-
Sec. 3 is incomplete (placeholders for figures/captions and generic filler text), so none of the central empirical claims (e.g., viscosity-dependent entropy/$\mathrm{ED}_{99}$ trends; approach to fixed-point-like behavior) stated in the Abstract, Introduction, and Conclusions can be verified (Sec. 1, Sec. 3, Sec. 4).
Recommendation: Populate Sec. 3 with the actual experimental results and make every key claim traceable to a figure/table: (i) plots of $\mathrm{ED}_{99}(s)$ and normalized entropy$(s)$ for representative viscosities spanning clearly defined regimes; (ii) heatmaps over $(\nu, s)$ for $\mathrm{ED}_{99}$ and entropy; (iii) representative PCA eigenvalue spectra at selected $(\nu, s)$. Include axis definitions, scale index $s$ meaning, and numerical summaries (e.g., large-scale values, slopes vs $s$, crossover scales). Then revise Abstract/Sec. 1/Sec. 4 to cite specific figures and quantify the reported trends rather than stating them qualitatively without evidence.
-
The governing PDE, variables, and dataset geometry are internally inconsistent: Sec. 2.1 writes a PDE with $x$ and $y$ derivatives and an advecting field $v$ ($\partial u/\partial t + u\,\partial u/\partial x + v\,\partial u/\partial y = \nu(\partial^2 u/\partial x^2+\partial^2 u/\partial y^2)$), but the dataset/analysis is defined on an $(x, t)$ grid only ($N_x\times N_t = 101\times 103$) with inputs $(x, t, \nu)$ and no $y$ dimension or definition of $v$. This prevents interpreting what physical system the latent vectors correspond to (Sec. 2.1).
Recommendation: Make the PDE and data tensor dimensions consistent, explicitly and unambiguously: either (a) revise to 1D-in-space Burgers ($x,t$) and remove $y$ and $v$ everywhere; or (b) present the full 2D-in-space Burgers system (typically coupled $u$ and $v$ equations), include $y$ (and $N_y$) in the dataset description and coarse-graining, and specify how $u/v$ are represented in the latent data. In either case, fully specify domain, boundary/initial conditions, and the viscosity set (list or range) in Sec. 2.1, and ensure the notation matches the actual grid used in Sec. 2.2–2.4.
-
The PINN and latent extraction are under-specified, undermining reproducibility and interpretation. The manuscript does not state whether there is one conditional PINN over $\nu$ or 25 separate models; where exactly the “10D latent vector” is taken (which layer; pre-/post-activation); training details (loss terms/weights, collocation and boundary sampling, optimizer, epochs); and baseline solution accuracy versus a reference solver. Representation geometry can depend strongly on these choices (Sec. 2.1).
Recommendation: Expand Sec. 2.1 (or add a dedicated ‘PINN and data generation’ subsection) to include: architecture (layers/widths/activations), how $\nu$ enters the network, definition/location of the 10D latent, whether the latent is shared/comparable across viscosities (single model) or model-specific (multiple models), full loss decomposition and weighting, training protocol and sampling counts, and quantitative accuracy metrics (e.g., relative $L_2$ error vs numerical reference) across $\nu$. If reusing a prior model/dataset (Auddy et al., 2023; Baty, 2024), identify the exact checkpoint/version and provide access information (repo/DOI) and any modifications.
-
The ‘RG flow/fixed point/attractor’ framing is currently qualitative and risks over-claiming. The transformation used is block averaging of latent vectors plus per-scale re-standardization before PCA; there is no explicit rescaling step, no parameter/coupling flow, and per-scale z-scoring can remove amplitude information and change how ‘flow’ should be interpreted (Sec. 2.2–2.5; Sec. 4).
Recommendation: In Sec. 2.5 and Sec. 4, tighten the conceptual claims: clearly state what aspects are RG-inspired rather than a formal RG. Provide operational definitions (e.g., what constitutes a ‘fixed point’: stabilization of $\mathrm{ED}_{99}$/entropy/spectrum within a tolerance over successive $s$). Justify why coarse-graining latents (rather than physical fields) is meaningful for ‘integrating out’ small scales. Add at least one baseline comparison in Sec. 3 to show the value of tracking multi-step flows (e.g., metrics on a single downsampled scale; or downsampling the latent field without iterative flow) and comment on the effect of per-scale standardization (e.g., repeat key plots with global standardization fixed at $s=0$).
-
Uncertainty/finite-sample effects are not addressed, even though sample size shrinks rapidly with coarse-graining, making PCA eigenvalues, entropy, and $\mathrm{ED}_{99}$ potentially unstable at large $s$. The manuscript also does not report the actual number of RG steps used and $N^{(s)}$ at each step for the $101\times 103$ grid (Sec. 2.2; Sec. 3).
Recommendation: In Sec. 2.2, tabulate $N_x^{(s)}, N_t^{(s)},$ and $N^{(s)}$ for $s=0,1,2,\ldots$ until termination for $N_x=101$, $N_t=103$ with $b_x=b_t=2$, and state the maximum $s$ retained (and the criterion $N^{(s)}\geq 20$). In Sec. 3, add uncertainty estimates: bootstrap or subsample grid points at each $(\nu, s)$ to put confidence intervals/error bars on eigenvalues, entropy, and $\mathrm{ED}_{99}$. Clearly mark in plots where $N^{(s)}$ becomes small and restrict ‘fixed point’ interpretation to scales with demonstrably stable estimates.
-
Robustness to design choices (block size, ED threshold, latent dimension, standardization choice, edge handling) is not tested, so regime-dependent conclusions may be artifacts of specific settings (Sec. 2.2–2.4; Sec. 4).
Recommendation: Add a compact sensitivity study in Sec. 3: (i) compare $2\times 2$ with at least one alternative coarse-graining (e.g., $3\times 3$, or anisotropic $2\times 1/1\times 2$) and report whether qualitative $\nu$-dependent trends persist; (ii) report ED at multiple thresholds (e.g., $\mathrm{ED}_{95}$ in addition to $\mathrm{ED}_{99}$) and/or an alternative effective-rank metric (participation ratio); (iii) explicitly compare per-$(\nu,s)$ standardization vs global/per-$\nu$ standardization (Sec. 2.3); (iv) justify the 10D latent choice and, if feasible, show at least one alternate latent size or clearly state it as a limitation.
-
The physical interpretation (latent ‘complexity’ tracking advection- vs diffusion-dominated regimes) is not supported by direct comparisons to physical-field diagnostics, making it unclear whether latent entropy/$\mathrm{ED}_{99}$ changes reflect actual solution multiscale structure or representation idiosyncrasies (Sec. 2.5; Sec. 4).
Recommendation: In Sec. 3, add a small set of physics-side diagnostics computed from the PINN solution (or reference solution) and relate them to latent metrics: e.g., gradient magnitude statistics/total variation/shock indicators/correlation length proxies versus $\nu$ and $s$. Include at least qualitative side-by-side plots (coarse-grained physical fields and latent metrics) for a few viscosities. If you claim a regime transition, define it numerically ($\nu$ ranges or a Reynolds-like proxy) and show where the transition appears in both physical and latent measures.