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REVIEW 5 major objections 4 minor 61 references

Multi-level datasets training method in Physics-Informed Neural Networks

T0 review · 5 major / 4 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read Multi-level dataset cycling cuts PINN errors by 30–60 percent and reaches Re=5000.

desk verdict A simple, plausible dataset-switching schedule for PINNs with real but modest epoch-matched gains; the larger 30–60% claims are not yet causally attributed to the method. read the letter →

arxiv 2504.21328 v1 pith:PEMGSP6W submitted 2025-04-30 cs.LG cs.CEphysics.flu-dyn

Yao-Hsuan Tsai , Hsiao-Tung Juan , Pao-Hsiung Chiu , Chao-An Lin This is my paper · ORCID
classification cs.LGcs.CEphysics.flu-dyn MSC 68T0765M9976D05
keywords physics-informedneuralnetworksmulti-leveldatasettrainingspectralbiasmultigrid-inspiredlid-drivencavityflowcurriculumtransferlearning
topics Navier-Stokes Regularity
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 tries to establish that Physics-Informed Neural Networks can be made substantially more accurate on stiff and high-frequency PDEs by a training-level trick: instead of feeding one fixed set of collocation points (locations where the PDE residual is evaluated), the network is trained successively on several datasets of different point densities, with the order and pace controlled by multigrid-inspired V- and W-cycles. The authors' numerical evidence covers a 1D high-frequency ODE, a 2D convection-diffusion benchmark, and the steady lid-driven cavity flow, where the switching method reduces errors by 30–60 percent relative to a single-dataset baseline and, combined with curriculum training over increasing Reynolds numbers, produces a usable solution at $Re=5000$. The payoff of the claim is practical: it offers a way to improve PINN accuracy without restructuring the network, tuning loss weights, or hand-crafting features, using only the ordering and density of the training collocations.

What carries the argument

The central object is a multi-level dataset training schedule: a fixed set of collocation datasets $D_1, D_2, D_3$ with decreasing numbers of points (e.g., 6670, 1770, 435 for the convection-diffusion case; 6565, 1685, 445 for the cavity at $Re=400$), cycled in sequences such as V-cycle ($D_1\to D_2\to D_3\to D_2\to D_1$) or W-cycle. The mechanism is that each level is trained with its own Adam optimizer and learning-rate scheduler and then connected to the next level through transfer learning, so that the network sees the same architecture but different collocation densities. For the cavity problem, the composite loss weights are fixed by the dimensional-analysis DWroot rule rather than by adaptive weighting. The intended function of the schedule is multigrid-like frequency separation: small datasets are supposed to teach low-frequency error components and large datasets high-frequency ones, so that alternation removes both kinds of error.

What would settle it

Hold total epochs and the sequence of learning-rate schedules fixed but replace the multi-dataset V-cycle with a single collocation set; if accuracy matches the multi-dataset run, dataset switching is not the cause. Then check the network's prediction spectrum after each level: the mechanism predicts low frequencies dominate on the smallest set and high frequencies appear on the largest.

Watch

Extended reading notes

Core claim

The paper's central claim is that spectral bias in PINN training can be counteracted by cycling through collocation datasets of different sizes, treating each switch as a transfer-learning step with its own optimizer and learning-rate scheduler. On the 1D equation $f(x)=\sin(10x)$ in $[0,2\pi]$, a two-level schedule (64 points, then 32, then 64) lowers the relative $L^2$ test error from $1.15\times10^{-1}$ to $4.16\times10^{-2}$ in the same 30,000 epochs. On the 2D convection-diffusion benchmark, a three-level V-cycle (6670, 1770, 435, 1770, 6670 points) improves the relative $L^2$ error from $2.54\times10^{-3}$ to $2.10\times10^{-3}$ at fixed 50,000-epoch cost. For the steady lid-driven cavity at $Re=400$, random V-cycling over three collocation sets reduces the best mean absolute error by 64 percent versus a single random dataset, and with curriculum training through $Re=400,1000,2000,3200,5000$ the model reaches about $7\times10^{-2}$ mean absolute error at $Re=5000$. All runs keep the same MLP with a sine first layer and Swish hidden layers; only the dataset sequence and per-level schedules change.

Load-bearing premise

The load-bearing premise is that training on a smaller collocation set exposes the network to lower-frequency components and that switching dataset sizes—rather than the extra training time, the per-level schedulers, or transfer-learning warm starts—is what produces the 30–60 percent gains.

