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The PDE/problem specification is not stated and the manuscript’s repeated claim of “2D Burgers” conflicts with the described data fields (x:101, t:103; no second spatial coordinate). This prevents readers from validating what equation is solved, what “2D” means (2 spatial dims vs 1D-in-space + time), and what physical regimes are expected as $\nu$ varies (Sec. 2.1, Sec. 3, Sec. 4). The dataset naming (e.g., “turbulence bundle”) further increases ambiguity about whether this is a single trajectory, a random-IC ensemble, forced/decaying Burgers, etc.
Recommendation: In Sec. 2.1, explicitly write the governing PDE(s) (including dimensionality, variables, and all terms), specify spatial/temporal domains, initial and boundary conditions, and clarify whether this is 1+1D Burgers (one space + time) or genuinely 2D-in-space Burgers (and if so, where the missing coordinate/fields are in the data). State whether each $\nu$ corresponds to a single IC/trajectory or an ensemble, and reconcile any “turbulence” terminology with the actual setup.
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The PINN and training setup are under-specified, making it impossible to assess whether the latent-space trends reflect physics/learning or artifacts of a particular model/run. Missing: architecture (depth/width/activations), loss terms and weights (PDE residual vs data/IC/BC), collocation and sampling strategy, optimizer schedule, stopping criteria, training diagnostics, and—crucially—whether a single multi-$\nu$ model was trained ($\nu$ as an input) or separate models per $\nu$ (Sec. 2.1, Sec. 4).
Recommendation: Expand Sec. 2.1 (or add a dedicated Methods subsection) to fully document the PINN: architecture (including where the 10D layer sits and whether it is pre/post activation), all loss components and weights, sampling/collocation details, optimizer and learning-rate schedule, training duration and convergence diagnostics, and whether training is joint across all $\nu$ or separate per $\nu$. Provide at least basic solution-quality validation across viscosities (e.g., PDE residual statistics, BC/IC error, or comparison to a reference solver for a few $\nu$) so the latent analysis is grounded in accurate solutions.
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The object of study (“10D latent space”) is not adequately defined or justified. The manuscript does not clearly identify which layer is used, why that layer is representative, whether it is a bottleneck vs simply a hidden layer of width 10, and whether similar conclusions hold for other layers (Sec. 2.1, Sec. 3.2–3.3). This limits interpretability and generality.
Recommendation: In Sec. 2.1, precisely identify the layer (index/depth; pre- vs post-nonlinearity; activation function) and justify why its activations are treated as “latent.” Add a minimal layer-wise comparison (e.g., one earlier and one later layer) to test whether (i) low effective dimension, (ii) “stable” PC orientations, and (iii) correlation-dimension behavior persist. If not feasible, explicitly scope conclusions to this layer in Sec. 4.
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Per-viscosity standardization (Sec. 2.2) materially changes what PCA measures and may affect cross-ν comparisons. Standardizing each $\nu$ separately makes PCA reflect correlation structure rather than absolute variance/scales, and can alter explained-variance trends and PC alignment across viscosities. The paper does not discuss these implications or provide sensitivity checks (Sec. 2.2–2.4, Sec. 3.2).
Recommendation: Explicitly state, in Sec. 2.2–2.4 and again in Sec. 3.2, that PCA is performed on per-$\nu$ standardized activations and interpret EVR trends accordingly. Add a sensitivity analysis comparing at least: (i) per-$\nu$ standardization (current), (ii) global standardization using pooled mean/std across all $\nu$, and ideally (iii) no standardization (with careful interpretation). Consider adding a ‘pooled PCA’ (fit PCA on all $\nu$ jointly) and then analyze how per-$\nu$ covariance/EVR projects onto this global basis; this directly tests the “stable basis” claim.
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PCA “stability” claims (cosine similarity of $\text{PC}_k$ between $\nu_i$ and $\nu_{i+1}$) are potentially confounded by eigenvalue near-degeneracies and ordering/permutation ambiguity. High cosine similarity can be misleading if PC2–PC4 eigenvalues are close and eigenvectors rotate within a near-degenerate subspace; comparing only successive viscosities can also mask cumulative drift (Sec. 2.4, Sec. 3.2.2, Table 3).
