Reviewer3 (AI review)

October 30, 2026

AI Review (reviewer3.com)

Overview

Region-Resolved Multi-Observable Assessment Protocol

Contribution

The authors establish a matched evaluation framework that separately quantifies velocity, gauge-aligned pressure, and sampled vorticity discrepancies of physics-informed neural networks across both strong upper circulation and weak floor recirculation in a deep cavity.

Quantification of Regional PINN Workflow Trade-Offs

Contribution

This study demonstrates that accumulating regional collocation sampling, transferred initialization, and residual scaling substantially reduces interior velocity and pressure discrepancies, while revealing that solver rankings vary across specific spatial regions and physical observables.

Comments (6)

1. Mismatch in pressure mean-centering symbol between Equation (31) and Figure 23 — Editorial

. The symbol p x ( h ) is the horizontal mean pressure at that height; its subscript x identifies the averaging direction rather than a derivative. The upper pressure view and full-depth vorticity

Section 5.5 introduces the symbol $\overline{p}_x(h)$ in prose and Equation (31) to represent the horizontal mean pressure at height $h$. However, Figure 23 on the preceding page plots the pressure variation about each horizontal cut's mean using the different notation $\langle p \rangle_x$ in the y-axis label $[p - \langle p \rangle_x]/(\rho U^2)$ without connecting the two symbols.

2. Heterogeneous neural evaluation procedures across Reynolds series — Major

. The retained Re = 100 arrays already use this grid. For Re = 250 and 500, we evaluate the completed neural weights in double precision on that grid; for Re = 1000 , we bilinearly sample the retained neural arrays. The CFD fields use the same bilinear sampling procedure. Each

In the common Reynolds-number series, Re = 250 and 500 are evaluated directly from the continuous neural weights in double precision on the 121 x 605 grid, whereas Re = 1000 is obtained by bilinearly interpolating pre-saved neural arrays onto that grid. Because bilinear interpolation introduces an O(dx^2) smoothing error that directly alters second-order finite-difference vorticity and small-amplitude floor fields, the Re = 1000 entry in the common series is subject to a postprocessing interpolation error that is absent from Re = 250 and 500.

3. Diagnostic conflates neural derivative error with CFD resampling error — Minor

The evaluation grid also affects a derivative comparison. As a diagnostic, we retain every second grid point in each direction and recompute vorticity from the coarser velocities. Relative to the fine-grid vorticity at those points, the interior change is 1.49%, 2.87%, 4.57% and 6.71% for CFD, and 8.82%, 23 7.99%, 11.09% and 17.54% for PINN in increasing Reynolds-number order. This is a postprocessing-grid

The grid-coarsening test computes finite-difference vorticity from velocity fields that were already bilinearly resampled onto the 121 x 605 grid rather than directly from the continuous neural network (via automatic differentiation) and the native polynomial basis in Nektar++. Without comparing the finite-difference vorticity against the exact analytical vorticity of each representation, it is unclear how much of the reported 8.82%–17.54% coarsening change reflects high-frequency neural oscillations versus truncation of the finite-difference stencil.

4. Unspecified training budget for prior floor-refined source model — Major

workflows use the balanced mixture in their final phase. The refined initialization comes from an earlier Re = 1000 model trained with 50% full-domain, 30% floor-strip and 20% lower-corner points. Control and soft-residual cases instead start from the same specified Re = 500 field. The half-envelope workflow includes additional

While the text notes that the refined workflow starts from a prior floor-refined Re = 1000 model, the iteration count, optimizer sequence, and residual scaling of that precursor Re = 1000 training phase are never specified. Providing the exact optimization history and computational budget of this precursor run is necessary so readers can gauge how much additional Re = 1000 optimization effort separates the refined workflow from the control.

5. Floor pressure RMS ranking driven by global offset — Major

floor velocity RMS discrepancy. The control has the smaller sampled floor-vorticity RMS discrepancy, 2 . 056 × 10 − 9 U / W compared with 3 . 025 × 10 − 9 U / W for the refined workflow, despite the latter’s smaller floor velocity RMS. The ordering depends on whether field amplitudes or their spatial derivatives are examined. Pressure is nearly constant across the lowest horizontal cuts. Figure 23 therefore shows pressure relative

In Section 5.1 (Table 8), the authors show that the globally gauge-aligned floor pressure RMS discrepancy is completely dominated by the constant regional mean offset b_f accumulated from the upper cavity rather than local floor pressure accuracy. However, in the Re = 1000 workflow comparison (Table 10), floor pressure rankings are based solely on this globally gauge-aligned metric without reporting the local-gauge shape RMS (e^{shape}_{p,f}), meaning the claim that the control retains the best floor pressure agreement may simply reflect fortuitous cancellation in its upper-cavity mean pressure rather than superior floor pressure reconstruction.

