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Insufficient specification of the PINN, PDE setup, and latent extraction makes the results hard to interpret and impossible to reproduce (Sec. 1, Sec. 2.1–2.2). The manuscript does not state the explicit Burgers equation form (including viscosity/parameters), domain, IC/BC, and whether “2D” refers to one space + time $(x,t)$ or two spatial dimensions. The PINN architecture and training details are also missing (layer sizes, activations, normalization, where the 10D latent is taken—pre/post nonlinearity, loss terms/weights, collocation strategy, optimizer/schedule, seeds/runs). The provenance of the provided NumPy array $(100,100,12)$ is unclear (who trained it, which checkpoint, whether multiple runs were analyzed).
Recommendation: Expand Sec. 2.1–2.2 (or add a dedicated subsection) to fully specify: (i) the exact PDE and parameters, plus IC/BC and domain, explicitly clarifying that the grid is spatio-temporal $(x,t)$ if that is the case; (ii) the PINN architecture (depth/width, activations, parameter count, where the 10D latent is extracted and whether it is before/after activation); (iii) the training objective (data terms, PDE residual, IC/BC terms) and weights, sampling strategy, optimizer and schedule, epochs/batches, and number of seeds; (iv) basic performance metrics versus a reference solution (e.g., $L^2$/NRMSE of $u(x,t)$ and/or PDE residual statistics). Add a short provenance/reproducibility statement describing how the $(100,100,12)$ tensor was produced and whether code/data will be released (Sec. 4).
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Wavelet analysis is currently internally inconsistent and under-specified in ways that can materially change all reported DWT results (Sec. 2.4 vs Sec. 3.1). (a) Wavelet family mismatch: Methods say ‘db1’ (Haar) (Sec. 2.4.1, p.3) while Results state ‘sym2’ (Sec. 3.1, p.5). (b) Energy definition mismatch: Methods include detail energies plus coarsest approximation energy (Sec. 2.4.2) while Results define per-level energy as detail-only sums (Sec. 3.1.2). (c) A $100\times100$ grid is non-dyadic, so DWT coefficient statistics depend on padding/extension mode (e.g., symmetric/periodization/zero padding) and the feasible maximum level; these choices are not documented. (d) It is unclear how subbands (H/V/D) are aggregated, whether energies are normalized by coefficient count per level, and what exactly the reported reconstruction NRMSE signifies (perfect reconstruction is expected for standard DWT implementations and is not, by itself, evidence of representation quality).
Recommendation: In Sec. 2.4.1–2.4.2, explicitly specify: the exact wavelet used in all reported figures/tables (or separate results by wavelet if multiple were tried), the PyWavelets (or equivalent) extension mode, the exact number of levels used and how it was determined for $100\times100$, and the precise definition of per-level energy (detail-only vs detail+approx, and how the approximation term is handled). State how H/V/D subbands are combined and whether energies are normalized (e.g., by coefficient counts) before cross-level or cross-component comparisons. If feasible, add a sensitivity check by repeating key metrics (energy fractions, kurtosis, fitted exponents) on a dyadic-cropped grid (e.g., $96\times96$) and/or with periodization to assess padding/boundary effects. Reframe or remove the reconstruction NRMSE as an interpretability result; if kept, clarify it is a numerical sanity check only (Sec. 3.1.1).
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Power-law/self-affinity/fractality claims are overstated relative to the evidence and the limited scale range, and the fitting procedure lacks diagnostics (Sec. 3.1.4, Sec. 3.3, Conclusion). Exponents are inferred from linear fits across only $\sim 5$ levels, without reporting goodness-of-fit ($R^2$), residuals, standard errors/confidence intervals, or sensitivity to the chosen fit range (coarsest/finest levels are often contaminated by finite-size and discretization effects). In addition, the slope-to-exponent conversion depends on the logarithm base, but the manuscript does not specify whether ‘log’ is $\ln$ or $\log_{10}$ (Sec. 3.1.4).
Recommendation: In Sec. 3.1.4, report for each component: the exact fit range (which levels), $R^2$ (or another fit metric), residual plots or a summary of residual structure, and uncertainty estimates (standard errors or confidence intervals) for the fitted slopes/exponents. Explicitly state the log base used in plots/fits and use the matching conversion $\alpha_i = m_i/\log_b(2)$ (Sec. 3.1.4). Add a short limitation statement that with $5$ levels the results indicate at most “approximate scaling over a narrow range,” not definitive fractality; adjust language in Sec. 3.3/Conclusion accordingly.