Recommendation: In Sec. 3.2.2, report eigenvalue gaps (e.g., $\lambda_k/\lambda_{k+1}$ or $\lambda_k-\lambda_{k+1}$) to show when individual PCs are well-defined. Replace/augment per-component cosine similarities with subspace similarity metrics (principal angles) for the span of the top-$m$ PCs. Add an all-pairs similarity heatmap or similarity-to-a-fixed-reference-$\nu$ plot to detect long-range drift. Keep the sign-handling (absolute cosine) but also handle potential component swaps by matching PCs via maximum absolute dot product when eigenvalues are close.
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The Grassberger–Procaccia correlation-dimension estimation is insufficiently specified and lacks uncertainty/robustness analysis, yet it supports a central claim (non-monotonic $D_2$ peak at intermediate $\nu$). Key missing details include: distance metric; whether standardized or raw latent points are used; $\epsilon$ range and sampling; how the scaling region is selected; handling of strong spatial/temporal correlations in the $(x,t)$ grid; and any computational approximations for $N\approx 10^4$ (Secs. 2.5, 3.3).
Recommendation: Substantially expand Sec. 2.5 to document the exact implementation: metric, $\epsilon$ grid (min/max, log spacing, number of radii), scaling-region selection procedure (e.g., sliding-window fits with $R^2$ thresholds), and computational approach (full pairs vs subsampled pairs/k-d tree). Because points come from a structured $(x,t)$ grid, incorporate correlation handling (e.g., Theiler window in time, spatial subsampling, or block bootstrap). In Sec. 3.3, add uncertainty quantification (bootstrap over points/blocks and over fit windows) and include representative $\log C(\epsilon)$ vs $\log \epsilon$ plots with the fitted scaling region for low/intermediate/high $\nu$ (appendix acceptable). Also test sensitivity of $D_2$ to $\epsilon$-range choices and to subsampling/resolution of the $(x,t)$ grid.
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The reported intrinsic dimension ($\approx 1.5$–$1.75$) and PCA effective dimension ($\approx 3$–4) are discussed as if directly comparable, but the sampling geometry strongly constrains the latent point cloud: for fixed $\nu$, $\mathrm{latent}(x,t;\nu)$ is the image of a 2D parameter domain $(x,t)$ under a smooth map, so intrinsic dimension is expected to be $\leq 2$ in generic settings. Without addressing this “domain-mapping” viewpoint, the correlation-dimension results may be an artifact of structured sampling rather than evidence of an emergent manifold complexity trend with $\nu$ (Sec. 3.3–3.4).
Recommendation: In Sec. 3.3–3.4, explicitly discuss that the point cloud is generated by a mapping from a 2D grid $(x,t)$ and is not i.i.d.; explain how this naturally yields $D_2$ near 2 (or below due to correlations/finite-size effects). To strengthen interpretation, test whether $D_2$ is stable under changes in grid resolution and under random subsampling of $(x,t)$ points. Clarify why PCA may require $>2$ components to capture variance (e.g., curved 2D surface embedded in 10D), and separate this geometric explanation from any stronger “RG-like” claims.
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The RG analogy is presented as a central interpretive lens but remains metaphorical; the manuscript does not define an explicit coarse-graining or a flow with RG-like properties (semigroup structure, fixed points, universality). Current wording risks over-claiming relative to the presented evidence (Secs. 2.6, 3.4, 4).
Recommendation: Revise Sec. 3.4 and Sec. 4 to clearly label the RG connection as heuristic and specify the limited mapping being proposed (e.g., $\nu$ as a control/scale-like parameter; leading latent modes as “effective” degrees of freedom). Remove or soften language suggesting a rigorous RG correspondence unless you add an explicit coarse-graining/transformation and demonstrate RG-like behavior (e.g., fixed-point-like stabilization, monotone flow of a quantity) on the latent statistics.
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Reproducibility is currently too limited for a scientific contribution: missing code/data availability statements; incomplete description of the exact pipeline from the .npy bundle to figures/tables; and corrupted/placeholder artifacts undermine confidence in reported numbers (Secs. 2.3–2.5, Sec. 3.1–3.2.1, Sec. 4).
Recommendation: Add a Data/Code Availability section (Sec. 4 or end matter) stating whether the trained model, activation bundles, and analysis scripts will be released. Provide a concise end-to-end pipeline description or pseudocode (loading, reshaping, standardization conventions, PCA implementation details, cosine-similarity computation, $D_2$ estimation settings). Ensure all tables/figures are regenerated from source outputs and remove placeholders/corrupt entries before submission.