6. Missing pressure and vorticity metrics for half-envelope variant — Major

. Reducing the envelope decay gives 4 . 908 × 10 − 3 U in the interior, 3 . 854 × 10 − 2 U in the upper-left region and 1 . 523 × 10 − 9 U at the

The half-envelope workflow is included in the Re = 1000 velocity comparison (Table 9) to separate field representation from residual emphasis, but it is omitted without explanation from the pressure and vorticity comparison in Table 10 and Figures 17–24. Including its pressure and vorticity discrepancies in Table 10 is needed to complete the multi-observable ranking across all four workflows.

XSci (AI review)

October 30, 2026

Summary

The manuscript assesses data-free physics-informed neural networks (PINNs) for steady, two-dimensional incompressible flow in a lid-driven cavity of depth-to-width ratio five at Re = 100, 250, 500 and 1000. An inherited hard-constrained streamfunction–pressure representation with a depth envelope is combined with a prescribed regional collocation mixture, transferred initialization and two scale variants, and trained by a staged SOAP then self-scaled Broyden procedure. Independent Nektar++ spectral/hp calculations of the same Gaussian-regularized-lid problem provide the reference. On a common 121 × 605 grid the interior velocity RMS discrepancies are 6.542 × 10⁻³U to 9.541 × 10⁻³U; in a separate 161 × 801 assessment at Re = 1000 a refined workflow lowers that discrepancy from 7.181 × 10⁻³U to 3.394 × 10⁻³U. The authors report that configuration rankings change with region and observable.

Contribution and Significance

The core contribution is a verification protocol rather than a new method: the PINN and the reference solve the identical boundary-value problem, including the same lid regularization (Eq. 4), and the discrepancies are decomposed by region (interior, upper-left, floor) and by observable (velocity, gauge-aligned pressure, sampled vorticity, with a common differentiation operator). Applied to a deep cavity this yields the paper's most useful result: the global error norms of a cavity PINN are almost entirely a lid-junction phenomenon. The strip h/W > 4.35 holds 12.69% of the interior samples and carries 92.41–99.91% of the squared velocity discrepancy, and the two 0.1W disks at the lid corners hold 0.168% of the samples yet carry 95.42–99.74% of the squared vorticity discrepancy (Sec. 5.1, Figs. 12–14). That is transferable: a single cavity-wide error figure, routinely reported in this literature, says little about where such a solution is actually right.

What is new relative to the closest prior work is narrower than the framing suggests. The representation, envelope, corner mask and optimizer stack are inherited from McDevitt et al. [39] and, as Sec. 3.1 states, were supplied by one of the authors; relative to Takao and Ii [41] the advance is a deeper geometry without interior data, and relative to Wang et al. [33] it is prescribed rather than adaptive sampling. After the authors' own disclaimers in Secs. 3.4.1, 5.2 and 6.1, the sampler itself is not shown to do anything; what remains is a case study plus a protocol.

Strengths

• The matched boundary-value problem removes the lid-treatment confound that the reviews cited as [43, 44] identify, and the reference is checked with polynomial order, mesh and time step varied separately (Table 4).
• The error-localization statistics (Sec. 5.1, Figs. 12–14) are precise and more informative than the aggregate norms they explain.
• The provenance discipline is unusual and welcome: Tables 2, 3, 5 and 6 record which state, grid and snapshot produced each number, and Sec. 5.2 states plainly that control versus refined compares accumulated workflows.
• The hard-constrained representation with the depth envelope (Eqs. 9–17) represents a field spanning seven decades of amplitude, as the ratio A_u(4.7W)/A_u(0.3W) ≈ 1.51 × 10⁷ in Table 11 shows.
• The floor pressure decomposition (Eq. 29, Table 8) separates a constant gauge offset from local shape: at Re = 100, an offset of 1.033 × 10⁻⁴ρU² against a shape RMS of 1.152 × 10⁻¹⁰ρU².

Recommendation

Major Revision for Computers & Fluids. The protocol and the error-localization result belong in the journal, but three things need closing: the reported discrepancies are never normalized, so the paper's central question about regional accuracy is not answered; the reference has not been shown to be accurate in the weak floor flow that motivates the study; and the headline improvement compares runs with unequal cumulative training on a single seed. Each is addressable within the present approach.

Major Concerns

1. Absolute lid-scaled metrics do not establish regional accuracy (Sec. 4.2, Tables 9 and 11). Every discrepancy is reported on the lid scale, and Sec. 4.2 explicitly declines "division by a reference-field RMS". The authors warn against misreading this, but a warning is not the measurement, and their own numbers invert the impression the abstract creates. Against Table 11, the Table 9 upper-left velocity discrepancies of 6.868 × 10⁻²U (control) and 2.819 × 10⁻²U (refined) are roughly 24% and 10% of A_u(4.7W)/U = 0.2856, whereas the floor discrepancies of 5.506 × 10⁻¹⁰U and 3.612 × 10⁻¹⁰U are roughly 3% and 2% of A_u(0.3W)/U = 1.889 × 10⁻⁸. On the lid scale the floor looks eight orders of magnitude better than the upper cavity; in local terms it is the better-resolved region. These ratios are indicative only, since A_u is a full-width average at one height, but the stated subject of the paper is regional accuracy: please report discrepancies normalized by the local reference amplitude, or a per-height relative profile alongside Fig. 24, and revise the abstract and conclusions accordingly.