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Geometric analysis risks conceptual over-interpretation and may be numerically fragile due to second-derivative noise and boundary handling (Sec. 2.5, Sec. 3.2). The computed “Ricci scalar” is the intrinsic curvature of the graph surface $z=L_i(x,t)\subset\mathbb{R}^3$, not curvature of a learned 10D latent manifold with a model-induced metric; the current phrasing can mislead readers about what is being measured (Sec. 2.5.3, Sec. 3.2). Numerically, curvature depends directly on $L_{xx}, L_{tt}, L_{xt}$, yet the derivative validation reports $\sim 15.7\%$ relative error for $f_{tt}$ on a smooth test function (Sec. 3.2.1), and boundary treatment (cropping vs one-sided differences vs keeping edges) is not described. These factors can substantially affect Ricci magnitude and possibly spatial patterns (Sec. 3.2.2–3.2.3).
Recommendation: Clarify terminology throughout Sec. 2.5 and Sec. 3.2: describe results as “Gaussian curvature / Ricci scalar of the graph surfaces $z=L_i(x,t)$” unless a latent-space metric is explicitly introduced. Strengthen numerical robustness in Sec. 2.5.1 and Sec. 3.2.1 by: (i) reporting errors not only for derivatives but also for $K$ and $R$ on the analytic test surface; (ii) specifying and justifying boundary handling and which points enter Table 2 statistics; (iii) adding a sensitivity analysis (e.g., higher-order finite differences, mild smoothing prior to differentiation, and/or varying grid spacing if available) and reporting how Table 2 statistics and representative Ricci maps change. If possible, compute derivatives via automatic differentiation from the original PINN rather than finite differences on a saved grid, and compare.
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Lack of baselines and limited scope (single model, single layer, single PDE setting) prevents determining what is PINN-specific versus generic, and limits the paper’s broader interpretability claims (Sec. 2.1–2.3, Sec. 3, Sec. 4). All findings are based on one 10D latent layer from one pre-trained network; no comparisons are provided to (i) a supervised network trained on the same $u(x,t)$ data, (ii) an untrained network, (iii) alternative PINN settings (loss weights, viscosity), or (iv) other layers/latent dimensions.
Recommendation: Add at least one baseline in Sec. 3 (new subsection): run the same wavelet + curvature pipeline on (i) a comparable supervised MLP trained on solution snapshots (no PDE residual) and/or (ii) an untrained network with the same architecture. Compare energy distributions, kurtosis, scaling exponents, and Ricci statistics. If feasible, analyze at least one additional layer (earlier vs later) or one additional PDE configuration (e.g., different viscosity) to assess stability. If new experiments are not possible, explicitly reframe the work in Sec. 4 as exploratory and avoid PINN-specific generalizations.
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Connections to physics, performance, and the paper’s key interpretations (“shock encoding”, “precisely the right properties”) are asserted but not directly tested (Sec. 3.3, Conclusion), and the hypothesized wavelet–curvature linkage is not quantified. The manuscript does not relate latent fine-scale energy or high $|R|$ regions to the actual Burgers solution $u(x,t)$, its gradients (e.g., $|u_x|$), or PDE residual/error maps, nor does it quantify whether components with more negative $\alpha_i$ also have larger curvature variance (claims in Sec. 3.3 appear only qualitative and potentially inconsistent with Table 2 ordering).
Recommendation: In Sec. 3.3, add direct, testable linkages: (i) plot $u(x,t)$ and a shock/steepness proxy such as $\left|\partial u/\partial x\right|$ (or PDE residual) alongside representative $L_i$, fine-scale wavelet magnitude maps, and $|R_i|$; (ii) compute spatial correlations/overlap metrics between high-gradient regions and high fine-scale energy / high $|R|$ regions; (iii) compute across-component correlations such as $\text{corr}(\alpha_i, \mathrm{Var}(R_i))$ and report uncertainty given $n=10$. If these additions are out of scope, soften claims in Sec. 3.3 and the Conclusion to “consistent with” and present shock/turbulence interpretations as hypotheses for future work.