction structure (3.6) and the amplitude estimate (2.8) carry over from integer-order multistage PINNs to fractional operators.
    The paper extends the method of Wang and Lai [19] to fractional PDEs without a convergence analysis.

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Cite this review

Pith. "Pith review of Multiprecision computing for multistage fractional physics-informed neural networks." pith.science (2026). https://pith.science/paper/U33FCDX3

@misc{pith2026250522377,
  author       = {Pith},
  title        = {Pith review of: Multiprecision computing for multistage fractional physics-informed neural networks},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/U33FCDX3}},
  note         = {Machine review of arXiv:2505.22377}
}
abstract

Fractional physics-informed neural networks (fPINNs) have been successfully introduced in [Pang, Lu and Karniadakis, SIAM J. Sci. Comput. 41 (2019) A2603-A2626], which observe relative errors of $10^{-3} \, \sim \, 10^{-4}$ for the subdiffusion equations. However their high-precision (multiprecision) numerical solution remains challenging, due to the limited regularity of the subdiffusion model caused by the nonlocal operator. To fill in the gap, we present the multistage fPINNs based on traditional multistage PINNs [Wang and Lai, J. Comput. Phys. 504 (2024) 112865]. Numerical experiments show that the relative errors improve to $10^{-7} \, \sim \, 10^{-8}$ for the subdiffusion equations on uniform or nouniform meshes.

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Reference graph

Works this paper leans on

21 extracted references · 14 canonical work pages

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