Editorial extensions

If this is right

  • Because the method only changes the collocation schedule, it can be added to any existing PINN architecture without new layers, loss terms, or feature engineering.
  • Layered onto curriculum training, it extends PINN solvability to Reynolds numbers like $Re=5000$ where single-dataset training does not converge, and with fewer parameter updates than the published comparison methods.
  • The reported 30–60 percent error reductions suggest the scheme can help on any PDE with multiple frequency scales, including sharp-front transport problems and oscillatory solutions.
  • Per-level learning-rate schedulers become the main tuning knob, giving practitioners a concrete substitute for hand-tuning a single global schedule.

Reading between the lines

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

  • A control that replays the same per-level optimizer and scheduler steps on a single dataset would determine whether dataset switching, rather than the added optimization structure, causes the gains; the paper does not report such a control.
  • If the multigrid analogy is exact, the optimal number of levels and the collocation ratios could be derived from the problem's frequency content or from the network's NTK spectrum, turning the schedule into an automatic design.
  • The approach should combine with residual-based adaptive sampling—using the coarse levels to locate regions needing refinement—though the paper does not test this combination.
  • The reported improvement is relative to single-dataset baselines with hand-picked schedulers, so in other applications the practical gain may depend on how well the baseline is already tuned.
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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 multi-level dataset training procedure for Physics-Informed Neural Networks (PINNs), inspired by multigrid ideas. Instead of training on a single collocation dataset, the method alternates between datasets of different collocation densities, each equipped with its own optimizer and learning-rate scheduler, using a transfer-learning-style handoff. The authors report accuracy gains on a 1D high-frequency ODE, the Smith-Hutton convection-diffusion problem, and steady lid-driven cavity flows at Re=400, 1000, and 5000, claiming 30--60% accuracy improvements and attributing the gains to mitigating spectral bias through exposure to different frequency scales. The paper also combines the approach with curriculum training for the higher-Reynolds-number cavity cases and compares against published PINN results and CFD references.

Significance. If the claimed gains are causal, the method is attractive because it is a training-level modification that leaves the network architecture and loss formulation unchanged, and it is compatible with transfer and curriculum learning. The paper's use of external benchmarks (Ghia et al., Smith-Hutton) and the detailed per-level training tables in the appendix are positive features. However, the current evidence does not establish the mechanism or the headline improvement magnitude: the central cavity comparison is confounded with total epoch count and schedule changes, the ODE benchmark contains an incorrect derivative that invalidates the experiment as written, run-to-run variability is not reported, and the spectral-bias mechanism is asserted rather than measured. The Smith-Hutton experiment provides some controlled evidence (equal total epochs, equal per-level optimizers and schedulers), but it is a single run and shows a modest improvement. With corrected benchmarks and controlled ablation experiments, the contribution would be of genuine interest to the PINN community.

major comments (5)
  1. [§III.B, Eq. (9)] The derivative in Eq. (9) is incorrect: d/dx sin(10x) = 10 cos(10x), not 10 sin(10x). As written, the residual equation is inconsistent with the stated exact solution f(x)=sin(10x), and the boundary conditions f(0)=f(2π)=0 do not determine the stated solution. The 64% test-error improvement in Table IV is therefore not a valid validation of the method until Eq. (9), the corresponding loss definition, and the reported experiment are corrected and rerun.
  2. [§III.D, Table IX] The headline Re=400 comparison is not controlled. The 'Random, 1 level' baseline runs 104,000 epochs, while 'Random, V-cycle' runs 163,000 epochs and the other variants run 162,000--184,000 epochs; additionally, the per-level schedulers in Table XV differ across levels. The claimed 64.1% error decrease could therefore be due to extra training, learning-rate cycling, or the combined schedule rather than to dataset switching. An epoch-matched baseline with the same optimizer/scheduler trajectory, or at least a matched total number of parameter updates, is required to attribute the gain to the proposed mechanism.
  3. [§II.A and §IV] The mechanistic premise that smaller collocation sets expose the network to lower-frequency components is never tested. Section II.A asserts this spectral relationship, but Section IV explicitly defers 'numerical analyses on training frequencies' to future work. Without frequency-content measurements or a mechanism-based ablation, the analogy to multigrid remains a hypothesis rather than a demonstrated explanation of the observed accuracy gains.
  4. [Abstract and §II.A] The abstract claims the method avoids time spent 'fine-tuning of neural network structures, loss weights as well as hyperparameters,' but the experiments hand-select per-level optimizers, learning rates, scheduler factors, stopping criteria, and network widths (Tables XIII--XVII), and different architectures are used for different problems. This overstates the turnkey character of the proposed approach.
  5. [§III.D, Table XII] The Re=5000 result is not evidence for the multi-dataset procedure: it compares the proposed pipeline against other published methods, but it does not include a same-architecture, same-epoch ablation trained without the V-cycle/curriculum schedule. The contribution of dataset switching at high Reynolds number is therefore untested.
minor comments (4)
  1. [§III.C, Eq. (11)] Eq. (11a) appears to contain a typo in the diffusion term: '∂²φ/∂xy' should be '∂²φ/∂x² + ∂²φ/∂y²'. Also, 'Pelect number' should be 'Péclet number'.
  2. [§III.C, Table VI] The text says the proposed method improves accuracy by 20%, but the relative L2 reduction in Table VI is (2.54e-3 - 2.10e-3)/2.54e-3 ≈ 17%; the reported percentage should be reconciled.
  3. [§III.D, Table IX] The column 'Error decrease' is reported with negative percentages, which is confusing because a decrease should be a positive number; the formula also averages the relative reductions of u and v without justification. Please clarify the sign convention and the aggregation.
  4. [§III.B--D] All experiments appear to be single runs without seed variability. Reporting mean and standard deviation over at least three seeds, particularly for the ODE and Smith-Hutton cases, would calibrate the magnitude of the reported gains.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: benchmark comparisons are external, and the self-cited DWroot weighting is an input, not a fitted prediction.