2. The reference is not validated where the motivating claim lies (Table 4, Sec. 4.1.1, Sec. 6.3). The sensitivity checks — 0.053769% for polynomial refinement, 0.065100% for mesh refinement, 0.000312% for time-step halving and 0.002886% between t ≈ 160 and 170 — are global relative sampled vector norms. By the paper's own accounting the upper 0.65W occupies 13% of the area and carries 77.37% of the sampled integral of u² + v² (Sec. 5.5), so these numbers are set by the upper flow and bound nothing at the floor, where amplitudes are about 1.5 × 10⁷ times smaller; a global difference of 5 × 10⁻⁴ is consistent with an order-unity relative error there. The floor entries of Tables 7 and 9 may therefore measure the reference's own discretization content rather than the PINN's error.

   The temporal situation compounds this: the reference is a transient snapshot (Eqs. 23–25) at tU/W = 317, compared against a steady PINN. The viscous relaxation time of a W-scale vortex at Re = 1000 is of order 10³ W/U and the deep vortices are driven through a chain of successively weaker ones, so the floor flow is unlikely to have equilibrated by then. Please recompute the Table 4 differences and the t = 160 → 170 drift in the floor-region norm of Table 7; if the floor amplitudes are not stationary at the comparison instant, present the floor results as bounds on agreement rather than measurements.

3. The headline improvement is confounded and unreplicated (Secs. 3.4.1, 5.2, Appendix B). The refined workflow starts from a prior floor-refined Re = 1000 model while the control starts from Re = 500, so it carries strictly more cumulative optimization, and Appendix B states that several histories were still decreasing at the retained stopping point. The reduction from 7.181 × 10⁻³U to 3.394 × 10⁻³U is therefore equally consistent with the refined model simply having been trained longer. The authors say as much, but the disclaimer leaves the result uninterpretable where one further run would settle it: continue the control to a budget matched to the refined workflow and report the same table. All runs also use seed 12345 without repetition, so the spread among refined, half envelope and soft residual (3.394, 4.908 and 5.059 × 10⁻³U) cannot be distinguished from run-to-run scatter. As the stated central result of Sec. 6.1 is that rankings change between regions and observables — including the control's floor advantage, 2.056 × 10⁻⁹ against 3.025 × 10⁻⁹ U/W — several seeds for the control and refined pair are needed before those orderings can be asserted.

Minor Comments

1. The expression following Eq. (28) defines E_ω(R) = (U/W)⟨e_u²⟩^{1/2}; this should be ⟨e_ω²⟩^{1/2}.
2. Reference [23] is cited in Sec. 1 ("[21–23, 34]") but is absent from the list, in which [25] appears twice (Krishnapriyan et al.; Sukumar and Srivastava).
3. Twelve listed references are never cited in the text ([15], [16], [24], [27], [29], [31], [32], [35], [36], [37], [38], [42]); SOAP [37] and the self-scaled Broyden studies [35, 36] in particular should be cited in Sec. 3.4.
4. The contour-level specifications in the captions of Figs. 7, 9 and 11 are garbled ("Levels are 1.4844 × 10⁻¹² 10⁵/6, j = 0, ..., 6") and the j-dependence is lost.
5. In Table 10 the refined upper-left and refined floor vorticity entries share the mantissa 3.025 (3.025 and 3.025 × 10⁻⁹ U/W); please confirm this is not a transcription error.
6. Please state whether the reported norms are converged with respect to the comparison grid, since the same control field gives 6.542 × 10⁻³U on 121 × 605 and 7.181 × 10⁻³U on 161 × 801.
7. In Sec. 5.1 the Re = 250 and 500 networks are evaluated directly on the comparison grid while the Re = 1000 arrays are bilinearly resampled, so one case carries an extra interpolation error.
8. The Reynolds-number series combines snapshots at tU/W = 120, 160, 160 and 317 with two neural formulations, so the four-Re ranges in the abstract should be labelled an inventory of retained states rather than a Reynolds-number result.
9. No computational cost is reported for the PINN workflows or the Nektar++ reference, so readers cannot judge what the neural solution offers.
10. The 0.02W wall band is excluded everywhere, so wall shear and the integrated lid force — the quantities most affected by the near-junction discrepancies — are never assessed.
11. Only one architecture (six tanh layers of width 64) and one lid smoothing length δ_x = √0.001 W are used; sensitivity to each would show whether the near-lid limitation is one of capacity, optimization or boundary-layer resolution.
12. Appendix A states that the repository is "not a complete release of the campaign training states"; releasing the retained weights and the Nektar++ session files would make the comparisons independently checkable.