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Core problem-specification ambiguity/inconsistency: the manuscript repeatedly says “2D Burgers’ equation,” but the described sampling grid and notation appear to be only $(x,t)$ with a single spatial coordinate $x$ (Secs. 1, 2.1–2.2; pp.2–3). The mention of predicted solution components “$u$ and $v$” further suggests either a vector-valued field or a different PDE than what is described. This ambiguity undermines interpretation of all geometric findings because the IC family and solution structure depend strongly on the PDE definition.
Recommendation: In Sec. 2.1 (and briefly in Sec. 1), state the exact PDE being solved: scalar vs vector Burgers, spatial dimensionality, full equation(s), viscosity and any forcing, domain $(x,t)$ (or $(x,y,t)$), and boundary/initial conditions. Then make terminology consistent throughout: either correct all “2D” mentions to “1D-in-space Burgers on a 2D spatiotemporal domain $(x,t)$,” or explicitly define the 2D spatial setting and show how $y$ is sampled/stored.
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Reproducibility gap: essential details of the PINN architecture, latent-vector definition, IC conditioning, training objective, and validation are missing (Secs. 2.1–2.2, 3). The paper states a pretrained model yields a 10D latent vector and a 13D feature vector, but does not specify (i) where the latent layer is (before/after activation/normalization), (ii) whether IC is an explicit input, an embedding, or handled by training separate networks, (iii) the loss terms and weights (PDE residual vs BC/IC/data), (iv) collocation/data sampling, optimizers/schedule, and (v) whether accuracy/generalization was validated across the 25 ICs. Without these, it is difficult to assess whether latent geometry reflects physics, architecture, or training artifacts.
Recommendation: Add a dedicated “PDE + PINN setup” subsection expanding Sec. 2.1/2.2 that includes: network diagram (inputs, layers, activations), explicit definition of the 10D latent vector (exact tensor taken, location), how IC information enters the model (conditional input, parameterization, or otherwise), the full loss with weights, training procedure (sampling, optimizer, epochs/stopping), and quantitative validation (e.g., PDE residual and/or error vs a reference solver) aggregated over ICs. If the model/data are from prior work, still summarize the essentials and cite precisely.
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Over-interpretation risk: claims of “3D affine manifolds,” “low-dimensional manifold structure,” and (implicitly) “disentanglement” are currently supported primarily by PCA variance-explained statistics (Secs. 3.1–3.3, 3.6–4). High explained variance by a few PCs does not rule out curved manifolds, regime mixtures (e.g., time-dominated variance), or nonlinear structure; and “disentanglement” has specific meanings in representation learning that are stronger than “dominant centroid direction + aligned subspaces.”
Recommendation: In Secs. 3.2–3.3 and 3.6–4, soften language to “approximately low-dimensional linear structure under PCA” unless additional tests are added. Strengthen the claim with at least two simple diagnostics: (i) reconstruction error curves vs number of PCs (per-IC and global), and (ii) a curvature/heterogeneity check such as time-sliced PCA (early vs late time, or several $t$-bins) to see whether the ‘3D’ claim holds uniformly. Optionally add one nonlinear intrinsic-dimension estimator (e.g., TwoNN, Levina–Bickel MLE, participation ratio) to corroborate that the effective dimension is not purely a linear artifact.
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Central “IC is mainly encoded as translation” interpretation is not directly tested (Secs. 3.3–3.6). The current evidence (near-1D centroid PCA + high subspace similarity) is suggestive but does not quantify how much of the between-IC variation is explained by centroid shift vs subspace rotation vs within-manifold shape changes.
Recommendation: Add a quantitative effect-size test in Sec. 3.3/3.6: (i) variance decomposition (ANOVA-style) into within-IC covariance vs between-IC centroid covariance in latent space, and report fractions along global PCs; and (ii) a “translated template” reconstruction experiment: choose a reference 3D basis $V_{\rm ref}$ (e.g., from one IC or pooled within-IC covariance), model each IC as $C_k + V_{\rm ref}z$, report reconstruction error; then allow per-IC bases $V_k$ and report improvement. This directly validates whether translation dominates and how much rotations matter.
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Subspace similarity metric and PCA methodology are underspecified, limiting reproducibility and interpretation (Secs. 2.2.1–2.2.2, 2.3.2, 3.4–3.5; Fig. 7 and related). It is unclear whether latent dimensions are only mean-centered or also scaled/standardized, which PCA algorithm is used (SVD vs covariance eigendecomposition), how dimensionality thresholds are applied, and the exact scalar similarity formula underlying the reported mean $\sim 0.986$ (e.g., $\|U^TV\|_F^2/d$, mean $\cos^2\theta_i$, product, min, etc.).
Recommendation: Expand Sec. 2.2 (and 2.3.2 if needed) with precise definitions: preprocessing (mean-centering; any scaling), PCA implementation (library + SVD/covariance), intrinsic-dimensionality definition $d=\min\{d: \mathrm{cvar}(d)\ge \tau\}$ with explicit $\tau$ values, and an explicit mathematical formula for the subspace similarity statistic with how principal angles are computed and aggregated. Ensure all reported similarity numbers (Secs. 3.4–3.5) reference this single definition.
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Robustness/generalization is not established: the study uses one pretrained network, one PDE parameter setting, and 25 ICs of unspecified diversity (Secs. 2.1, 3.1–3.7, 4). Yet parts of Sec. 3.6 and Sec. 4 read broadly (as if generic to PINNs). Without reruns across seeds/architectures or variation in PDE parameters (e.g., viscosity) and IC families, it is unclear whether the 6D/3D/1D geometry is stable or idiosyncratic.
Recommendation: Narrow claims in Secs. 3.6–4 to the studied configuration unless robustness is added. If feasible, add a small robustness section: retrain at least one additional model with a different seed and/or modest architecture change, and (optionally) vary viscosity or IC family; then report whether key summary statistics (global effective dimension, per-IC dimension, centroid PC1 variance, similarity distribution) remain consistent. If out of scope, state this explicitly as a limitation in Sec. 3.7.
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Connection to physics and IC parameterization is currently too indirect to support interpretability claims (Secs. 2.1, 3.3, 3.6). The near-linear centroid ordering along CPC1 is hard to interpret without knowing how ICs are generated/ordered; it may be a trivial reflection of a single scalar IC parameter (e.g., amplitude) or even arbitrary indexing.
Recommendation: In Sec. 2.1, describe the 25 ICs concretely (functional form, parameters, ranges, and how IC index $0$–$24$ is assigned). In Sec. 3.3/3.6, correlate centroid coordinates (CPC1 or global PC1 projection of centroids) with physically meaningful IC descriptors (amplitude, wavenumber, phase, energy, etc.), and report correlation/regression results. Optionally relate within-IC PCs to solution statistics (e.g., gradient norms/shock indicators) to interpret what the “3D” variation corresponds to physically.