full rationale

The paper's central accuracy claims are validated against external references, not against quantities produced by the method itself. The 1D ODE and Smith-Hutton cases are compared with ground-truth/CFD reference solutions, the lid-driven cavity results are compared with Ghia et al.'s reference data, and Table XII compares the proposed model with independently published PINN results. No fitted parameter is renamed as a prediction: the dimensional-analysis weighting DWroot is adopted from a prior same-group thesis, but it is fully specified in Eqs. (17)-(18) of this paper and used as a fixed input for the cavity experiments; it is not inferred from the benchmark outputs, and the reported accuracy gains are not explained by that weighting alone. The multilevel dataset sequences (V-cycle, W-cycle) are design choices inspired by multigrid and MscaleDNN, not derived from the target results. The spectral-bias premise in Section II.A is an unmeasured design assumption, and the paper itself defers numerical analyses of training frequencies to future work; this is a rigor or attribution concern, not circularity, because no claim reduces by construction to its own input. The absence of epoch-matched controls means the 30-60% improvement may be confounded by longer training or per-level scheduler settings, but that is an alternative explanation for a benchmark result, not an equivalence between input and output. Accordingly, no circular step meeting the required evidentiary standard is present.

Assumptions & free parameters 5 free parameters · 3 assumptions · 0 invented entities

The central claim rests on the standard PINN loss-minimization setup, plus two paper-specific assumptions: the frequency-dataset analogy and the correctness of the 1D ODE equation. Many hyperparameters (dataset sizes, level durations, scheduler settings) are hand-chosen and effectively free parameters, though they are not fitted to the target outputs.

free parameters (5)
  • h length scale in DWroot = 1/81 for 6565 collocations; varies as 1/Round(sqrt(N))
    Sets the dimensional-analysis loss weights in Eq. 18; hand-chosen relation between h and collocation count, not derived or validated.
  • Dataset sizes per level = (16,64) ODE; (435,1770,6670) SH; (445,1685,6565) Re=400/1000; (965,3725,14645) Re=5000
    The number of collocation points per level is selected by hand for each problem; no rule for choosing level sizes is given.
  • Training epochs per level = e.g., 5000/10000/15000 for ODE; 10000 per level for SH V-cycle; 20000 per level for cavity
    Level durations are manually assigned in Tables III, V, VIII; the benefit may depend on these durations and no sensitivity analysis is provided.
  • Per-level scheduler hyperparameters = e.g., factor=0.65/0.8, patience=100 for ODE; factor=0.73/0.80/0.88 for cavity curriculum
    ReduceLROnPlateau/StepLR factors, patience and min_lr are set per level and per Reynolds case (Appendix), contradicting the no-fine-tuning claim.
  • Initial learning rates per level = 1e-3 ODE/cavity; 5e-3 SH; 1e-4/4e-5 in Re=5000 curriculum
    Learning rates differ across levels and problems; performance is likely sensitive to them.
assumptions (3)
  • domain assumption Minimizing the MSE-based PINN loss (Eq. 4-6) with Adam yields an approximate strong solution of the PDE on the training collocation points.
    Standard PINN assumption; the paper relies on it throughout without proving convergence for its specific architectures or problems.
  • ad hoc to paper Smaller collocation datasets expose the network to different (lower) frequency components, and switching between datasets during training alleviates spectral bias.
    Core premise of the method, asserted in Section II.A via analogy to multigrid and MscaleDNN; no measurement or ablation supports it. This is the load-bearing assumption identified as weakest_assumption.
  • ad hoc to paper The derivative of sin(10x) is 10 sin(10x) as written in Eq. 9.
    Eq. 9 states g(x)=10 sin(10x) for f(x)=sin(10x). The correct derivative is 10 cos(10x). The 1D benchmark is therefore inconsistent as written, and the reported 'test error' is computed against the true solution sin(10x) which the PDE residual does not encode.

