-
The PDE problem definition, the PINN architecture/training, and the quality/accuracy of the learned solution are insufficiently specified, limiting interpretability and reproducibility (Sec. 1, Sec. 2.1, Sec. 3). The manuscript refers to “2D Burgers” while using a 2D $(x,t)$ grid, but does not explicitly state the governing equation (viscous vs. inviscid; $\nu$ value; any forcing), domain, and initial/boundary conditions. Likewise, key PINN details are missing (layer sizes/activations; where the $10$D “latent layer” is taken—pre/post nonlinearity; loss terms and weights; collocation/boundary sampling; optimizer/schedule/epochs; checkpoint used for latent extraction). Without validation (e.g., error vs. reference solver; PDE residual maps), it is unclear whether observed LID patterns reflect physics, architecture choices, or training artifacts.
Recommendation: Add a dedicated setup section (expand Sec. 2.1 or add a new Sec. 2.0) that (i) writes the exact Burgers’ equation solved, including $\nu$ (and forcing if any), domain, and initial/boundary conditions; (ii) clarifies that this is 1D-in-space Burgers’ with time on a 2D spatio-temporal grid (or explicitly states if truly 2D in space); (iii) fully specifies the PINN (layers/widths/activations; location/definition of the $10$D latent vector; loss components and weights; training data/collocation strategy; optimizer and stopping criteria; random seed). In Sec. 3, include basic solution-quality evidence: plots of $u(x,t)$ and PDE residual, plus $L^2$/relative error against a standard numerical reference (or at least residual/BC-IC satisfaction metrics). State precisely which checkpoint produced the analyzed latent vectors.
-
The central physical interpretation (“low LID bands correspond to shocks/high complexity; higher LID corresponds to smooth regions”) is currently qualitative and not directly validated against physical fields (Sec. 3.3–3.4, Sec. 4). The paper does not present $u(x,t)$ itself, nor gradient/curvature/shock indicators (e.g., $|\partial u/\partial x|$, $|\partial^2 u/\partial x^2|$, total variation, entropy production), nor quantitative association tests. This leaves the main claim speculative.
Recommendation: Augment Sec. 3.3–3.4 with quantitative comparisons between $D(x,t)$ and physics-derived fields: (1) plot $u(x,t)$ and derived measures such as $|\partial u/\partial x|$ (and optionally $|\partial^2 u/\partial x^2|$, $|\partial u/\partial t|$, PDE residual magnitude) alongside $D(x,t)$ at representative times; (2) define a “shock/steep-layer region” via a standard threshold on $|\partial u/\partial x|$ or total variation and compare the LID distributions inside vs. outside; (3) report correlation/MI/scatter analyses between $D(x,t)$ and these complexity metrics over all grid points. If the solution is viscous (finite $\nu$), rephrase ‘shock’ as ‘steep gradient layer’ and interpret accordingly. Update Sec. 4 conclusions to separate validated findings from hypotheses.
-
Robustness and reliability of the LID estimates are not established, despite extreme values (LID < 1 and LID > 10, with maxima $\approx 14.4$ exceeding the $10$D embedding) and potential sensitivity to $k$-range, metric, anisotropy, and regression fit quality (Sec. 2.3.1–2.3.2, Sec. 3.2–3.3). Using Euclidean distance on raw latent coordinates together with a narrow $k$ range ($5$–$20$) can yield unstable estimates, particularly if latent dimensions have very different scales/heavy tails (as suggested by the EDA). The manuscript also does not report regression goodness-of-fit (e.g., $R^2$) or uncertainty for $D_p$.
Recommendation: Extend Sec. 2.3 and Sec. 3.2–3.3 with robustness/diagnostics: (1) sensitivity to $k$ choices (e.g., $[3,10]$, $[5,15]$, $[10,30]$) and report how global stats and spatial patterns of $D(x,t)$ change; (2) compare distances computed on raw $z$ vs. standardized/whitened $z$ (and state the chosen preprocessing); (3) report regression diagnostics per point ($R^2$ distribution, slope $m_p$ distribution) and flag or mask unreliable fits (e.g., low $R^2$); (4) consider a second estimator on a subset (e.g., Levina–Bickel MLE LID, TWO-NN) or bootstrap/subsampling to quantify variance. In Sec. 3.3–3.4, explicitly interpret extreme values as potential estimator/finite-sample artifacts unless shown robust, and report the fraction of points with LID$>10$ or LID$<1$.
-
Methodological/implementation details are not sufficient for faithful reproduction of the LID pipeline and latent extraction (Sec. 2.2–2.3). Critical ambiguities include: kNN implementation and parameters; whether the query point is included as its own nearest neighbor; how ties/duplicate distances and near-zero distances are handled before taking logs; the log base; the exact regression routine; and whether any filtering (e.g., $m_p\leq 0$, NaNs) was necessary. The text also mentions possible negative slopes ($m_p<0$), which should not occur if $r_k$ is defined as the $k$-th nearest-neighbor distance with $k$ increasing and distances sorted, suggesting either a definitional mismatch or an implementation detail that must be clarified (Sec. 2.3.2).
Recommendation: In Sec. 2.3.1–2.3.2, add implementation-level specifics (and optionally short pseudocode): name the library/function used for kNN (e.g., sklearn NearestNeighbors), metric, whether self-neighbors are excluded, and how $r_k$ and $k$ are indexed; specify log base; define handling of $r_k=0$ or ties (epsilon, jitter, or skipping); specify the regression implementation (e.g., numpy.polyfit/OLS) and whether any robust fitting is used. Correct the $m_p<0$ discussion: under the stated definition it should be $m_p\geq 0$; if negatives occurred, explain what differs (e.g., unsorted distances, including self, numerical issues). Report NaN counts and any validity filtering explicitly in Sec. 3.2.
-
The manuscript occasionally overreaches from a geometric descriptor (LID of a latent point cloud) to causal claims about “adaptive compression strategies” and broader generality, while evidence comes from a single model/run and a single PDE setting (Sec. 3.5, Sec. 4). Low LID does not uniquely imply compression in an information-theoretic or capacity-allocation sense; it can also arise from activation saturation, symmetries/degeneracies, sampling density issues, or layer choice.
Recommendation: Reframe Sec. 3.5 and Sec. 4 to clearly distinguish observation from interpretation (e.g., ‘consistent with’ rather than ‘demonstrates’). Add at least one minimal generality check where feasible: compute $D(x,t)$ for (i) a different hidden layer (layer-wise LID), and/or (ii) another seed or slight architecture/latent-dimension variant. If possible, complement LID with a Jacobian-based local rank/effective-rank measure (e.g., rank of $\partial z/\partial(x,t)$) to strengthen the ‘compression/representation’ narrative. Expand related work in Sec. 1 to better situate the contribution within PINN interpretability and intrinsic-dimension analysis in deep networks.