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Cite this review

Pith. "Pith review of Multi-level datasets training method in Physics-Informed Neural Networks." pith.science (2026). https://pith.science/paper/PEMGSP6W

@misc{pith2026250421328,
  author       = {Pith},
  title        = {Pith review of: Multi-level datasets training method in Physics-Informed Neural Networks},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PEMGSP6W}},
  note         = {Machine review of arXiv:2504.21328}
}
read the original abstract

Physics-Informed Neural Networks have emerged as a promising methodology for solving PDEs, gaining significant attention in computer science and various physics-related fields. Despite being demonstrated the ability to incorporate the physics of laws for versatile applications, PINNs still struggle with the challenging problems which are stiff to be solved and/or have high-frequency components in the solutions, resulting in accuracy and convergence issues. It may not only increase computational costs, but also lead to accuracy loss or solution divergence. In this study, an alternative approach is proposed to mitigate the above-mentioned problems. Inspired by the multi-grid method in CFD community, the underlying idea of the current approach is to efficiently remove different frequency errors via training with different levels of training samples, resulting in a simpler way to improve the training accuracy without spending time in fine-tuning of neural network structures, loss weights as well as hyperparameters. To demonstrate the efficacy of current approach, we first investigate canonical 1D ODE with high-frequency component and 2D convection-diffusion equation with V-cycle training strategy. Finally, the current method is employed for the classical benchmark problem of steady Lid-driven cavity flows at different Reynolds numbers, to investigate the applicability and efficacy for the problem involved multiple modes of high and low frequency. By virtue of various training sequence modes, improvement through predictions lead to 30% to 60% accuracy improvement. We also investigate the synergies between current method and transfer learning techniques for more challenging problems (i.e., higher Re). From the present results, it also revealed that the current framework can produce good predictions even for the case of Re=5000, demonstrating the ability to solve complex high-frequency PDEs.

Figures

Figures reproduced from arXiv: 2504.21328 by the authors.

Figure 1
Figure 1. FIG. 1: Training result (blue dot), Training ground truth (solid line) and Absolute error (dashed [PITH_FULL_IMAGE:figures/full_fig_p012_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: Test result (blue dot), Test ground truth (solid line) and Absolute error (dashed line) after [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: Comparisons of training result (top) and absolute error (bottom) for different training [PITH_FULL_IMAGE:figures/full_fig_p015_3.png] view at source ↗
Figures from the paper (9 more)
Figure 4
Figure 4. Figure 4: FIG. 4: NN structure for Lid-driven cavity [PITH_FULL_IMAGE:figures/full_fig_p018_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5: (a)Re=400 Model training result [PITH_FULL_IMAGE:figures/full_fig_p020_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6: (a)Re=1000 Model training result [PITH_FULL_IMAGE:figures/full_fig_p020_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7: (a)Re=5000 Model training result [PITH_FULL_IMAGE:figures/full_fig_p021_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8: Gradients value graphs after training for 30,000 epochs [PITH_FULL_IMAGE:figures/full_fig_p029_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9: Loss history graphs after training for 30,000 epochs [PITH_FULL_IMAGE:figures/full_fig_p030_9.png]
Figure 10
Figure 10. Figure 10: FIG. 10: plot of training history for 2D Smith-Hutton problem with different training settings. [PITH_FULL_IMAGE:figures/full_fig_p031_10.png]
Figure 11
Figure 11. Figure 11: FIG. 11: (a)Loss history and (b)Error history for Re=400 (fig.5) [PITH_FULL_IMAGE:figures/full_fig_p032_11.png]
Figure 12
Figure 12. Figure 12: FIG. 12: (a)Loss history and (b)Error history for Re=1000 (fig.6) [PITH_FULL_IMAGE:figures/full_fig_p033_12.png]

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Pith tools

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